285:
1749:
5942:
6323:
1340:
6628:
6060:
1077:
1833:, one physical interpretation of this problem is as follows: fix the temperature on the boundary of the domain according to the given specification of the boundary condition. Allow heat to flow until a stationary state is reached in which the temperature at each point on the domain does not change anymore. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem.
6375:
64:
5784:
1026:
8858:
6318:{\displaystyle \nabla ^{2}f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}=0.}
1335:{\displaystyle \nabla ^{2}f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}=0.}
6623:{\displaystyle {\frac {1}{R}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)=\lambda ,\qquad {\frac {1}{Y}}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial Y}{\partial \theta }}\right)+{\frac {1}{Y}}{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}Y}{\partial \varphi ^{2}}}=-\lambda .}
9417:
1715:
1526:
5524:
843:
830:
6861:
8703:
7723:
2014:
9115:
1531:
4341:
1356:
5888:
7341:
1877:
within the domain where the equation is satisfied. If any two functions are solutions to
Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution. This property, called the
4772:
688:
6742:
5779:{\displaystyle u(P)={\frac {1}{4\pi }}a^{3}\left(1-{\frac {\rho ^{2}}{a^{2}}}\right)\int _{0}^{2\pi }\int _{0}^{\pi }{\frac {g(\theta ',\varphi ')\sin \theta '}{(a^{2}+\rho ^{2}-2a\rho \cos \Theta )^{\frac {3}{2}}}}d\theta '\,d\varphi '}
7849:
5113:
7217:
5510:
denotes the angle with the vertical axis, which is contrary to the usual
American mathematical notation, but agrees with standard European and physical practice. Then the solution of the Laplace equation with Dirichlet boundary values
4070:
3678:
7581:
5268:
1021:{\displaystyle \nabla ^{2}f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}=0.}
8311:
1893:
8121:
6738:
4187:
3192:
9088:
4837:
5926:
at the center of the sphere is the mean value of its values on the sphere. This mean value property immediately implies that a non-constant harmonic function cannot assume its maximum value at an interior point.
5455:
2528:
8415:
3554:
Thus every analytic function corresponds to a steady incompressible, irrotational, inviscid fluid flow in the plane. The real part is the velocity potential, and the imaginary part is the stream function.
2355:
8853:{\displaystyle {\begin{aligned}\mathbf {g} &=-\nabla V,\\\nabla \cdot \mathbf {g} &=\nabla \cdot (-\nabla V)=-\nabla ^{2}V,\\\implies \nabla ^{2}V&=-\nabla \cdot \mathbf {g} .\end{aligned}}}
4414:
2603:
4105:). It is common to take a different sign convention for this equation than one typically does when defining fundamental solutions. This choice of sign is often convenient to work with because −Δ is a
3337:
4168:
460:
8696:
8357:
8216:
7222:
4101:
rather than a function; but it can be thought of as a limit of functions whose integrals over space are unity, and whose support (the region where the function is non-zero) shrinks to a point (see
3854:
3552:
5375:
2896:
8708:
3912:
7519:
9412:{\displaystyle R(r)=(-1)^{l}{\frac {(l!)^{2}r_{s}^{l}}{(2l)!}}P_{l}\left(1-{\frac {2r}{r_{s}}}\right)+(-1)^{l+1}{\frac {2(2l+1)!}{(l)!^{2}r_{s}^{l+1}}}Q_{l}\left(1-{\frac {2r}{r_{s}}}\right).}
1710:{\displaystyle \nabla ^{2}f={\frac {1}{\sqrt {|g|}}}{\frac {\partial }{\partial \xi ^{i}}}\!\left({\sqrt {|g|}}g^{ij}{\frac {\partial f}{\partial \xi ^{j}}}\right)=0,\qquad (g=\det\{g_{ij}\})}
4668:
5789:
3468:
3230:
be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions. The continuity condition for an incompressible flow is that
2455:
2216:
1521:{\displaystyle \nabla ^{2}f={\frac {\partial }{\partial \xi ^{j}}}\left({\frac {\partial f}{\partial \xi ^{k}}}g^{kj}\right)+{\frac {\partial f}{\partial \xi ^{j}}}g^{jm}\Gamma _{mn}^{n}=0,}
4542:
2786:
4172:
The
Laplace equation is unchanged under a rotation of coordinates, and hence we can expect that a fundamental solution may be obtained among solutions that only depend upon the distance
8908:
4890:
3725:
8180:
3786:
4462:
2664:
9588:
7753:
4919:
2797:
The close connection between the
Laplace equation and analytic functions implies that any solution of the Laplace equation has derivatives of all orders, and can be expanded in a
2116:
1072:
7121:
3395:
2955:
4181:
5971:(left to right). Zonal, sectoral, and tesseral harmonics are depicted along the left-most column, the main diagonal, and elsewhere, respectively. (The negative order harmonics
8066:
2721:
383:
8976:
8514:
8490:
6004:
3277:
7989:
6045:
3582:
1860:
alone. For the example of the heat equation it amounts to prescribing the heat flux through the boundary. In particular, at an adiabatic boundary, the normal derivative of
487:
8607:
8238:
8019:
415:
4673:
629:
7927:
600:
556:
8937:
8443:
8245:
511:
7877:
8627:
8534:
8039:
8075:
7957:
6666:
2960:
2676:
implies that the value of the line integral connecting two points is independent of the path. The resulting pair of solutions of the
Laplace equation are called
8647:
8574:
8554:
8466:
8143:
7897:
3938:
9009:
4776:
5154:
2460:
825:{\displaystyle \nabla ^{2}f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}=0.}
6856:{\displaystyle \lambda \sin ^{2}\theta +{\frac {\sin \theta }{\Theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d\Theta }{d\theta }}\right)=m^{2}}
315:
8364:
5392:
2240:
10022:
4348:
2539:
3282:
4118:
8652:
8315:
7718:{\displaystyle f(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell }^{m}r^{\ell }Y_{\ell }^{m}(\theta ,\varphi ),}
9982:
3797:
3495:
2825:
3859:
3201:
7457:
2009:{\displaystyle {\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}\equiv \psi _{xx}+\psi _{yy}=0.}
5331:
4336:{\displaystyle -1=\iiint _{V}\nabla \cdot \nabla u\,dV=\iint _{S}{\frac {du}{dr}}\,dS=\left.4\pi a^{2}{\frac {du}{dr}}\right|_{r=a}.}
3414:
2383:
2148:
9652:. 8th edition / ed., Brooks/Cole, Cengage Learning, 2013. Chapter 12: Boundary-value Problems in Rectangular Coordinates. p. 462.
4489:
2726:
558:
is a twice-differentiable real-valued function. The
Laplace operator therefore maps a scalar function to another scalar function.
10027:
9473:
8579:
A potential that does not satisfy
Laplace's equation together with the boundary condition is an invalid electrostatic potential.
8700:
The gravitational field is conservative and can therefore be expressed as the negative gradient of the gravitational potential:
8865:
4854:
3683:
3402:
3213:
638:
308:
1854:. Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of
420:
9942:
9914:
9747:
9726:
9685:
9657:
9636:
8185:
8148:
3918:. The Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions.
3736:
4423:
2620:
5936:
2369:
also satisfies the
Laplace equation. Conversely, given a harmonic function, it is the real part of an analytic function,
2044:
3397:
then the continuity condition is the integrability condition for this differential: the resulting function is called the
3348:
2901:
2680:. This construction is only valid locally, or provided that the path does not loop around a singularity. For example, if
10008:
Find out how boundary value problems governed by
Laplace's equation may be solved numerically by boundary element method
4599:
2691:
671:. In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.
351:
9888:
8942:
301:
166:
3233:
4569:
17:
9676:
9627:
6974:
637:, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of
388:
6661:. Applying separation of variables again to the second equation gives way to the pair of differential equations
605:
7372:
7336:{\displaystyle r^{2}\nabla ^{2}Y_{\ell }^{m}(\theta ,\varphi )=-\ell (\ell +1)Y_{\ell }^{m}(\theta ,\varphi ).}
6996:
31:
2801:, at least inside a circle that does not enclose a singularity. This is in sharp contrast to solutions of the
10042:
9969:
1882:, is very useful. For example, solutions to complex problems can be constructed by summing simple solutions.
341:
7554:
6992:
2142:
171:
161:
133:
9544:
1039:
10032:
9964:
6938:
6052:
5883:{\displaystyle \cos \Theta =\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos(\varphi -\varphi ')}
9541:
The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example,
4098:
1837:
9959:
8044:
3733:
is the charge density. The first
Maxwell equation is the integrability condition for the differential
10037:
6978:
6367:
225:
118:
109:
8495:
8471:
5974:
7962:
6015:
5480:
469:
8590:
8221:
8002:
1879:
1346:
836:
147:
656:, which are important in multiple branches of physics, notably electrostatics, gravitation, and
7906:
7550:
7111:
3564:
1800:
564:
520:
9512:
8916:
8422:
7844:{\displaystyle r<R={\frac {1}{\limsup _{\ell \to \infty }|f_{\ell }^{m}|^{{1}/{\ell }}}}.}
5497:
5108:{\displaystyle \iiint _{V}\left\,dV=\iiint _{V}\nabla \cdot \left\,dV=\iint _{S}\left\,dS.\,}
4911:
1753:
1032:
681:
634:
496:
241:
43:
8125:
Now, the electric field can be expressed as the negative gradient of the electric potential
7856:
7212:{\displaystyle Y_{\ell }^{m}(\theta ,\varphi )=Ne^{im\varphi }P_{\ell }^{m}(\cos {\theta })}
9821:
9523:
9515:
uses the Laplace equation to show that stable static ferromagnetic suspension is impossible
9469:
9453:
8987:
8612:
8519:
8024:
4073:
3932:
345:
251:
232:
186:
128:
8:
9976:
9763:
Chicone, C.; Mashhoon, B. (2011-11-20). "Nonlocal Gravity: Modified Poisson's Equation".
9631:. 7th ed., Brooks/Cole, Cengage Learning, 2012. Chapter 14: Partial Derivatives. p. 908.
9109:
7936:
4585:
4573:
1890:
Laplace's equation in two independent variables in rectangular coordinates has the form
1737:
138:
100:
9825:
9790:
9772:
9492:
9487:
9481:
9440:
8632:
8559:
8539:
8451:
8361:
Plugging this relation into Gauss's law, we obtain Poisson's equation for electricity,
8128:
7899:
are chosen instead. In that case, one needs to expand the solution of known regions in
7882:
7748:
7569:
7522:
7115:
4479:
3489:
2673:
642:
289:
196:
9993:
3673:{\displaystyle \nabla \times (u,v,0)=(v_{x}-u_{y}){\hat {\mathbf {k} }}=\mathbf {0} ,}
9990:
9938:
9910:
9884:
9834:
9809:
9794:
9743:
9722:
9681:
9653:
9632:
9507:
8649:
the gravitational constant. Then Gauss's law for gravitation in differential form is
6882:
4553:
4106:
2024:
1874:
1870:
1851:
1806:
1748:
653:
284:
211:
123:
95:
4109:. The definition of the fundamental solution thus implies that, if the Laplacian of
9829:
9782:
9518:
9497:
4767:{\displaystyle \nabla \cdot \nabla G=-\delta (x-x',y-y',z-z')\qquad {\text{in }}V,}
4588:
is a fundamental solution that also satisfies a suitable condition on the boundary
4483:
3915:
649:
463:
271:
266:
256:
85:
55:
9853:
8445:
and Poisson's equation reduces to Laplace's equation for the electric potential.
7744:
7568:
The general solution to Laplace's equation in a ball centered at the origin is a
3398:
661:
191:
156:
10007:
9599:
Physical applications often take the solution that vanishes at infinity, making
8306:{\displaystyle \nabla \cdot \mathbf {E} =\nabla \cdot (-\nabla V)=-\nabla ^{2}V}
2898:
with suitably defined coefficients whose real and imaginary parts are given by
652:. The twice continuously differentiable solutions of Laplace's equation are the
9502:
8069:
7900:
7572:
of the spherical harmonic functions multiplied by the appropriate scale factor
5941:
4565:
4561:
4475:
2809:
657:
246:
181:
176:
71:
9935:
Handbook of Linear Partial Differential Equations for Engineers and Scientists
10016:
9857:
7930:
4102:
2802:
1830:
1729:
668:
206:
201:
90:
8116:{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}.}
6733:{\displaystyle {\frac {1}{\Phi }}{\frac {d^{2}\Phi }{d\varphi ^{2}}}=-m^{2}}
5322:
from the center of the sphere is reflected along its radial line to a point
3187:{\displaystyle f(z)=\sum _{n=0}^{\infty }\left+i\sum _{n=1}^{\infty }\left,}
2798:
1829:
is equal to some given function. Since the Laplace operator appears in the
665:
9590:. Its use to represent the Laplacian should not be confused with this use.
329:
261:
80:
9606:. This does not affect the angular portion of the spherical harmonics.
8516:
is surrounded by a conducting material with a specified charge density
8072:
for electricity (Maxwell's first equation) in differential form states
7390:
4065:{\displaystyle \Delta u=u_{xx}+u_{yy}+u_{zz}=-\delta (x-x',y-y',z-z'),}
490:
9786:
9083:{\displaystyle \Psi (r,\theta ,\varphi )=R(r)Y_{l}(\theta ,\varphi ),}
4832:{\displaystyle G=0\quad {\text{if}}\quad (x,y,z)\qquad {\text{on }}S.}
4176:
from the source point. If we choose the volume to be a ball of radius
9998:
7414:
7389:
represent colatitude and longitude, respectively. In particular, the
5263:{\displaystyle u(x',y',z')=\iiint _{V}Gf\,dV+\iint _{S}G_{n}g\,dS.\,}
4557:
4471:
2523:{\displaystyle \psi _{x}=-\varphi _{y},\quad \psi _{y}=\varphi _{x}.}
6370:, two differential equations result by imposing Laplace's equation:
5478:′. A consequence of this expression for the Green's function is the
2794:
is single-valued only in a region that does not enclose the origin.
9977:
Laplace Equation (particular solutions and boundary value problems)
8862:
Using the differential form of Gauss's law of gravitation, we have
4115:
is integrated over any volume that encloses the source point, then
3200:. These trigonometric functions can themselves be expanded, using
2672:
may be defined by a line integral. The integrability condition and
514:
9777:
5389:
will be outside the sphere. The Green's function is then given by
648:
The general theory of solutions to Laplace's equation is known as
9810:"The Laplace and poisson equations in Schwarzschild's space-time"
7421:
6937:
of the second equation at the boundary points of the domain is a
5306:, the Green's function may be obtained by means of a reflection (
4467:
2363:
satisfies the Laplace equation. A similar calculation shows that
333:
9881:
Introduction to Partial Differential Equations with Applications
8410:{\displaystyle \nabla ^{2}V=-{\frac {\rho }{\varepsilon _{0}}}.}
6632:
The second equation can be simplified under the assumption that
5450:{\displaystyle {\frac {1}{4\pi R}}-{\frac {a}{4\pi \rho R'}},\,}
9988:
63:
9721:. 4th ed., Pearson, 2013. Chapter 2: Electrostatics. p. 83-4.
4914:, (a consequence of the divergence theorem) which states that
3579:
in two space dimensions that is independent of time satisfies
2457:
then the Cauchy–Riemann equations will be satisfied if we set
2350:{\displaystyle u_{yy}=(-v_{x})_{y}=-(v_{y})_{x}=-(u_{x})_{x}.}
9742:. 4th ed., Pearson, 2013. Chapter 3: Potentials. p. 119-121.
9472:, a coordinate system under which Laplace's equation becomes
348:, who first studied its properties. This is often written as
9697:
The approach to spherical harmonics taken here is found in (
4556:. Note that, with the opposite sign convention, this is the
4274:
4409:{\displaystyle {\frac {du}{dr}}=-{\frac {1}{4\pi r^{2}}},}
2598:{\displaystyle d\psi =-\varphi _{y}\,dx+\varphi _{x}\,dy.}
7089:
independent solutions of this form, one for each integer
3332:{\displaystyle \nabla \times \mathbf {V} =v_{x}-u_{y}=0.}
2808:
There is an intimate connection between power series and
561:
If the right-hand side is specified as a given function,
5937:
Spherical harmonics § Laplace's spherical harmonics
4163:{\displaystyle \iiint _{V}\nabla \cdot \nabla u\,dV=-1.}
3279:
and the condition that the flow be irrotational is that
674:
455:{\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}
8691:{\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho .}
8352:{\displaystyle \nabla ^{2}V=-\nabla \cdot \mathbf {E} }
8211:{\displaystyle \nabla \times \mathbf {E} =\mathbf {0} }
7052:
Here the solution was assumed to have the special form
8978:
which is Laplace's equation for gravitational fields.
8910:
which is Poisson's equation for gravitational fields.
6327:
Consider the problem of finding solutions of the form
4466:
Note that, with the opposite sign convention (used in
3849:{\displaystyle \varphi _{x}=-u,\quad \varphi _{y}=-v.}
3547:{\displaystyle \varphi _{x}=-u,\quad \varphi _{y}=-v.}
3476:
satisfies the Laplace equation. The harmonic function
1809:
for Laplace's equation consists of finding a solution
9547:
9118:
9012:
8945:
8919:
8868:
8706:
8655:
8635:
8615:
8593:
8562:
8542:
8522:
8498:
8474:
8454:
8425:
8367:
8318:
8248:
8224:
8188:
8151:
8131:
8078:
8047:
8027:
8005:
7965:
7939:
7909:
7885:
7859:
7756:
7584:
7460:
7225:
7124:
6745:
6669:
6378:
6063:
6018:
5977:
5792:
5527:
5395:
5334:
5300:. For the case of the interior of a sphere of radius
5272:
Thus the Green's function describes the influence at
5157:
4922:
4857:
4779:
4676:
4602:
4492:
4426:
4351:
4190:
4121:
3941:
3862:
3856:
The second of Maxwell's equations then implies that
3800:
3739:
3686:
3585:
3498:
3417:
3351:
3285:
3236:
2963:
2904:
2891:{\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}z^{n},}
2828:
2729:
2694:
2623:
2542:
2463:
2386:
2243:
2151:
2047:
1896:
1534:
1359:
1080:
1042:
846:
691:
608:
567:
523:
499:
472:
423:
391:
354:
6902:
is a linear combination of the complex exponentials
9650:
Differential Equations with Boundary-Value Problems
7557:), and so counting dimensions shows that there are
5914:. A simple consequence of this formula is that if
3907:{\displaystyle \varphi _{xx}+\varphi _{yy}=-\rho ,}
641:. Laplace's equation is also a special case of the
9985:using Laplace's equation from exampleproblems.com.
9582:
9443:of the first and second kind, respectively, while
9411:
9082:
8970:
8931:
8902:
8852:
8690:
8641:
8621:
8601:
8568:
8548:
8528:
8508:
8484:
8460:
8437:
8409:
8351:
8305:
8232:
8210:
8174:
8137:
8115:
8060:
8033:
8013:
7983:
7951:
7921:
7891:
7871:
7843:
7717:
7514:{\displaystyle r^{2}\nabla ^{2}Y=-\ell (\ell +1)Y}
7513:
7354:is called a spherical harmonic function of degree
7335:
7211:
7036:; requiring the solution to be regular throughout
6855:
6732:
6622:
6317:
6039:
5998:
5882:
5778:
5449:
5369:
5262:
5107:
4884:
4831:
4766:
4662:
4536:
4486:. A similar argument shows that in two dimensions
4456:
4408:
4335:
4162:
4064:
3906:
3848:
3780:
3719:
3672:
3546:
3462:
3389:
3331:
3271:
3186:
2949:
2890:
2780:
2715:
2658:
2597:
2522:
2449:
2349:
2210:
2110:
2008:
1709:
1520:
1334:
1066:
1020:
824:
623:
594:
550:
505:
481:
454:
409:
377:
9814:Journal of Mathematical Analysis and Applications
4097:. No function has this property: in fact it is a
1595:
365:
10014:
9979:at EqWorld: The World of Mathematical Equations.
9878:
8419:In the particular case of a source-free region,
7776:
7549:is the expression in spherical coordinates of a
5930:
5518:
5370:{\displaystyle \rho '={\frac {a^{2}}{\rho }}.\,}
4420:that is centered on the source point, and hence
4081:denotes a unit source concentrated at the point
1840:for Laplace's equation specify not the function
1682:
32:Theory of tides § Laplace's tidal equations
9762:
3470:and the irrotationality condition implies that
3463:{\displaystyle \psi _{x}=v,\quad \psi _{y}=-u,}
2450:{\displaystyle f(z)=\varphi (x,y)+i\psi (x,y),}
2211:{\displaystyle u_{x}=v_{y},\quad v_{x}=-u_{y}.}
2027:both satisfy the Laplace equation. That is, if
9852:
9698:
8981:
8240:is also known as the electrostatic condition.
7879:, the solid harmonics with negative powers of
9909:. Providence: American Mathematical Society.
9713:
9711:
9709:
9707:
9680:. 4th ed., Pearson, 2013. Inner front cover.
6923:is regular at the poles of the sphere, where
2611:implies that the integrability condition for
309:
9937:. Boca Raton: Chapman & Hall/CRC Press.
7110:. These angular solutions are a product of
6933:. Imposing this regularity in the solution
5474:denotes the distance to the reflected point
4537:{\displaystyle u=-{\frac {\log(r)}{2\pi }}.}
3339:If we define the differential of a function
2818:in a power series inside a circle of radius
2781:{\displaystyle f(z)=\log z=\log r+i\theta .}
1701:
1685:
9879:Zachmanoglou, E. C.; Thoe, Dale W. (1986).
9670:
9668:
9666:
8986:S. Persides solved the Laplace equation in
4847:is any solution of the Poisson equation in
1869:Solutions of Laplace's equation are called
9869:
9732:
9704:
8808:
8804:
5920:is a harmonic function, then the value of
5307:
3492:. The Cauchy–Riemann equations imply that
2723:then a corresponding analytic function is
2023:The real and imaginary parts of a complex
316:
302:
27:Second-order partial differential equation
9923:
9872:Partial Differential Equations in Physics
9862:Methods of Mathematical Physics, Volume I
9833:
9776:
8903:{\displaystyle \nabla ^{2}V=4\pi G\rho ,}
8468:is specified on the boundary of a region
6047:with respect to the positive order ones.)
5764:
5461:denotes the distance to the source point
5446:
5366:
5259:
5249:
5216:
5137:. In view of the conditions satisfied by
5104:
5094:
5035:
4978:
4960:
4941:
4885:{\displaystyle \nabla \cdot \nabla u=-f,}
4262:
4222:
4144:
3768:
3755:
3720:{\displaystyle \nabla \cdot (u,v)=\rho ,}
3377:
3364:
2585:
2565:
9932:
9807:
9663:
8068:be the permittivity of free space. Then
5940:
2805:, which generally have less regularity.
1747:
10023:Elliptic partial differential equations
9983:Example initial-boundary value problems
9484:, a general case of Laplace's equation.
8175:{\displaystyle \mathbf {E} =-\nabla V,}
7565:linearly independent such polynomials.
7118:, and associated Legendre polynomials:
3926:
3781:{\displaystyle d\varphi =-u\,dx-v\,dy,}
2380:(at least locally). If a trial form is
639:elliptic partial differential equations
517:operator (also symbolized "grad"), and
14:
10015:
9648:Zill, Dennis G, and Michael R Cullen.
9460:is an arbitrary non-negative integer.
6995:, whose solution is a multiple of the
6981:. Furthermore, a change of variables
4457:{\displaystyle u={\frac {1}{4\pi r}}.}
3921:
3214:Laplace equation for irrotational flow
2659:{\displaystyle \psi _{xy}=\psi _{yx},}
1743:
9989:
9904:
9628:Calculus : Early Transcendentals
8492:, then it is uniquely determined. If
2018:
1770:) with Dirichlet boundary conditions
675:Forms in different coordinate systems
9758:
9756:
9583:{\displaystyle \Delta x=x_{1}-x_{2}}
8041:be the electric charge density, and
7747:. Such an expansion is valid in the
2111:{\displaystyle f(z)=u(x,y)+iv(x,y),}
1885:
1732:relative to the new coordinates and
1067:{\displaystyle (r,\theta ,\varphi )}
7994:
7534:. In fact, for any such solution,
6962:for some non-negative integer with
6877:is a complex constant, but because
5945:Real (Laplace) spherical harmonics
5890:is the cosine of the angle between
4579:
3390:{\displaystyle d\psi =v\,dx-u\,dy,}
2950:{\displaystyle c_{n}=a_{n}+ib_{n}.}
2227:is the first partial derivative of
30:For Laplace's tidal equations, see
24:
9898:
9548:
9013:
8947:
8870:
8832:
8810:
8785:
8769:
8757:
8739:
8726:
8656:
8501:
8477:
8369:
8338:
8320:
8291:
8275:
8263:
8249:
8189:
8163:
8079:
7916:
7786:
7628:
7472:
7237:
6991:transforms this equation into the
6821:
6781:
6693:
6675:
6592:
6578:
6522:
6514:
6488:
6484:
6293:
6279:
6223:
6215:
6189:
6185:
6138:
6130:
6103:
6099:
6065:
5799:
5732:
5024:
5012:
4998:
4967:
4961:
4948:
4942:
4864:
4858:
4683:
4677:
4482:force, arising in the solution of
4216:
4210:
4138:
4132:
3942:
3687:
3586:
3286:
3099:
2995:
2860:
2118:then the necessary condition that
1952:
1938:
1915:
1901:
1642:
1634:
1579:
1575:
1536:
1492:
1462:
1454:
1414:
1406:
1382:
1378:
1361:
1310:
1296:
1240:
1232:
1206:
1202:
1155:
1147:
1120:
1116:
1082:
996:
982:
959:
945:
907:
899:
879:
875:
848:
800:
786:
763:
749:
726:
712:
693:
609:
500:
493:operator (also symbolized "div"),
473:
443:
436:
430:
424:
392:
356:
25:
10054:
9952:
9753:
7381:is a normalization constant, and
6006:would be shown rotated about the
4663:{\displaystyle G(x,y,z;x',y',z')}
3558:
2530:This relation does not determine
8994:. Using the canonical variables
8839:
8746:
8712:
8663:
8595:
8345:
8256:
8226:
8204:
8196:
8153:
8086:
8061:{\displaystyle \varepsilon _{0}}
8007:
4568:), which is the solution of the
3935:of Laplace's equation satisfies
3663:
3649:
3293:
2716:{\displaystyle \varphi =\log r,}
378:{\displaystyle \nabla ^{2}\!f=0}
283:
62:
9928:. Philadelphia: W. B. Saunders.
9801:
9765:Journal of Mathematical Physics
9740:Introduction to Electrodynamics
9719:Introduction to Electrodynamics
9677:Introduction to Electrodynamics
8971:{\displaystyle \nabla ^{2}V=0,}
8448:If the electrostatic potential
7959:), to match the terms and find
6453:
4817:
4795:
4789:
4752:
3823:
3521:
3437:
2493:
2178:
2141:be differentiable and that the
1672:
10028:Eponymous equations of physics
9926:Partial Differential Equations
9907:Partial Differential Equations
9691:
9642:
9619:
9593:
9535:
9321:
9315:
9307:
9292:
9271:
9261:
9201:
9192:
9166:
9156:
9144:
9134:
9128:
9122:
9074:
9062:
9049:
9043:
9034:
9016:
8805:
8775:
8763:
8582:
8509:{\displaystyle {\mathcal {R}}}
8485:{\displaystyle {\mathcal {R}}}
8281:
8269:
8182:if the field is irrotational,
7813:
7792:
7783:
7734:are constants and the factors
7709:
7697:
7606:
7588:
7505:
7493:
7396:, or polar angle, ranges from
7373:associated Legendre polynomial
7327:
7315:
7297:
7285:
7273:
7261:
7206:
7192:
7152:
7140:
6997:associated Legendre polynomial
6898:is necessarily an integer and
5999:{\displaystyle Y_{\ell }^{-m}}
5877:
5860:
5736:
5688:
5669:
5647:
5537:
5531:
5517:inside the sphere is given by(
5194:
5161:
4814:
4796:
4749:
4698:
4657:
4606:
4517:
4511:
4180:around the source point, then
4056:
4005:
3794:may be constructed to satisfy
3705:
3693:
3653:
3642:
3616:
3610:
3592:
3272:{\displaystyle u_{x}+v_{y}=0,}
3194:which is a Fourier series for
2973:
2967:
2838:
2832:
2739:
2733:
2441:
2429:
2417:
2405:
2396:
2390:
2335:
2321:
2306:
2292:
2277:
2260:
2102:
2090:
2078:
2066:
2057:
2051:
1704:
1673:
1612:
1604:
1566:
1558:
1061:
1043:
664:, the Laplace equation is the
589:
571:
545:
527:
13:
1:
9612:
8990:on hypersurfaces of constant
7984:{\displaystyle f_{\ell }^{m}}
7424:, may assume all values with
7012:. Finally, the equation for
6040:{\displaystyle 90^{\circ }/m}
5931:Laplace's spherical harmonics
5131:denote normal derivatives on
3401:because it is constant along
3207:
1344:More generally, in arbitrary
482:{\displaystyle \nabla \cdot }
342:partial differential equation
9835:10.1016/0022-247X(73)90277-1
8609:be the gravitational field,
8602:{\displaystyle \mathbf {g} }
8233:{\displaystyle \mathbf {E} }
8014:{\displaystyle \mathbf {E} }
6885:whose period evenly divides
5519:Zachmanoglou & Thoe 1986
5149:, this result simplifies to
4900:assumes the boundary values
2678:conjugate harmonic functions
7:
9965:Encyclopedia of Mathematics
9874:. New York: Academic Press.
9463:
9110:spherical harmonic function
8982:In the Schwarzschild metric
7413:at the South Pole, and the
5385:is inside the sphere, then
3405:. The first derivatives of
2536:, but only its increments:
1838:Neumann boundary conditions
410:{\displaystyle \Delta f=0,}
10:
10059:
9846:
9699:Courant & Hilbert 1962
8536:, and if the total charge
7454:of the eigenvalue problem
7018:has solutions of the form
6941:that forces the parameter
5934:
4182:Gauss's divergence theorem
3788:so the electric potential
3211:
2812:. If we expand a function
2688:are polar coordinates and
1880:principle of superposition
1846:itself on the boundary of
1798:
624:{\displaystyle \Delta f=h}
29:
9924:Petrovsky, I. G. (1967).
8218:. The irrotationality of
7922:{\displaystyle r=\infty }
6973:; this is also explained
4560:generated by a pointlike
2605:The Laplace equation for
1752:Laplace's equation on an
226:Geometric function theory
172:Cauchy's integral formula
162:Cauchy's integral theorem
9933:Polyanin, A. D. (2002).
9529:
7114:, here represented as a
7075:. For a given value of
6979:orbital angular momentum
6908:. The solution function
5481:Poisson integral formula
2143:Cauchy–Riemann equations
595:{\displaystyle h(x,y,z)}
551:{\displaystyle f(x,y,z)}
134:Cauchy–Riemann equations
9870:Sommerfeld, A. (1949).
8988:Schwarzschild spacetime
8932:{\displaystyle \rho =0}
8438:{\displaystyle \rho =0}
8021:be the electric field,
7435:. For a fixed integer
7112:trigonometric functions
6939:Sturm–Liouville problem
6368:separation of variables
3202:multiple angle formulae
1347:curvilinear coordinates
837:cylindrical coordinates
682:rectangular coordinates
506:{\displaystyle \nabla }
119:Complex-valued function
9584:
9413:
9084:
8972:
8933:
8904:
8854:
8692:
8643:
8629:the mass density, and
8623:
8603:
8570:
8550:
8530:
8510:
8486:
8462:
8439:
8411:
8353:
8307:
8234:
8212:
8176:
8139:
8117:
8062:
8035:
8015:
7985:
7953:
7923:
7893:
7873:
7872:{\displaystyle r>R}
7845:
7719:
7656:
7632:
7553:that is harmonic (see
7551:homogeneous polynomial
7515:
7400:at the North Pole, to
7337:
7213:
6857:
6734:
6624:
6319:
6051:Laplace's equation in
6048:
6041:
6000:
5884:
5780:
5451:
5371:
5326:that is at a distance
5264:
5109:
4886:
4833:
4768:
4664:
4538:
4458:
4416:on a sphere of radius
4410:
4337:
4164:
4066:
3908:
3850:
3782:
3721:
3674:
3548:
3464:
3391:
3333:
3273:
3188:
3103:
2999:
2951:
2892:
2864:
2782:
2717:
2660:
2599:
2524:
2451:
2351:
2212:
2112:
2010:
1801:Boundary value problem
1796:
1711:
1522:
1336:
1068:
1022:
826:
625:
596:
552:
507:
483:
456:
411:
379:
290:Mathematics portal
9905:Evans, L. C. (1998).
9808:Persides, S. (1973).
9585:
9414:
9085:
8973:
8934:
8905:
8855:
8693:
8644:
8624:
8622:{\displaystyle \rho }
8604:
8571:
8551:
8531:
8529:{\displaystyle \rho }
8511:
8487:
8463:
8440:
8412:
8354:
8308:
8235:
8213:
8177:
8140:
8118:
8063:
8036:
8034:{\displaystyle \rho }
8016:
7986:
7954:
7924:
7894:
7874:
7846:
7720:
7633:
7612:
7516:
7338:
7214:
6858:
6735:
6625:
6320:
6053:spherical coordinates
6042:
6001:
5944:
5885:
5781:
5500:for the source point
5498:spherical coordinates
5452:
5372:
5265:
5110:
4887:
4834:
4769:
4665:
4539:
4459:
4411:
4338:
4165:
4067:
3909:
3851:
3783:
3722:
3675:
3549:
3482:that is conjugate to
3465:
3392:
3334:
3274:
3189:
3083:
2979:
2952:
2893:
2844:
2783:
2718:
2661:
2600:
2525:
2452:
2352:
2213:
2113:
2011:
1751:
1712:
1523:
1337:
1069:
1033:spherical coordinates
1023:
827:
626:
597:
553:
508:
484:
457:
412:
380:
242:Augustin-Louis Cauchy
44:Mathematical analysis
10043:Pierre-Simon Laplace
9994:"Laplace's Equation"
9864:, Wiley-Interscience
9738:Griffiths, David J.
9717:Griffiths, David J.
9674:Griffiths, David J.
9545:
9524:Fundamental solution
9470:6-sphere coordinates
9454:Schwarzschild radius
9116:
9010:
8943:
8917:
8866:
8704:
8653:
8633:
8613:
8591:
8560:
8540:
8520:
8496:
8472:
8452:
8423:
8365:
8316:
8246:
8222:
8186:
8149:
8129:
8076:
8045:
8025:
8003:
7963:
7937:
7907:
7883:
7857:
7754:
7582:
7458:
7223:
7122:
6743:
6667:
6376:
6061:
6016:
5975:
5961:(top to bottom) and
5790:
5525:
5393:
5332:
5310:): the source point
5155:
4920:
4910:, then we may apply
4855:
4777:
4674:
4600:
4490:
4424:
4349:
4188:
4119:
4074:Dirac delta function
3939:
3933:fundamental solution
3927:Fundamental solution
3860:
3798:
3737:
3684:
3583:
3567:, an electric field
3496:
3415:
3349:
3283:
3234:
2961:
2902:
2826:
2727:
2692:
2621:
2540:
2461:
2384:
2241:
2149:
2129:be analytic is that
2045:
1894:
1532:
1357:
1078:
1040:
844:
689:
606:
565:
521:
497:
470:
421:
389:
352:
346:Pierre-Simon Laplace
252:Carl Friedrich Gauss
187:Isolated singularity
129:Holomorphic function
9883:. New York: Dover.
9826:1973JMAA...43..571P
9354:
9189:
7980:
7952:{\displaystyle r=0}
7810:
7696:
7671:
7407:at the Equator, to
7314:
7260:
7191:
7139:
7116:complex exponential
5995:
5640:
5625:
4574:incompressible flow
4572:in two-dimensional
3922:In three dimensions
3565:Maxwell's equations
3218:Let the quantities
2790:However, the angle
2237:. It follows that
1825:on the boundary of
1744:Boundary conditions
1738:Christoffel symbols
1508:
139:Formal power series
101:Unit complex number
10033:Harmonic functions
9991:Weisstein, Eric W.
9960:"Laplace equation"
9580:
9513:Earnshaw's theorem
9493:Quadrature domains
9488:Spherical harmonic
9482:Helmholtz equation
9441:Legendre functions
9409:
9334:
9175:
9080:
8968:
8929:
8900:
8850:
8848:
8688:
8639:
8619:
8599:
8566:
8546:
8526:
8506:
8482:
8458:
8435:
8407:
8349:
8303:
8230:
8208:
8172:
8135:
8113:
8058:
8031:
8011:
7981:
7966:
7949:
7919:
7889:
7869:
7841:
7796:
7790:
7715:
7682:
7657:
7570:linear combination
7523:linear combination
7511:
7333:
7300:
7246:
7209:
7177:
7125:
6947:to be of the form
6853:
6730:
6620:
6315:
6049:
6037:
5996:
5978:
5880:
5776:
5626:
5608:
5447:
5367:
5260:
5105:
4882:
4829:
4764:
4660:
4534:
4480:inverse-square law
4454:
4406:
4333:
4160:
4062:
3904:
3846:
3778:
3717:
3670:
3544:
3490:velocity potential
3460:
3387:
3329:
3269:
3184:
2947:
2888:
2822:, this means that
2778:
2713:
2656:
2595:
2520:
2447:
2347:
2208:
2108:
2019:Analytic functions
2006:
1871:harmonic functions
1797:
1707:
1518:
1491:
1332:
1064:
1018:
822:
660:. In the study of
654:harmonic functions
643:Helmholtz equation
635:Poisson's equation
621:
592:
548:
503:
479:
452:
407:
375:
340:is a second-order
338:Laplace's equation
217:Laplace's equation
197:Argument principle
9944:978-1-58488-299-2
9916:978-0-8218-0772-9
9787:10.1063/1.3702449
9748:978-1-108-42041-9
9727:978-1-108-42041-9
9686:978-1-108-42041-9
9658:978-1-111-82706-9
9637:978-0-538-49790-9
9508:Bateman transform
9399:
9356:
9251:
9208:
8642:{\displaystyle G}
8569:{\displaystyle V}
8549:{\displaystyle Q}
8461:{\displaystyle V}
8402:
8138:{\displaystyle V}
8108:
7892:{\displaystyle r}
7836:
7775:
7439:, every solution
6993:Legendre equation
6883:periodic function
6833:
6799:
6784:
6712:
6678:
6606:
6572:
6547:
6529:
6495:
6480:
6462:
6437:
6402:
6387:
6307:
6273:
6230:
6196:
6181:
6145:
6110:
6095:
5751:
5747:
5601:
5556:
5441:
5412:
5361:
4821:
4793:
4756:
4596:. For instance,
4554:natural logarithm
4529:
4449:
4401:
4370:
4345:It follows that
4311:
4260:
4107:positive operator
3656:
2025:analytic function
1966:
1929:
1886:In two dimensions
1852:normal derivative
1807:Dirichlet problem
1763:and outer radius
1728:is the Euclidean
1656:
1616:
1593:
1571:
1570:
1476:
1428:
1396:
1324:
1290:
1247:
1213:
1198:
1162:
1127:
1112:
1010:
973:
939:
914:
886:
871:
814:
777:
740:
326:
325:
212:Harmonic function
124:Analytic function
110:Complex functions
96:Complex conjugate
16:(Redirected from
10050:
10038:Fourier analysis
10004:
10003:
9973:
9948:
9929:
9920:
9894:
9875:
9865:
9854:Courant, Richard
9840:
9839:
9837:
9805:
9799:
9798:
9780:
9760:
9751:
9736:
9730:
9715:
9702:
9701:, §V.8, §VII.5).
9695:
9689:
9672:
9661:
9646:
9640:
9625:Stewart, James.
9623:
9607:
9605:
9597:
9591:
9589:
9587:
9586:
9581:
9579:
9578:
9566:
9565:
9539:
9519:Vector Laplacian
9498:Potential theory
9459:
9456:. The parameter
9451:
9438:
9429:
9418:
9416:
9415:
9410:
9405:
9401:
9400:
9398:
9397:
9388:
9380:
9367:
9366:
9357:
9355:
9353:
9342:
9333:
9332:
9313:
9287:
9285:
9284:
9257:
9253:
9252:
9250:
9249:
9240:
9232:
9219:
9218:
9209:
9207:
9190:
9188:
9183:
9174:
9173:
9154:
9152:
9151:
9107:
9089:
9087:
9086:
9081:
9061:
9060:
9006:the solution is
9005:
9001:
8997:
8993:
8977:
8975:
8974:
8969:
8955:
8954:
8938:
8936:
8935:
8930:
8913:In empty space,
8909:
8907:
8906:
8901:
8878:
8877:
8859:
8857:
8856:
8851:
8849:
8842:
8818:
8817:
8793:
8792:
8749:
8715:
8697:
8695:
8694:
8689:
8666:
8648:
8646:
8645:
8640:
8628:
8626:
8625:
8620:
8608:
8606:
8605:
8600:
8598:
8576:is also unique.
8575:
8573:
8572:
8567:
8555:
8553:
8552:
8547:
8535:
8533:
8532:
8527:
8515:
8513:
8512:
8507:
8505:
8504:
8491:
8489:
8488:
8483:
8481:
8480:
8467:
8465:
8464:
8459:
8444:
8442:
8441:
8436:
8416:
8414:
8413:
8408:
8403:
8401:
8400:
8388:
8377:
8376:
8358:
8356:
8355:
8350:
8348:
8328:
8327:
8312:
8310:
8309:
8304:
8299:
8298:
8259:
8239:
8237:
8236:
8231:
8229:
8217:
8215:
8214:
8209:
8207:
8199:
8181:
8179:
8178:
8173:
8156:
8144:
8142:
8141:
8136:
8122:
8120:
8119:
8114:
8109:
8107:
8106:
8094:
8089:
8067:
8065:
8064:
8059:
8057:
8056:
8040:
8038:
8037:
8032:
8020:
8018:
8017:
8012:
8010:
7990:
7988:
7987:
7982:
7979:
7974:
7958:
7956:
7955:
7950:
7928:
7926:
7925:
7920:
7898:
7896:
7895:
7890:
7878:
7876:
7875:
7870:
7850:
7848:
7847:
7842:
7837:
7835:
7834:
7833:
7832:
7827:
7822:
7816:
7809:
7804:
7795:
7789:
7770:
7742:
7733:
7724:
7722:
7721:
7716:
7695:
7690:
7681:
7680:
7670:
7665:
7655:
7650:
7631:
7626:
7577:
7564:
7548:
7533:
7520:
7518:
7517:
7512:
7480:
7479:
7470:
7469:
7453:
7438:
7434:
7419:
7412:
7406:
7399:
7395:
7388:
7384:
7380:
7370:
7361:
7357:
7353:
7342:
7340:
7339:
7334:
7313:
7308:
7259:
7254:
7245:
7244:
7235:
7234:
7218:
7216:
7215:
7210:
7205:
7190:
7185:
7176:
7175:
7138:
7133:
7109:
7094:
7088:
7080:
7074:
7048:
7041:
7035:
7017:
7011:
6990:
6977:in terms of the
6972:
6961:
6946:
6936:
6932:
6922:
6907:
6901:
6897:
6891:
6880:
6876:
6870:
6865:for some number
6862:
6860:
6859:
6854:
6852:
6851:
6839:
6835:
6834:
6832:
6824:
6816:
6800:
6798:
6787:
6785:
6780:
6769:
6758:
6757:
6739:
6737:
6736:
6731:
6729:
6728:
6713:
6711:
6710:
6709:
6696:
6692:
6691:
6681:
6679:
6671:
6660:
6637:
6629:
6627:
6626:
6621:
6607:
6605:
6604:
6603:
6590:
6586:
6585:
6575:
6573:
6571:
6564:
6563:
6550:
6548:
6540:
6535:
6531:
6530:
6528:
6520:
6512:
6496:
6494:
6483:
6481:
6479:
6465:
6463:
6455:
6443:
6439:
6438:
6436:
6428:
6420:
6418:
6417:
6403:
6401:
6390:
6388:
6380:
6365:
6324:
6322:
6321:
6316:
6308:
6306:
6305:
6304:
6291:
6287:
6286:
6276:
6274:
6272:
6265:
6264:
6255:
6254:
6241:
6236:
6232:
6231:
6229:
6221:
6213:
6197:
6195:
6184:
6182:
6180:
6170:
6169:
6156:
6151:
6147:
6146:
6144:
6136:
6128:
6126:
6125:
6111:
6109:
6098:
6096:
6094:
6093:
6081:
6073:
6072:
6046:
6044:
6043:
6038:
6033:
6028:
6027:
6011:
6005:
6003:
6002:
5997:
5994:
5986:
5970:
5960:
5953:
5925:
5919:
5913:
5901:
5889:
5887:
5886:
5881:
5876:
5853:
5827:
5785:
5783:
5782:
5777:
5775:
5763:
5752:
5750:
5749:
5748:
5740:
5713:
5712:
5700:
5699:
5686:
5685:
5668:
5657:
5642:
5639:
5634:
5624:
5616:
5607:
5603:
5602:
5600:
5599:
5590:
5589:
5580:
5567:
5566:
5557:
5555:
5544:
5516:
5509:
5505:
5495:
5491:
5487:
5473:
5466:
5460:
5456:
5454:
5453:
5448:
5442:
5440:
5439:
5418:
5413:
5411:
5397:
5384:
5376:
5374:
5373:
5368:
5362:
5357:
5356:
5347:
5342:
5321:
5315:
5305:
5299:
5293:
5287:
5269:
5267:
5266:
5261:
5245:
5244:
5235:
5234:
5209:
5208:
5193:
5182:
5171:
5148:
5142:
5136:
5114:
5112:
5111:
5106:
5093:
5089:
5088:
5087:
5072:
5071:
5054:
5053:
5034:
5030:
4997:
4996:
4977:
4973:
4932:
4931:
4912:Green's identity
4909:
4905:
4899:
4891:
4889:
4888:
4883:
4850:
4846:
4838:
4836:
4835:
4830:
4822:
4819:
4794:
4791:
4773:
4771:
4770:
4765:
4757:
4754:
4748:
4731:
4714:
4669:
4667:
4666:
4661:
4656:
4645:
4634:
4595:
4591:
4586:Green's function
4580:Green's function
4551:
4543:
4541:
4540:
4535:
4530:
4528:
4520:
4503:
4484:Poisson equation
4463:
4461:
4460:
4455:
4450:
4448:
4434:
4419:
4415:
4413:
4412:
4407:
4402:
4400:
4399:
4398:
4379:
4371:
4369:
4361:
4353:
4342:
4340:
4339:
4334:
4329:
4328:
4317:
4313:
4312:
4310:
4302:
4294:
4292:
4291:
4261:
4259:
4251:
4243:
4241:
4240:
4209:
4208:
4179:
4175:
4169:
4167:
4166:
4161:
4131:
4130:
4114:
4096:
4080:
4071:
4069:
4068:
4063:
4055:
4038:
4021:
3995:
3994:
3979:
3978:
3963:
3962:
3916:Poisson equation
3913:
3911:
3910:
3905:
3891:
3890:
3875:
3874:
3855:
3853:
3852:
3847:
3833:
3832:
3810:
3809:
3793:
3787:
3785:
3784:
3779:
3732:
3726:
3724:
3723:
3718:
3679:
3677:
3676:
3671:
3666:
3658:
3657:
3652:
3647:
3641:
3640:
3628:
3627:
3578:
3553:
3551:
3550:
3545:
3531:
3530:
3508:
3507:
3487:
3481:
3475:
3469:
3467:
3466:
3461:
3447:
3446:
3427:
3426:
3410:
3396:
3394:
3393:
3388:
3344:
3338:
3336:
3335:
3330:
3322:
3321:
3309:
3308:
3296:
3278:
3276:
3275:
3270:
3259:
3258:
3246:
3245:
3229:
3223:
3199:
3193:
3191:
3190:
3185:
3180:
3176:
3163:
3162:
3153:
3152:
3128:
3127:
3118:
3117:
3102:
3097:
3076:
3072:
3059:
3058:
3049:
3048:
3024:
3023:
3014:
3013:
2998:
2993:
2956:
2954:
2953:
2948:
2943:
2942:
2927:
2926:
2914:
2913:
2897:
2895:
2894:
2889:
2884:
2883:
2874:
2873:
2863:
2858:
2821:
2817:
2793:
2787:
2785:
2784:
2779:
2722:
2720:
2719:
2714:
2687:
2683:
2671:
2665:
2663:
2662:
2657:
2652:
2651:
2636:
2635:
2616:
2610:
2604:
2602:
2601:
2596:
2584:
2583:
2564:
2563:
2535:
2529:
2527:
2526:
2521:
2516:
2515:
2503:
2502:
2489:
2488:
2473:
2472:
2456:
2454:
2453:
2448:
2379:
2368:
2362:
2356:
2354:
2353:
2348:
2343:
2342:
2333:
2332:
2314:
2313:
2304:
2303:
2285:
2284:
2275:
2274:
2256:
2255:
2236:
2233:with respect to
2232:
2226:
2217:
2215:
2214:
2209:
2204:
2203:
2188:
2187:
2174:
2173:
2161:
2160:
2140:
2134:
2128:
2117:
2115:
2114:
2109:
2040:
2015:
2013:
2012:
2007:
1999:
1998:
1983:
1982:
1967:
1965:
1964:
1963:
1950:
1946:
1945:
1935:
1930:
1928:
1927:
1926:
1913:
1909:
1908:
1898:
1865:
1859:
1849:
1845:
1828:
1824:
1818:
1814:
1795:
1780:
1769:
1762:
1735:
1727:
1716:
1714:
1713:
1708:
1700:
1699:
1662:
1658:
1657:
1655:
1654:
1653:
1640:
1632:
1630:
1629:
1617:
1615:
1607:
1602:
1594:
1592:
1591:
1590:
1574:
1572:
1569:
1561:
1556:
1552:
1544:
1543:
1527:
1525:
1524:
1519:
1507:
1502:
1490:
1489:
1477:
1475:
1474:
1473:
1460:
1452:
1447:
1443:
1442:
1441:
1429:
1427:
1426:
1425:
1412:
1404:
1397:
1395:
1394:
1393:
1377:
1369:
1368:
1352:
1341:
1339:
1338:
1333:
1325:
1323:
1322:
1321:
1308:
1304:
1303:
1293:
1291:
1289:
1282:
1281:
1272:
1271:
1258:
1253:
1249:
1248:
1246:
1238:
1230:
1214:
1212:
1201:
1199:
1197:
1187:
1186:
1173:
1168:
1164:
1163:
1161:
1153:
1145:
1143:
1142:
1128:
1126:
1115:
1113:
1111:
1110:
1098:
1090:
1089:
1073:
1071:
1070:
1065:
1027:
1025:
1024:
1019:
1011:
1009:
1008:
1007:
994:
990:
989:
979:
974:
972:
971:
970:
957:
953:
952:
942:
940:
938:
937:
925:
920:
916:
915:
913:
905:
897:
887:
885:
874:
872:
864:
856:
855:
831:
829:
828:
823:
815:
813:
812:
811:
798:
794:
793:
783:
778:
776:
775:
774:
761:
757:
756:
746:
741:
739:
738:
737:
724:
720:
719:
709:
701:
700:
650:potential theory
630:
628:
627:
622:
601:
599:
598:
593:
557:
555:
554:
549:
512:
510:
509:
504:
488:
486:
485:
480:
464:Laplace operator
461:
459:
458:
453:
451:
450:
416:
414:
413:
408:
384:
382:
381:
376:
364:
363:
318:
311:
304:
288:
287:
272:Karl Weierstrass
267:Bernhard Riemann
257:Jacques Hadamard
86:Imaginary number
66:
56:Complex analysis
50:
48:Complex analysis
39:
38:
21:
18:Laplace equation
10058:
10057:
10053:
10052:
10051:
10049:
10048:
10047:
10013:
10012:
9958:
9955:
9945:
9917:
9901:
9899:Further reading
9891:
9849:
9844:
9843:
9806:
9802:
9761:
9754:
9737:
9733:
9716:
9705:
9696:
9692:
9673:
9664:
9647:
9643:
9624:
9620:
9615:
9610:
9600:
9598:
9594:
9574:
9570:
9561:
9557:
9546:
9543:
9542:
9540:
9536:
9532:
9466:
9457:
9449:
9444:
9436:
9431:
9427:
9422:
9393:
9389:
9381:
9379:
9372:
9368:
9362:
9358:
9343:
9338:
9328:
9324:
9314:
9288:
9286:
9274:
9270:
9245:
9241:
9233:
9231:
9224:
9220:
9214:
9210:
9191:
9184:
9179:
9169:
9165:
9155:
9153:
9147:
9143:
9117:
9114:
9113:
9096:
9091:
9056:
9052:
9011:
9008:
9007:
9003:
8999:
8995:
8991:
8984:
8950:
8946:
8944:
8941:
8940:
8918:
8915:
8914:
8873:
8869:
8867:
8864:
8863:
8847:
8846:
8838:
8822:
8813:
8809:
8801:
8800:
8788:
8784:
8750:
8745:
8736:
8735:
8716:
8711:
8707:
8705:
8702:
8701:
8662:
8654:
8651:
8650:
8634:
8631:
8630:
8614:
8611:
8610:
8594:
8592:
8589:
8588:
8585:
8561:
8558:
8557:
8556:is known, then
8541:
8538:
8537:
8521:
8518:
8517:
8500:
8499:
8497:
8494:
8493:
8476:
8475:
8473:
8470:
8469:
8453:
8450:
8449:
8424:
8421:
8420:
8396:
8392:
8387:
8372:
8368:
8366:
8363:
8362:
8344:
8323:
8319:
8317:
8314:
8313:
8294:
8290:
8255:
8247:
8244:
8243:
8225:
8223:
8220:
8219:
8203:
8195:
8187:
8184:
8183:
8152:
8150:
8147:
8146:
8130:
8127:
8126:
8102:
8098:
8093:
8085:
8077:
8074:
8073:
8052:
8048:
8046:
8043:
8042:
8026:
8023:
8022:
8006:
8004:
8001:
8000:
7997:
7975:
7970:
7964:
7961:
7960:
7938:
7935:
7934:
7908:
7905:
7904:
7884:
7881:
7880:
7858:
7855:
7854:
7828:
7823:
7818:
7817:
7812:
7811:
7805:
7800:
7791:
7779:
7774:
7769:
7755:
7752:
7751:
7745:solid harmonics
7740:
7735:
7731:
7726:
7691:
7686:
7676:
7672:
7666:
7661:
7651:
7637:
7627:
7616:
7583:
7580:
7579:
7573:
7558:
7535:
7531:
7526:
7475:
7471:
7465:
7461:
7459:
7456:
7455:
7440:
7436:
7425:
7417:
7408:
7401:
7397:
7393:
7386:
7382:
7376:
7368:
7363:
7359:
7355:
7351:
7346:
7309:
7304:
7255:
7250:
7240:
7236:
7230:
7226:
7224:
7221:
7220:
7201:
7186:
7181:
7165:
7161:
7134:
7129:
7123:
7120:
7119:
7096:
7090:
7082:
7076:
7053:
7043:
7037:
7019:
7013:
7004:
6999:
6982:
6963:
6948:
6942:
6934:
6924:
6909:
6903:
6899:
6893:
6886:
6878:
6872:
6866:
6847:
6843:
6825:
6817:
6815:
6805:
6801:
6791:
6786:
6770:
6768:
6753:
6749:
6744:
6741:
6740:
6724:
6720:
6705:
6701:
6697:
6687:
6683:
6682:
6680:
6670:
6668:
6665:
6664:
6639:
6633:
6599:
6595:
6591:
6581:
6577:
6576:
6574:
6559:
6555:
6554:
6549:
6539:
6521:
6513:
6511:
6501:
6497:
6487:
6482:
6469:
6464:
6454:
6429:
6421:
6419:
6413:
6409:
6408:
6404:
6394:
6389:
6379:
6377:
6374:
6373:
6328:
6300:
6296:
6292:
6282:
6278:
6277:
6275:
6260:
6256:
6250:
6246:
6245:
6240:
6222:
6214:
6212:
6202:
6198:
6188:
6183:
6165:
6161:
6160:
6155:
6137:
6129:
6127:
6121:
6117:
6116:
6112:
6102:
6097:
6089:
6085:
6080:
6068:
6064:
6062:
6059:
6058:
6029:
6023:
6019:
6017:
6014:
6013:
6007:
5987:
5982:
5976:
5973:
5972:
5962:
5955:
5951:
5946:
5939:
5933:
5921:
5915:
5903:
5891:
5869:
5846:
5820:
5791:
5788:
5787:
5768:
5756:
5739:
5735:
5708:
5704:
5695:
5691:
5687:
5678:
5661:
5650:
5643:
5641:
5635:
5630:
5617:
5612:
5595:
5591:
5585:
5581:
5579:
5572:
5568:
5562:
5558:
5548:
5543:
5526:
5523:
5522:
5512:
5507:
5501:
5493:
5489:
5485:
5468:
5462:
5458:
5432:
5422:
5417:
5401:
5396:
5394:
5391:
5390:
5380:
5352:
5348:
5346:
5335:
5333:
5330:
5329:
5317:
5311:
5308:Sommerfeld 1949
5301:
5295:
5289:
5273:
5240:
5236:
5230:
5226:
5204:
5200:
5186:
5175:
5164:
5156:
5153:
5152:
5144:
5138:
5132:
5129:
5122:
5083:
5079:
5067:
5063:
5059:
5055:
5049:
5045:
5008:
5004:
4992:
4988:
4937:
4933:
4927:
4923:
4921:
4918:
4917:
4907:
4901:
4895:
4856:
4853:
4852:
4848:
4842:
4818:
4790:
4778:
4775:
4774:
4753:
4741:
4724:
4707:
4675:
4672:
4671:
4649:
4638:
4627:
4601:
4598:
4597:
4593:
4589:
4582:
4570:Euler equations
4545:
4521:
4504:
4502:
4491:
4488:
4487:
4474:generated by a
4470:), this is the
4438:
4433:
4425:
4422:
4421:
4417:
4394:
4390:
4383:
4378:
4362:
4354:
4352:
4350:
4347:
4346:
4318:
4303:
4295:
4293:
4287:
4283:
4276:
4273:
4272:
4252:
4244:
4242:
4236:
4232:
4204:
4200:
4189:
4186:
4185:
4177:
4173:
4126:
4122:
4120:
4117:
4116:
4110:
4082:
4076:
4048:
4031:
4014:
3987:
3983:
3971:
3967:
3955:
3951:
3940:
3937:
3936:
3929:
3924:
3883:
3879:
3867:
3863:
3861:
3858:
3857:
3828:
3824:
3805:
3801:
3799:
3796:
3795:
3789:
3738:
3735:
3734:
3728:
3685:
3682:
3681:
3662:
3648:
3646:
3645:
3636:
3632:
3623:
3619:
3584:
3581:
3580:
3568:
3561:
3526:
3522:
3503:
3499:
3497:
3494:
3493:
3483:
3477:
3471:
3442:
3438:
3422:
3418:
3416:
3413:
3412:
3406:
3399:stream function
3350:
3347:
3346:
3340:
3317:
3313:
3304:
3300:
3292:
3284:
3281:
3280:
3254:
3250:
3241:
3237:
3235:
3232:
3231:
3225:
3219:
3216:
3210:
3195:
3158:
3154:
3148:
3144:
3123:
3119:
3113:
3109:
3108:
3104:
3098:
3087:
3054:
3050:
3044:
3040:
3019:
3015:
3009:
3005:
3004:
3000:
2994:
2983:
2962:
2959:
2958:
2938:
2934:
2922:
2918:
2909:
2905:
2903:
2900:
2899:
2879:
2875:
2869:
2865:
2859:
2848:
2827:
2824:
2823:
2819:
2813:
2791:
2728:
2725:
2724:
2693:
2690:
2689:
2685:
2681:
2674:Stokes' theorem
2667:
2644:
2640:
2628:
2624:
2622:
2619:
2618:
2612:
2606:
2579:
2575:
2559:
2555:
2541:
2538:
2537:
2531:
2511:
2507:
2498:
2494:
2484:
2480:
2468:
2464:
2462:
2459:
2458:
2385:
2382:
2381:
2370:
2364:
2358:
2338:
2334:
2328:
2324:
2309:
2305:
2299:
2295:
2280:
2276:
2270:
2266:
2248:
2244:
2242:
2239:
2238:
2234:
2228:
2224:
2219:
2199:
2195:
2183:
2179:
2169:
2165:
2156:
2152:
2150:
2147:
2146:
2136:
2130:
2119:
2046:
2043:
2042:
2028:
2021:
1991:
1987:
1975:
1971:
1959:
1955:
1951:
1941:
1937:
1936:
1934:
1922:
1918:
1914:
1904:
1900:
1899:
1897:
1895:
1892:
1891:
1888:
1873:; they are all
1861:
1855:
1847:
1841:
1826:
1820:
1816:
1815:on some domain
1810:
1803:
1782:
1771:
1764:
1757:
1746:
1733:
1726:
1718:
1692:
1688:
1649:
1645:
1641:
1633:
1631:
1622:
1618:
1611:
1603:
1601:
1600:
1596:
1586:
1582:
1578:
1573:
1565:
1557:
1551:
1539:
1535:
1533:
1530:
1529:
1503:
1495:
1482:
1478:
1469:
1465:
1461:
1453:
1451:
1434:
1430:
1421:
1417:
1413:
1405:
1403:
1402:
1398:
1389:
1385:
1381:
1376:
1364:
1360:
1358:
1355:
1354:
1350:
1317:
1313:
1309:
1299:
1295:
1294:
1292:
1277:
1273:
1267:
1263:
1262:
1257:
1239:
1231:
1229:
1219:
1215:
1205:
1200:
1182:
1178:
1177:
1172:
1154:
1146:
1144:
1138:
1134:
1133:
1129:
1119:
1114:
1106:
1102:
1097:
1085:
1081:
1079:
1076:
1075:
1041:
1038:
1037:
1003:
999:
995:
985:
981:
980:
978:
966:
962:
958:
948:
944:
943:
941:
933:
929:
924:
906:
898:
896:
892:
888:
878:
873:
863:
851:
847:
845:
842:
841:
807:
803:
799:
789:
785:
784:
782:
770:
766:
762:
752:
748:
747:
745:
733:
729:
725:
715:
711:
710:
708:
696:
692:
690:
687:
686:
677:
662:heat conduction
633:This is called
607:
604:
603:
566:
563:
562:
522:
519:
518:
498:
495:
494:
471:
468:
467:
446:
442:
422:
419:
418:
390:
387:
386:
359:
355:
353:
350:
349:
322:
282:
192:Residue theorem
167:Local primitive
157:Zeros and poles
72:Complex numbers
42:
35:
28:
23:
22:
15:
12:
11:
5:
10056:
10046:
10045:
10040:
10035:
10030:
10025:
10011:
10010:
10005:
9986:
9980:
9974:
9954:
9953:External links
9951:
9950:
9949:
9943:
9930:
9921:
9915:
9900:
9897:
9896:
9895:
9889:
9876:
9867:
9858:Hilbert, David
9848:
9845:
9842:
9841:
9820:(3): 571–578.
9800:
9752:
9731:
9703:
9690:
9662:
9641:
9617:
9616:
9614:
9611:
9609:
9608:
9592:
9577:
9573:
9569:
9564:
9560:
9556:
9553:
9550:
9533:
9531:
9528:
9527:
9526:
9521:
9516:
9510:
9505:
9503:Potential flow
9500:
9495:
9490:
9485:
9479:
9465:
9462:
9447:
9434:
9425:
9408:
9404:
9396:
9392:
9387:
9384:
9378:
9375:
9371:
9365:
9361:
9352:
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9341:
9337:
9331:
9327:
9323:
9320:
9317:
9312:
9309:
9306:
9303:
9300:
9297:
9294:
9291:
9283:
9280:
9277:
9273:
9269:
9266:
9263:
9260:
9256:
9248:
9244:
9239:
9236:
9230:
9227:
9223:
9217:
9213:
9206:
9203:
9200:
9197:
9194:
9187:
9182:
9178:
9172:
9168:
9164:
9161:
9158:
9150:
9146:
9142:
9139:
9136:
9133:
9130:
9127:
9124:
9121:
9094:
9079:
9076:
9073:
9070:
9067:
9064:
9059:
9055:
9051:
9048:
9045:
9042:
9039:
9036:
9033:
9030:
9027:
9024:
9021:
9018:
9015:
8983:
8980:
8967:
8964:
8961:
8958:
8953:
8949:
8928:
8925:
8922:
8899:
8896:
8893:
8890:
8887:
8884:
8881:
8876:
8872:
8845:
8841:
8837:
8834:
8831:
8828:
8825:
8823:
8821:
8816:
8812:
8807:
8803:
8802:
8799:
8796:
8791:
8787:
8783:
8780:
8777:
8774:
8771:
8768:
8765:
8762:
8759:
8756:
8753:
8751:
8748:
8744:
8741:
8738:
8737:
8734:
8731:
8728:
8725:
8722:
8719:
8717:
8714:
8710:
8709:
8687:
8684:
8681:
8678:
8675:
8672:
8669:
8665:
8661:
8658:
8638:
8618:
8597:
8584:
8581:
8565:
8545:
8525:
8503:
8479:
8457:
8434:
8431:
8428:
8406:
8399:
8395:
8391:
8386:
8383:
8380:
8375:
8371:
8347:
8343:
8340:
8337:
8334:
8331:
8326:
8322:
8302:
8297:
8293:
8289:
8286:
8283:
8280:
8277:
8274:
8271:
8268:
8265:
8262:
8258:
8254:
8251:
8228:
8206:
8202:
8198:
8194:
8191:
8171:
8168:
8165:
8162:
8159:
8155:
8134:
8112:
8105:
8101:
8097:
8092:
8088:
8084:
8081:
8055:
8051:
8030:
8009:
7996:
7995:Electrostatics
7993:
7978:
7973:
7969:
7948:
7945:
7942:
7929:), instead of
7918:
7915:
7912:
7901:Laurent series
7888:
7868:
7865:
7862:
7840:
7831:
7826:
7821:
7815:
7808:
7803:
7799:
7794:
7788:
7785:
7782:
7778:
7777:lim sup
7773:
7768:
7765:
7762:
7759:
7738:
7729:
7714:
7711:
7708:
7705:
7702:
7699:
7694:
7689:
7685:
7679:
7675:
7669:
7664:
7660:
7654:
7649:
7646:
7643:
7640:
7636:
7630:
7625:
7622:
7619:
7615:
7611:
7608:
7605:
7602:
7599:
7596:
7593:
7590:
7587:
7529:
7510:
7507:
7504:
7501:
7498:
7495:
7492:
7489:
7486:
7483:
7478:
7474:
7468:
7464:
7366:
7349:
7332:
7329:
7326:
7323:
7320:
7317:
7312:
7307:
7303:
7299:
7296:
7293:
7290:
7287:
7284:
7281:
7278:
7275:
7272:
7269:
7266:
7263:
7258:
7253:
7249:
7243:
7239:
7233:
7229:
7219:which fulfill
7208:
7204:
7200:
7197:
7194:
7189:
7184:
7180:
7174:
7171:
7168:
7164:
7160:
7157:
7154:
7151:
7148:
7145:
7142:
7137:
7132:
7128:
7002:
6850:
6846:
6842:
6838:
6831:
6828:
6823:
6820:
6814:
6811:
6808:
6804:
6797:
6794:
6790:
6783:
6779:
6776:
6773:
6767:
6764:
6761:
6756:
6752:
6748:
6727:
6723:
6719:
6716:
6708:
6704:
6700:
6695:
6690:
6686:
6677:
6674:
6619:
6616:
6613:
6610:
6602:
6598:
6594:
6589:
6584:
6580:
6570:
6567:
6562:
6558:
6553:
6546:
6543:
6538:
6534:
6527:
6524:
6519:
6516:
6510:
6507:
6504:
6500:
6493:
6490:
6486:
6478:
6475:
6472:
6468:
6461:
6458:
6452:
6449:
6446:
6442:
6435:
6432:
6427:
6424:
6416:
6412:
6407:
6400:
6397:
6393:
6386:
6383:
6314:
6311:
6303:
6299:
6295:
6290:
6285:
6281:
6271:
6268:
6263:
6259:
6253:
6249:
6244:
6239:
6235:
6228:
6225:
6220:
6217:
6211:
6208:
6205:
6201:
6194:
6191:
6187:
6179:
6176:
6173:
6168:
6164:
6159:
6154:
6150:
6143:
6140:
6135:
6132:
6124:
6120:
6115:
6108:
6105:
6101:
6092:
6088:
6084:
6079:
6076:
6071:
6067:
6036:
6032:
6026:
6022:
5993:
5990:
5985:
5981:
5949:
5935:Main article:
5932:
5929:
5879:
5875:
5872:
5868:
5865:
5862:
5859:
5856:
5852:
5849:
5845:
5842:
5839:
5836:
5833:
5830:
5826:
5823:
5819:
5816:
5813:
5810:
5807:
5804:
5801:
5798:
5795:
5774:
5771:
5767:
5762:
5759:
5755:
5746:
5743:
5738:
5734:
5731:
5728:
5725:
5722:
5719:
5716:
5711:
5707:
5703:
5698:
5694:
5690:
5684:
5681:
5677:
5674:
5671:
5667:
5664:
5660:
5656:
5653:
5649:
5646:
5638:
5633:
5629:
5623:
5620:
5615:
5611:
5606:
5598:
5594:
5588:
5584:
5578:
5575:
5571:
5565:
5561:
5554:
5551:
5547:
5542:
5539:
5536:
5533:
5530:
5445:
5438:
5435:
5431:
5428:
5425:
5421:
5416:
5410:
5407:
5404:
5400:
5365:
5360:
5355:
5351:
5345:
5341:
5338:
5258:
5255:
5252:
5248:
5243:
5239:
5233:
5229:
5225:
5222:
5219:
5215:
5212:
5207:
5203:
5199:
5196:
5192:
5189:
5185:
5181:
5178:
5174:
5170:
5167:
5163:
5160:
5127:
5120:
5117:The notations
5103:
5100:
5097:
5092:
5086:
5082:
5078:
5075:
5070:
5066:
5062:
5058:
5052:
5048:
5044:
5041:
5038:
5033:
5029:
5026:
5023:
5020:
5017:
5014:
5011:
5007:
5003:
5000:
4995:
4991:
4987:
4984:
4981:
4976:
4972:
4969:
4966:
4963:
4959:
4956:
4953:
4950:
4947:
4944:
4940:
4936:
4930:
4926:
4881:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4828:
4825:
4816:
4813:
4810:
4807:
4804:
4801:
4798:
4788:
4785:
4782:
4763:
4760:
4751:
4747:
4744:
4740:
4737:
4734:
4730:
4727:
4723:
4720:
4717:
4713:
4710:
4706:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4679:
4659:
4655:
4652:
4648:
4644:
4641:
4637:
4633:
4630:
4626:
4623:
4620:
4617:
4614:
4611:
4608:
4605:
4581:
4578:
4566:point particle
4533:
4527:
4524:
4519:
4516:
4513:
4510:
4507:
4501:
4498:
4495:
4476:point particle
4453:
4447:
4444:
4441:
4437:
4432:
4429:
4405:
4397:
4393:
4389:
4386:
4382:
4377:
4374:
4368:
4365:
4360:
4357:
4332:
4327:
4324:
4321:
4316:
4309:
4306:
4301:
4298:
4290:
4286:
4282:
4279:
4275:
4271:
4268:
4265:
4258:
4255:
4250:
4247:
4239:
4235:
4231:
4228:
4225:
4221:
4218:
4215:
4212:
4207:
4203:
4199:
4196:
4193:
4159:
4156:
4153:
4150:
4147:
4143:
4140:
4137:
4134:
4129:
4125:
4061:
4058:
4054:
4051:
4047:
4044:
4041:
4037:
4034:
4030:
4027:
4024:
4020:
4017:
4013:
4010:
4007:
4004:
4001:
3998:
3993:
3990:
3986:
3982:
3977:
3974:
3970:
3966:
3961:
3958:
3954:
3950:
3947:
3944:
3928:
3925:
3923:
3920:
3903:
3900:
3897:
3894:
3889:
3886:
3882:
3878:
3873:
3870:
3866:
3845:
3842:
3839:
3836:
3831:
3827:
3822:
3819:
3816:
3813:
3808:
3804:
3777:
3774:
3771:
3767:
3764:
3761:
3758:
3754:
3751:
3748:
3745:
3742:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3669:
3665:
3661:
3655:
3651:
3644:
3639:
3635:
3631:
3626:
3622:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3560:
3559:Electrostatics
3557:
3543:
3540:
3537:
3534:
3529:
3525:
3520:
3517:
3514:
3511:
3506:
3502:
3488:is called the
3459:
3456:
3453:
3450:
3445:
3441:
3436:
3433:
3430:
3425:
3421:
3386:
3383:
3380:
3376:
3373:
3370:
3367:
3363:
3360:
3357:
3354:
3328:
3325:
3320:
3316:
3312:
3307:
3303:
3299:
3295:
3291:
3288:
3268:
3265:
3262:
3257:
3253:
3249:
3244:
3240:
3212:Main article:
3209:
3206:
3183:
3179:
3175:
3172:
3169:
3166:
3161:
3157:
3151:
3147:
3143:
3140:
3137:
3134:
3131:
3126:
3122:
3116:
3112:
3107:
3101:
3096:
3093:
3090:
3086:
3082:
3079:
3075:
3071:
3068:
3065:
3062:
3057:
3053:
3047:
3043:
3039:
3036:
3033:
3030:
3027:
3022:
3018:
3012:
3008:
3003:
2997:
2992:
2989:
2986:
2982:
2978:
2975:
2972:
2969:
2966:
2946:
2941:
2937:
2933:
2930:
2925:
2921:
2917:
2912:
2908:
2887:
2882:
2878:
2872:
2868:
2862:
2857:
2854:
2851:
2847:
2843:
2840:
2837:
2834:
2831:
2810:Fourier series
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2712:
2709:
2706:
2703:
2700:
2697:
2655:
2650:
2647:
2643:
2639:
2634:
2631:
2627:
2617:is satisfied:
2594:
2591:
2588:
2582:
2578:
2574:
2571:
2568:
2562:
2558:
2554:
2551:
2548:
2545:
2519:
2514:
2510:
2506:
2501:
2497:
2492:
2487:
2483:
2479:
2476:
2471:
2467:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2346:
2341:
2337:
2331:
2327:
2323:
2320:
2317:
2312:
2308:
2302:
2298:
2294:
2291:
2288:
2283:
2279:
2273:
2269:
2265:
2262:
2259:
2254:
2251:
2247:
2222:
2207:
2202:
2198:
2194:
2191:
2186:
2182:
2177:
2172:
2168:
2164:
2159:
2155:
2145:be satisfied:
2107:
2104:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2020:
2017:
2005:
2002:
1997:
1994:
1990:
1986:
1981:
1978:
1974:
1970:
1962:
1958:
1954:
1949:
1944:
1940:
1933:
1925:
1921:
1917:
1912:
1907:
1903:
1887:
1884:
1790:=4) = 4 sin(5
1756:(inner radius
1745:
1742:
1722:
1706:
1703:
1698:
1695:
1691:
1687:
1684:
1681:
1678:
1675:
1671:
1668:
1665:
1661:
1652:
1648:
1644:
1639:
1636:
1628:
1625:
1621:
1614:
1610:
1606:
1599:
1589:
1585:
1581:
1577:
1568:
1564:
1560:
1555:
1550:
1547:
1542:
1538:
1517:
1514:
1511:
1506:
1501:
1498:
1494:
1488:
1485:
1481:
1472:
1468:
1464:
1459:
1456:
1450:
1446:
1440:
1437:
1433:
1424:
1420:
1416:
1411:
1408:
1401:
1392:
1388:
1384:
1380:
1375:
1372:
1367:
1363:
1331:
1328:
1320:
1316:
1312:
1307:
1302:
1298:
1288:
1285:
1280:
1276:
1270:
1266:
1261:
1256:
1252:
1245:
1242:
1237:
1234:
1228:
1225:
1222:
1218:
1211:
1208:
1204:
1196:
1193:
1190:
1185:
1181:
1176:
1171:
1167:
1160:
1157:
1152:
1149:
1141:
1137:
1132:
1125:
1122:
1118:
1109:
1105:
1101:
1096:
1093:
1088:
1084:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1017:
1014:
1006:
1002:
998:
993:
988:
984:
977:
969:
965:
961:
956:
951:
947:
936:
932:
928:
923:
919:
912:
909:
904:
901:
895:
891:
884:
881:
877:
870:
867:
862:
859:
854:
850:
821:
818:
810:
806:
802:
797:
792:
788:
781:
773:
769:
765:
760:
755:
751:
744:
736:
732:
728:
723:
718:
714:
707:
704:
699:
695:
676:
673:
658:fluid dynamics
620:
617:
614:
611:
591:
588:
585:
582:
579:
576:
573:
570:
547:
544:
541:
538:
535:
532:
529:
526:
502:
478:
475:
449:
445:
441:
438:
435:
432:
429:
426:
406:
403:
400:
397:
394:
374:
371:
368:
362:
358:
324:
323:
321:
320:
313:
306:
298:
295:
294:
293:
292:
277:
276:
275:
274:
269:
264:
259:
254:
249:
247:Leonhard Euler
244:
236:
235:
229:
228:
222:
221:
220:
219:
214:
209:
204:
199:
194:
189:
184:
182:Laurent series
179:
177:Winding number
174:
169:
164:
159:
151:
150:
144:
143:
142:
141:
136:
131:
126:
121:
113:
112:
106:
105:
104:
103:
98:
93:
88:
83:
75:
74:
68:
67:
59:
58:
52:
51:
26:
9:
6:
4:
3:
2:
10055:
10044:
10041:
10039:
10036:
10034:
10031:
10029:
10026:
10024:
10021:
10020:
10018:
10009:
10006:
10001:
10000:
9995:
9992:
9987:
9984:
9981:
9978:
9975:
9971:
9967:
9966:
9961:
9957:
9956:
9946:
9940:
9936:
9931:
9927:
9922:
9918:
9912:
9908:
9903:
9902:
9892:
9890:9780486652511
9886:
9882:
9877:
9873:
9868:
9863:
9859:
9855:
9851:
9850:
9836:
9831:
9827:
9823:
9819:
9815:
9811:
9804:
9796:
9792:
9788:
9784:
9779:
9774:
9771:(4): 042501.
9770:
9766:
9759:
9757:
9749:
9745:
9741:
9735:
9728:
9724:
9720:
9714:
9712:
9710:
9708:
9700:
9694:
9687:
9683:
9679:
9678:
9671:
9669:
9667:
9659:
9655:
9651:
9645:
9638:
9634:
9630:
9629:
9622:
9618:
9603:
9596:
9575:
9571:
9567:
9562:
9558:
9554:
9551:
9538:
9534:
9525:
9522:
9520:
9517:
9514:
9511:
9509:
9506:
9504:
9501:
9499:
9496:
9494:
9491:
9489:
9486:
9483:
9480:
9478:
9476:
9471:
9468:
9467:
9461:
9455:
9450:
9442:
9437:
9428:
9419:
9406:
9402:
9394:
9390:
9385:
9382:
9376:
9373:
9369:
9363:
9359:
9350:
9347:
9344:
9339:
9335:
9329:
9325:
9318:
9310:
9304:
9301:
9298:
9295:
9289:
9281:
9278:
9275:
9267:
9264:
9258:
9254:
9246:
9242:
9237:
9234:
9228:
9225:
9221:
9215:
9211:
9204:
9198:
9195:
9185:
9180:
9176:
9170:
9162:
9159:
9148:
9140:
9137:
9131:
9125:
9119:
9111:
9105:
9101:
9097:
9077:
9071:
9068:
9065:
9057:
9053:
9046:
9040:
9037:
9031:
9028:
9025:
9022:
9019:
8989:
8979:
8965:
8962:
8959:
8956:
8951:
8926:
8923:
8920:
8911:
8897:
8894:
8891:
8888:
8885:
8882:
8879:
8874:
8860:
8843:
8835:
8829:
8826:
8824:
8819:
8814:
8797:
8794:
8789:
8781:
8778:
8772:
8766:
8760:
8754:
8752:
8742:
8732:
8729:
8723:
8720:
8718:
8698:
8685:
8682:
8679:
8676:
8673:
8670:
8667:
8659:
8636:
8616:
8580:
8577:
8563:
8543:
8523:
8455:
8446:
8432:
8429:
8426:
8417:
8404:
8397:
8393:
8389:
8384:
8381:
8378:
8373:
8359:
8341:
8335:
8332:
8329:
8324:
8300:
8295:
8287:
8284:
8278:
8272:
8266:
8260:
8252:
8241:
8200:
8192:
8169:
8166:
8160:
8157:
8132:
8123:
8110:
8103:
8099:
8095:
8090:
8082:
8071:
8053:
8049:
8028:
7992:
7976:
7971:
7967:
7946:
7943:
7940:
7932:
7931:Taylor series
7913:
7910:
7902:
7886:
7866:
7863:
7860:
7851:
7838:
7829:
7824:
7819:
7806:
7801:
7797:
7780:
7771:
7766:
7763:
7760:
7757:
7750:
7746:
7743:are known as
7741:
7732:
7712:
7706:
7703:
7700:
7692:
7687:
7683:
7677:
7673:
7667:
7662:
7658:
7652:
7647:
7644:
7641:
7638:
7634:
7623:
7620:
7617:
7613:
7609:
7603:
7600:
7597:
7594:
7591:
7585:
7576:
7571:
7566:
7562:
7556:
7552:
7546:
7542:
7538:
7532:
7524:
7508:
7502:
7499:
7496:
7490:
7487:
7484:
7481:
7476:
7466:
7462:
7451:
7447:
7443:
7433:
7429:
7423:
7416:
7411:
7404:
7392:
7379:
7374:
7369:
7352:
7343:
7330:
7324:
7321:
7318:
7310:
7305:
7301:
7294:
7291:
7288:
7282:
7279:
7276:
7270:
7267:
7264:
7256:
7251:
7247:
7241:
7231:
7227:
7202:
7198:
7195:
7187:
7182:
7178:
7172:
7169:
7166:
7162:
7158:
7155:
7149:
7146:
7143:
7135:
7130:
7126:
7117:
7113:
7108:
7104:
7100:
7093:
7086:
7079:
7072:
7068:
7064:
7060:
7056:
7050:
7046:
7040:
7034:
7030:
7026:
7022:
7016:
7009:
7005:
6998:
6994:
6989:
6985:
6980:
6976:
6970:
6966:
6959:
6955:
6951:
6945:
6940:
6931:
6927:
6920:
6916:
6912:
6906:
6896:
6890:
6884:
6875:
6869:
6863:
6848:
6844:
6840:
6836:
6829:
6826:
6818:
6812:
6809:
6806:
6802:
6795:
6792:
6788:
6777:
6774:
6771:
6765:
6762:
6759:
6754:
6750:
6746:
6725:
6721:
6717:
6714:
6706:
6702:
6698:
6688:
6684:
6672:
6662:
6658:
6654:
6650:
6646:
6642:
6638:has the form
6636:
6630:
6617:
6614:
6611:
6608:
6600:
6596:
6587:
6582:
6568:
6565:
6560:
6556:
6551:
6544:
6541:
6536:
6532:
6525:
6517:
6508:
6505:
6502:
6498:
6491:
6476:
6473:
6470:
6466:
6459:
6456:
6450:
6447:
6444:
6440:
6433:
6430:
6425:
6422:
6414:
6410:
6405:
6398:
6395:
6391:
6384:
6381:
6371:
6369:
6363:
6359:
6355:
6351:
6347:
6343:
6339:
6335:
6331:
6325:
6312:
6309:
6301:
6297:
6288:
6283:
6269:
6266:
6261:
6257:
6251:
6247:
6242:
6237:
6233:
6226:
6218:
6209:
6206:
6203:
6199:
6192:
6177:
6174:
6171:
6166:
6162:
6157:
6152:
6148:
6141:
6133:
6122:
6118:
6113:
6106:
6090:
6086:
6082:
6077:
6074:
6069:
6056:
6054:
6034:
6030:
6024:
6020:
6010:
5991:
5988:
5983:
5979:
5969:
5965:
5958:
5952:
5943:
5938:
5928:
5924:
5918:
5911:
5907:
5899:
5895:
5873:
5870:
5866:
5863:
5857:
5854:
5850:
5847:
5843:
5840:
5837:
5834:
5831:
5828:
5824:
5821:
5817:
5814:
5811:
5808:
5805:
5802:
5796:
5793:
5772:
5769:
5765:
5760:
5757:
5753:
5744:
5741:
5729:
5726:
5723:
5720:
5717:
5714:
5709:
5705:
5701:
5696:
5692:
5682:
5679:
5675:
5672:
5665:
5662:
5658:
5654:
5651:
5644:
5636:
5631:
5627:
5621:
5618:
5613:
5609:
5604:
5596:
5592:
5586:
5582:
5576:
5573:
5569:
5563:
5559:
5552:
5549:
5545:
5540:
5534:
5528:
5520:
5515:
5504:
5499:
5483:
5482:
5477:
5471:
5465:
5443:
5436:
5433:
5429:
5426:
5423:
5419:
5414:
5408:
5405:
5402:
5398:
5388:
5383:
5379:Note that if
5377:
5363:
5358:
5353:
5349:
5343:
5339:
5336:
5327:
5325:
5320:
5314:
5309:
5304:
5298:
5292:
5285:
5281:
5277:
5270:
5256:
5253:
5250:
5246:
5241:
5237:
5231:
5227:
5223:
5220:
5217:
5213:
5210:
5205:
5201:
5197:
5190:
5187:
5183:
5179:
5176:
5172:
5168:
5165:
5158:
5150:
5147:
5141:
5135:
5130:
5123:
5115:
5101:
5098:
5095:
5090:
5084:
5080:
5076:
5073:
5068:
5064:
5060:
5056:
5050:
5046:
5042:
5039:
5036:
5031:
5027:
5021:
5018:
5015:
5009:
5005:
5001:
4993:
4989:
4985:
4982:
4979:
4974:
4970:
4964:
4957:
4954:
4951:
4945:
4938:
4934:
4928:
4924:
4915:
4913:
4904:
4898:
4892:
4879:
4876:
4873:
4870:
4867:
4861:
4845:
4839:
4826:
4823:
4811:
4808:
4805:
4802:
4799:
4786:
4783:
4780:
4761:
4758:
4745:
4742:
4738:
4735:
4732:
4728:
4725:
4721:
4718:
4715:
4711:
4708:
4704:
4701:
4695:
4692:
4689:
4686:
4680:
4653:
4650:
4646:
4642:
4639:
4635:
4631:
4628:
4624:
4621:
4618:
4615:
4612:
4609:
4603:
4587:
4577:
4575:
4571:
4567:
4563:
4559:
4555:
4549:
4531:
4525:
4522:
4514:
4508:
4505:
4499:
4496:
4493:
4485:
4481:
4477:
4473:
4469:
4464:
4451:
4445:
4442:
4439:
4435:
4430:
4427:
4403:
4395:
4391:
4387:
4384:
4380:
4375:
4372:
4366:
4363:
4358:
4355:
4343:
4330:
4325:
4322:
4319:
4314:
4307:
4304:
4299:
4296:
4288:
4284:
4280:
4277:
4269:
4266:
4263:
4256:
4253:
4248:
4245:
4237:
4233:
4229:
4226:
4223:
4219:
4213:
4205:
4201:
4197:
4194:
4191:
4184:implies that
4183:
4170:
4157:
4154:
4151:
4148:
4145:
4141:
4135:
4127:
4123:
4113:
4108:
4104:
4103:weak solution
4100:
4094:
4090:
4086:
4079:
4075:
4059:
4052:
4049:
4045:
4042:
4039:
4035:
4032:
4028:
4025:
4022:
4018:
4015:
4011:
4008:
4002:
3999:
3996:
3991:
3988:
3984:
3980:
3975:
3972:
3968:
3964:
3959:
3956:
3952:
3948:
3945:
3934:
3919:
3917:
3914:which is the
3901:
3898:
3895:
3892:
3887:
3884:
3880:
3876:
3871:
3868:
3864:
3843:
3840:
3837:
3834:
3829:
3825:
3820:
3817:
3814:
3811:
3806:
3802:
3792:
3775:
3772:
3769:
3765:
3762:
3759:
3756:
3752:
3749:
3746:
3743:
3740:
3731:
3714:
3711:
3708:
3702:
3699:
3696:
3690:
3667:
3659:
3637:
3633:
3629:
3624:
3620:
3613:
3607:
3604:
3601:
3598:
3595:
3589:
3576:
3572:
3566:
3563:According to
3556:
3541:
3538:
3535:
3532:
3527:
3523:
3518:
3515:
3512:
3509:
3504:
3500:
3491:
3486:
3480:
3474:
3457:
3454:
3451:
3448:
3443:
3439:
3434:
3431:
3428:
3423:
3419:
3411:are given by
3409:
3404:
3400:
3384:
3381:
3378:
3374:
3371:
3368:
3365:
3361:
3358:
3355:
3352:
3343:
3326:
3323:
3318:
3314:
3310:
3305:
3301:
3297:
3289:
3266:
3263:
3260:
3255:
3251:
3247:
3242:
3238:
3228:
3222:
3215:
3205:
3203:
3198:
3181:
3177:
3173:
3170:
3167:
3164:
3159:
3155:
3149:
3145:
3141:
3138:
3135:
3132:
3129:
3124:
3120:
3114:
3110:
3105:
3094:
3091:
3088:
3084:
3080:
3077:
3073:
3069:
3066:
3063:
3060:
3055:
3051:
3045:
3041:
3037:
3034:
3031:
3028:
3025:
3020:
3016:
3010:
3006:
3001:
2990:
2987:
2984:
2980:
2976:
2970:
2964:
2944:
2939:
2935:
2931:
2928:
2923:
2919:
2915:
2910:
2906:
2885:
2880:
2876:
2870:
2866:
2855:
2852:
2849:
2845:
2841:
2835:
2829:
2816:
2811:
2806:
2804:
2803:wave equation
2800:
2795:
2788:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2736:
2730:
2710:
2707:
2704:
2701:
2698:
2695:
2679:
2675:
2670:
2653:
2648:
2645:
2641:
2637:
2632:
2629:
2625:
2615:
2609:
2592:
2589:
2586:
2580:
2576:
2572:
2569:
2566:
2560:
2556:
2552:
2549:
2546:
2543:
2534:
2517:
2512:
2508:
2504:
2499:
2495:
2490:
2485:
2481:
2477:
2474:
2469:
2465:
2444:
2438:
2435:
2432:
2426:
2423:
2420:
2414:
2411:
2408:
2402:
2399:
2393:
2387:
2377:
2373:
2367:
2361:
2344:
2339:
2329:
2325:
2318:
2315:
2310:
2300:
2296:
2289:
2286:
2281:
2271:
2267:
2263:
2257:
2252:
2249:
2245:
2231:
2225:
2205:
2200:
2196:
2192:
2189:
2184:
2180:
2175:
2170:
2166:
2162:
2157:
2153:
2144:
2139:
2133:
2126:
2122:
2105:
2099:
2096:
2093:
2087:
2084:
2081:
2075:
2072:
2069:
2063:
2060:
2054:
2048:
2039:
2035:
2031:
2026:
2016:
2003:
2000:
1995:
1992:
1988:
1984:
1979:
1976:
1972:
1968:
1960:
1956:
1947:
1942:
1931:
1923:
1919:
1910:
1905:
1883:
1881:
1876:
1872:
1867:
1864:
1858:
1853:
1844:
1839:
1834:
1832:
1831:heat equation
1823:
1813:
1808:
1802:
1793:
1789:
1785:
1778:
1774:
1767:
1760:
1755:
1750:
1741:
1739:
1731:
1730:metric tensor
1725:
1721:
1696:
1693:
1689:
1679:
1676:
1669:
1666:
1663:
1659:
1650:
1646:
1637:
1626:
1623:
1619:
1608:
1597:
1587:
1583:
1562:
1553:
1548:
1545:
1540:
1515:
1512:
1509:
1504:
1499:
1496:
1486:
1483:
1479:
1470:
1466:
1457:
1448:
1444:
1438:
1435:
1431:
1422:
1418:
1409:
1399:
1390:
1386:
1373:
1370:
1365:
1349:
1348:
1342:
1329:
1326:
1318:
1314:
1305:
1300:
1286:
1283:
1278:
1274:
1268:
1264:
1259:
1254:
1250:
1243:
1235:
1226:
1223:
1220:
1216:
1209:
1194:
1191:
1188:
1183:
1179:
1174:
1169:
1165:
1158:
1150:
1139:
1135:
1130:
1123:
1107:
1103:
1099:
1094:
1091:
1086:
1058:
1055:
1052:
1049:
1046:
1035:
1034:
1028:
1015:
1012:
1004:
1000:
991:
986:
975:
967:
963:
954:
949:
934:
930:
926:
921:
917:
910:
902:
893:
889:
882:
868:
865:
860:
857:
852:
839:
838:
832:
819:
816:
808:
804:
795:
790:
779:
771:
767:
758:
753:
742:
734:
730:
721:
716:
705:
702:
697:
685:
683:
672:
670:
669:heat equation
667:
663:
659:
655:
651:
646:
644:
640:
636:
631:
618:
615:
612:
586:
583:
580:
577:
574:
568:
559:
542:
539:
536:
533:
530:
524:
516:
492:
476:
465:
447:
439:
433:
427:
404:
401:
398:
395:
372:
369:
366:
360:
347:
343:
339:
335:
331:
319:
314:
312:
307:
305:
300:
299:
297:
296:
291:
286:
281:
280:
279:
278:
273:
270:
268:
265:
263:
260:
258:
255:
253:
250:
248:
245:
243:
240:
239:
238:
237:
234:
231:
230:
227:
224:
223:
218:
215:
213:
210:
208:
207:Schwarz lemma
205:
203:
202:Conformal map
200:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
154:
153:
152:
149:
146:
145:
140:
137:
135:
132:
130:
127:
125:
122:
120:
117:
116:
115:
114:
111:
108:
107:
102:
99:
97:
94:
92:
91:Complex plane
89:
87:
84:
82:
79:
78:
77:
76:
73:
70:
69:
65:
61:
60:
57:
54:
53:
49:
45:
41:
40:
37:
33:
19:
9997:
9963:
9934:
9925:
9906:
9880:
9871:
9861:
9817:
9813:
9803:
9768:
9764:
9739:
9734:
9718:
9693:
9675:
9649:
9644:
9626:
9621:
9601:
9595:
9537:
9474:
9445:
9432:
9423:
9420:
9103:
9099:
9092:
8985:
8939:and we have
8912:
8861:
8699:
8586:
8578:
8447:
8418:
8360:
8242:
8124:
7998:
7852:
7736:
7727:
7574:
7567:
7560:
7544:
7540:
7536:
7527:
7449:
7445:
7441:
7431:
7427:
7409:
7402:
7377:
7364:
7347:
7344:
7106:
7102:
7098:
7091:
7084:
7081:, there are
7077:
7070:
7066:
7062:
7058:
7054:
7051:
7044:
7038:
7032:
7028:
7024:
7020:
7014:
7007:
7000:
6987:
6983:
6968:
6964:
6957:
6953:
6949:
6943:
6929:
6925:
6918:
6914:
6910:
6904:
6894:
6888:
6873:
6871:. A priori,
6867:
6864:
6663:
6656:
6652:
6648:
6644:
6640:
6634:
6631:
6372:
6361:
6357:
6353:
6349:
6345:
6341:
6337:
6333:
6329:
6326:
6057:
6050:
6008:
5967:
5963:
5956:
5947:
5922:
5916:
5909:
5905:
5897:
5893:
5513:
5502:
5479:
5475:
5469:
5463:
5386:
5381:
5378:
5328:
5323:
5318:
5316:at distance
5312:
5302:
5296:
5290:
5288:of the data
5283:
5279:
5275:
5271:
5151:
5145:
5139:
5133:
5125:
5118:
5116:
4916:
4902:
4896:
4893:
4843:
4840:
4670:may satisfy
4592:of a volume
4583:
4552:denotes the
4547:
4465:
4344:
4171:
4111:
4099:distribution
4092:
4088:
4084:
4077:
3930:
3790:
3729:
3574:
3570:
3562:
3484:
3478:
3472:
3407:
3341:
3226:
3220:
3217:
3196:
2814:
2807:
2799:power series
2796:
2789:
2677:
2668:
2613:
2607:
2532:
2375:
2371:
2365:
2359:
2229:
2220:
2137:
2131:
2124:
2120:
2037:
2033:
2029:
2022:
1889:
1868:
1862:
1856:
1842:
1835:
1821:
1811:
1804:
1791:
1787:
1783:
1776:
1772:
1765:
1758:
1736:denotes its
1723:
1719:
1345:
1343:
1074:convention,
1036:, using the
1031:
1029:
835:
833:
680:
678:
666:steady-state
647:
632:
560:
344:named after
337:
327:
216:
148:Basic theory
47:
36:
8583:Gravitation
8070:Gauss's law
5959:= 0, ..., 4
330:mathematics
262:Kiyoshi Oka
81:Real number
10017:Categories
9613:References
9477:-separable
7725:where the
7391:colatitude
7358:and order
6881:must be a
5966:= 0, ...,
5521:, p. 228)
4072:where the
3403:flow lines
3208:Fluid flow
2957:Therefore
2357:Therefore
2041:, and if
1819:such that
1799:See also:
602:, we have
491:divergence
9999:MathWorld
9970:EMS Press
9795:118707082
9778:1111.4702
9568:−
9549:Δ
9377:−
9265:−
9229:−
9138:−
9072:φ
9066:θ
9032:φ
9026:θ
9014:Ψ
8948:∇
8921:ρ
8895:ρ
8889:π
8871:∇
8836:⋅
8833:∇
8830:−
8811:∇
8806:⟹
8786:∇
8782:−
8770:∇
8767:−
8761:⋅
8758:∇
8743:⋅
8740:∇
8727:∇
8724:−
8683:ρ
8677:π
8671:−
8660:⋅
8657:∇
8617:ρ
8524:ρ
8427:ρ
8394:ε
8390:ρ
8385:−
8370:∇
8342:⋅
8339:∇
8336:−
8321:∇
8292:∇
8288:−
8276:∇
8273:−
8267:⋅
8264:∇
8253:⋅
8250:∇
8193:×
8190:∇
8164:∇
8161:−
8100:ε
8096:ρ
8083:⋅
8080:∇
8050:ε
8029:ρ
7972:ℓ
7917:∞
7830:ℓ
7802:ℓ
7787:∞
7784:→
7781:ℓ
7707:φ
7701:θ
7688:ℓ
7678:ℓ
7663:ℓ
7653:ℓ
7648:ℓ
7645:−
7635:∑
7629:∞
7618:ℓ
7614:∑
7604:φ
7598:θ
7497:ℓ
7491:ℓ
7488:−
7473:∇
7415:longitude
7325:φ
7319:θ
7306:ℓ
7289:ℓ
7283:ℓ
7280:−
7271:φ
7265:θ
7252:ℓ
7238:∇
7203:θ
7199:
7183:ℓ
7173:φ
7150:φ
7144:θ
7131:ℓ
6830:θ
6822:Θ
6813:θ
6810:
6796:θ
6782:Θ
6778:θ
6775:
6763:θ
6760:
6747:λ
6718:−
6703:φ
6694:Φ
6676:Φ
6615:λ
6612:−
6597:φ
6593:∂
6579:∂
6569:θ
6566:
6526:θ
6523:∂
6515:∂
6509:θ
6506:
6492:θ
6489:∂
6485:∂
6477:θ
6474:
6448:λ
6298:φ
6294:∂
6280:∂
6270:θ
6267:
6227:θ
6224:∂
6216:∂
6210:θ
6207:
6193:θ
6190:∂
6186:∂
6178:θ
6175:
6139:∂
6131:∂
6104:∂
6100:∂
6066:∇
6025:∘
5989:−
5984:ℓ
5871:φ
5867:−
5864:φ
5858:
5848:θ
5844:
5838:θ
5835:
5822:θ
5818:
5812:θ
5809:
5800:Θ
5797:
5770:φ
5758:θ
5733:Θ
5730:
5724:ρ
5715:−
5706:ρ
5680:θ
5676:
5663:φ
5652:θ
5637:π
5628:∫
5622:π
5610:∫
5583:ρ
5577:−
5553:π
5430:ρ
5427:π
5415:−
5406:π
5359:ρ
5337:ρ
5228:∬
5202:∭
5074:−
5047:∬
5025:∇
5019:−
5013:∇
5002:⋅
4999:∇
4990:∭
4968:∇
4965:⋅
4962:∇
4955:−
4949:∇
4946:⋅
4943:∇
4925:∭
4874:−
4865:∇
4862:⋅
4859:∇
4739:−
4722:−
4705:−
4696:δ
4693:−
4684:∇
4681:⋅
4678:∇
4558:potential
4526:π
4509:
4500:−
4478:, for an
4472:potential
4443:π
4388:π
4376:−
4281:π
4234:∬
4217:∇
4214:⋅
4211:∇
4202:∭
4192:−
4155:−
4139:∇
4136:⋅
4133:∇
4124:∭
4046:−
4029:−
4012:−
4003:δ
4000:−
3943:Δ
3899:ρ
3896:−
3881:φ
3865:φ
3838:−
3826:φ
3815:−
3803:φ
3763:−
3750:−
3744:φ
3712:ρ
3691:⋅
3688:∇
3654:^
3630:−
3590:×
3587:∇
3536:−
3524:φ
3513:−
3501:φ
3452:−
3440:ψ
3420:ψ
3372:−
3356:ψ
3311:−
3290:×
3287:∇
3174:θ
3168:
3139:θ
3133:
3100:∞
3085:∑
3070:θ
3064:
3038:−
3035:θ
3029:
2996:∞
2981:∑
2861:∞
2846:∑
2773:θ
2761:
2749:
2705:
2696:φ
2666:and thus
2642:ψ
2626:ψ
2577:φ
2557:φ
2553:−
2547:ψ
2509:φ
2496:ψ
2482:φ
2478:−
2466:ψ
2427:ψ
2403:φ
2319:−
2290:−
2264:−
2193:−
1989:ψ
1973:ψ
1969:≡
1953:∂
1948:ψ
1939:∂
1916:∂
1911:ψ
1902:∂
1866:is zero.
1647:ξ
1643:∂
1635:∂
1584:ξ
1580:∂
1576:∂
1537:∇
1493:Γ
1467:ξ
1463:∂
1455:∂
1419:ξ
1415:∂
1407:∂
1387:ξ
1383:∂
1379:∂
1362:∇
1315:φ
1311:∂
1297:∂
1287:θ
1284:
1244:θ
1241:∂
1233:∂
1227:θ
1224:
1210:θ
1207:∂
1203:∂
1195:θ
1192:
1156:∂
1148:∂
1121:∂
1117:∂
1083:∇
1059:φ
1053:θ
997:∂
983:∂
964:ϕ
960:∂
946:∂
908:∂
900:∂
880:∂
876:∂
849:∇
801:∂
787:∂
764:∂
750:∂
727:∂
713:∂
694:∇
610:Δ
501:∇
477:⋅
474:∇
444:∇
437:∇
434:⋅
431:∇
425:Δ
393:Δ
357:∇
9860:(1962),
9464:See also
6012:axis by
5874:′
5851:′
5825:′
5773:′
5761:′
5683:′
5666:′
5655:′
5506:. Here
5437:′
5387:P′
5340:′
5191:′
5180:′
5169:′
4820:on
4755:in
4746:′
4729:′
4712:′
4654:′
4643:′
4632:′
4053:′
4036:′
4019:′
1875:analytic
1850:but its
1351:(ξ)
515:gradient
9972:, 2001
9847:Sources
9822:Bibcode
9452:is the
7933:(about
7903:(about
7422:azimuth
7042:forces
4841:Now if
4468:physics
1779:=2) = 0
1754:annulus
513:is the
489:is the
462:is the
334:physics
9941:
9913:
9887:
9793:
9746:
9725:
9684:
9656:
9635:
9112:, and
9090:where
7430:< 2
7371:is an
7065:) = Θ(
6986:= cos
6651:) = Θ(
6366:. By
5786:where
5492:, and
5484:. Let
5457:where
4544:where
3727:where
2218:where
1734:Γ
1717:where
417:where
233:People
9791:S2CID
9773:arXiv
9530:Notes
9421:Here
9108:is a
7555:below
7521:is a
7420:, or
7345:Here
7095:with
7006:(cos
6975:below
6928:= 0,
4564:(see
9939:ISBN
9911:ISBN
9885:ISBN
9744:ISBN
9723:ISBN
9682:ISBN
9654:ISBN
9633:ISBN
9439:are
9430:and
8587:Let
7999:Let
7864:>
7853:For
7761:<
7749:ball
7426:0 ≤
7385:and
7069:) Φ(
7027:) =
6960:+ 1)
6655:) Φ(
6344:) =
6055:is:
5954:for
5902:and
5467:and
5294:and
5143:and
5124:and
4894:and
4562:sink
4546:log(
3680:and
3224:and
2684:and
2135:and
1836:The
1805:The
1781:and
1528:or
332:and
9830:doi
9783:doi
9604:= 0
7737:r Y
7563:+ 1
7537:r Y
7525:of
7196:cos
7087:+ 1
7047:= 0
7033:B r
7029:A r
6967:≥ |
6807:sin
6772:sin
6751:sin
6557:sin
6503:sin
6471:sin
6258:sin
6204:sin
6172:sin
5908:′,
5855:cos
5841:sin
5832:sin
5815:cos
5806:cos
5794:cos
5727:cos
5673:sin
5496:be
5282:′,
5278:′,
4906:on
4506:log
4091:′,
4087:′,
3345:by
3165:cos
3130:sin
3061:sin
3026:cos
2758:log
2746:log
2702:log
1768:= 4
1761:= 2
1683:det
1275:sin
1221:sin
1189:sin
1030:In
834:In
679:In
385:or
328:In
10019::
9996:.
9968:,
9962:,
9856:;
9828:.
9818:43
9816:.
9812:.
9789:.
9781:.
9769:53
9767:.
9755:^
9706:^
9665:^
9102:,
9002:,
8998:,
8145:,
7991:.
7578:,
7543:,
7448:,
7405:/2
7375:,
7362:,
7105:≤
7101:≤
7061:,
7049:.
7031:+
6952:=
6917:,
6892:,
6647:,
6360:,
6352:)
6340:,
6336:,
6313:0.
6021:90
5912:′)
5896:,
5488:,
5324:P'
5286:′)
4851::
4792:if
4584:A
4576:.
4158:1.
4095:′)
3931:A
3573:,
3327:0.
3204:.
2038:iy
2036:+
2032:=
2004:0.
1740:.
1724:ij
1353:,
1330:0.
1016:0.
840:,
820:0.
645:.
466:,
336:,
46:→
10002:.
9947:.
9919:.
9893:.
9866:.
9838:.
9832::
9824::
9797:.
9785::
9775::
9750:.
9729:.
9688:.
9660:.
9639:.
9602:A
9576:2
9572:x
9563:1
9559:x
9555:=
9552:x
9475:R
9458:l
9448:s
9446:r
9435:l
9433:Q
9426:l
9424:P
9407:.
9403:)
9395:s
9391:r
9386:r
9383:2
9374:1
9370:(
9364:l
9360:Q
9351:1
9348:+
9345:l
9340:s
9336:r
9330:2
9326:!
9322:)
9319:l
9316:(
9311:!
9308:)
9305:1
9302:+
9299:l
9296:2
9293:(
9290:2
9282:1
9279:+
9276:l
9272:)
9268:1
9262:(
9259:+
9255:)
9247:s
9243:r
9238:r
9235:2
9226:1
9222:(
9216:l
9212:P
9205:!
9202:)
9199:l
9196:2
9193:(
9186:l
9181:s
9177:r
9171:2
9167:)
9163:!
9160:l
9157:(
9149:l
9145:)
9141:1
9135:(
9132:=
9129:)
9126:r
9123:(
9120:R
9106:)
9104:φ
9100:θ
9098:(
9095:l
9093:Y
9078:,
9075:)
9069:,
9063:(
9058:l
9054:Y
9050:)
9047:r
9044:(
9041:R
9038:=
9035:)
9029:,
9023:,
9020:r
9017:(
9004:φ
9000:θ
8996:r
8992:t
8966:,
8963:0
8960:=
8957:V
8952:2
8927:0
8924:=
8898:,
8892:G
8886:4
8883:=
8880:V
8875:2
8844:.
8840:g
8827:=
8820:V
8815:2
8798:,
8795:V
8790:2
8779:=
8776:)
8773:V
8764:(
8755:=
8747:g
8733:,
8730:V
8721:=
8713:g
8686:.
8680:G
8674:4
8668:=
8664:g
8637:G
8596:g
8564:V
8544:Q
8502:R
8478:R
8456:V
8433:0
8430:=
8405:.
8398:0
8382:=
8379:V
8374:2
8346:E
8333:=
8330:V
8325:2
8301:V
8296:2
8285:=
8282:)
8279:V
8270:(
8261:=
8257:E
8227:E
8205:0
8201:=
8197:E
8170:,
8167:V
8158:=
8154:E
8133:V
8111:.
8104:0
8091:=
8087:E
8054:0
8008:E
7977:m
7968:f
7947:0
7944:=
7941:r
7914:=
7911:r
7887:r
7867:R
7861:r
7839:.
7825:/
7820:1
7814:|
7807:m
7798:f
7793:|
7772:1
7767:=
7764:R
7758:r
7739:ℓ
7730:ℓ
7728:f
7713:,
7710:)
7704:,
7698:(
7693:m
7684:Y
7674:r
7668:m
7659:f
7642:=
7639:m
7624:0
7621:=
7610:=
7607:)
7601:,
7595:,
7592:r
7589:(
7586:f
7575:r
7561:ℓ
7559:2
7547:)
7545:φ
7541:θ
7539:(
7530:ℓ
7528:Y
7509:Y
7506:)
7503:1
7500:+
7494:(
7485:=
7482:Y
7477:2
7467:2
7463:r
7452:)
7450:φ
7446:θ
7444:(
7442:Y
7437:ℓ
7432:π
7428:φ
7418:φ
7410:π
7403:π
7398:0
7394:θ
7387:φ
7383:θ
7378:N
7367:ℓ
7365:P
7360:m
7356:ℓ
7350:ℓ
7348:Y
7331:.
7328:)
7322:,
7316:(
7311:m
7302:Y
7298:)
7295:1
7292:+
7286:(
7277:=
7274:)
7268:,
7262:(
7257:m
7248:Y
7242:2
7232:2
7228:r
7207:)
7193:(
7188:m
7179:P
7170:m
7167:i
7163:e
7159:N
7156:=
7153:)
7147:,
7141:(
7136:m
7127:Y
7107:ℓ
7103:m
7099:ℓ
7097:−
7092:m
7085:ℓ
7083:2
7078:ℓ
7073:)
7071:φ
7067:θ
7063:φ
7059:θ
7057:(
7055:Y
7045:B
7039:R
7025:r
7023:(
7021:R
7015:R
7010:)
7008:θ
7003:ℓ
7001:P
6988:θ
6984:t
6971:|
6969:m
6965:ℓ
6958:ℓ
6956:(
6954:ℓ
6950:λ
6944:λ
6935:Θ
6930:π
6926:θ
6921:)
6919:φ
6915:θ
6913:(
6911:Y
6905:e
6900:Φ
6895:m
6889:π
6887:2
6879:Φ
6874:m
6868:m
6849:2
6845:m
6841:=
6837:)
6827:d
6819:d
6803:(
6793:d
6789:d
6766:+
6755:2
6726:2
6722:m
6715:=
6707:2
6699:d
6689:2
6685:d
6673:1
6659:)
6657:φ
6653:θ
6649:φ
6645:θ
6643:(
6641:Y
6635:Y
6618:.
6609:=
6601:2
6588:Y
6583:2
6561:2
6552:1
6545:Y
6542:1
6537:+
6533:)
6518:Y
6499:(
6467:1
6460:Y
6457:1
6451:,
6445:=
6441:)
6434:r
6431:d
6426:R
6423:d
6415:2
6411:r
6406:(
6399:r
6396:d
6392:d
6385:R
6382:1
6364:)
6362:φ
6358:θ
6356:(
6354:Y
6350:r
6348:(
6346:R
6342:φ
6338:θ
6334:r
6332:(
6330:f
6310:=
6302:2
6289:f
6284:2
6262:2
6252:2
6248:r
6243:1
6238:+
6234:)
6219:f
6200:(
6167:2
6163:r
6158:1
6153:+
6149:)
6142:r
6134:f
6123:2
6119:r
6114:(
6107:r
6091:2
6087:r
6083:1
6078:=
6075:f
6070:2
6035:m
6031:/
6009:z
5992:m
5980:Y
5968:ℓ
5964:m
5957:ℓ
5950:ℓ
5948:Y
5923:u
5917:u
5910:φ
5906:θ
5904:(
5900:)
5898:φ
5894:θ
5892:(
5878:)
5861:(
5829:+
5803:=
5766:d
5754:d
5745:2
5742:3
5737:)
5721:a
5718:2
5710:2
5702:+
5697:2
5693:a
5689:(
5670:)
5659:,
5648:(
5645:g
5632:0
5619:2
5614:0
5605:)
5597:2
5593:a
5587:2
5574:1
5570:(
5564:3
5560:a
5550:4
5546:1
5541:=
5538:)
5535:P
5532:(
5529:u
5514:g
5508:θ
5503:P
5494:φ
5490:θ
5486:ρ
5476:P
5472:′
5470:R
5464:P
5459:R
5444:,
5434:R
5424:4
5420:a
5409:R
5403:4
5399:1
5382:P
5364:.
5354:2
5350:a
5344:=
5319:ρ
5313:P
5303:a
5297:g
5291:f
5284:z
5280:y
5276:x
5274:(
5257:.
5254:S
5251:d
5247:g
5242:n
5238:G
5232:S
5224:+
5221:V
5218:d
5214:f
5211:G
5206:V
5198:=
5195:)
5188:z
5184:,
5177:y
5173:,
5166:x
5162:(
5159:u
5146:G
5140:u
5134:S
5128:n
5126:G
5121:n
5119:u
5102:.
5099:S
5096:d
5091:]
5085:n
5081:G
5077:u
5069:n
5065:u
5061:G
5057:[
5051:S
5043:=
5040:V
5037:d
5032:]
5028:G
5022:u
5016:u
5010:G
5006:[
4994:V
4986:=
4983:V
4980:d
4975:]
4971:G
4958:u
4952:u
4939:G
4935:[
4929:V
4908:S
4903:g
4897:u
4880:,
4877:f
4871:=
4868:u
4849:V
4844:u
4827:.
4824:S
4815:)
4812:z
4809:,
4806:y
4803:,
4800:x
4797:(
4787:0
4784:=
4781:G
4762:,
4759:V
4750:)
4743:z
4736:z
4733:,
4726:y
4719:y
4716:,
4709:x
4702:x
4699:(
4690:=
4687:G
4658:)
4651:z
4647:,
4640:y
4636:,
4629:x
4625:;
4622:z
4619:,
4616:y
4613:,
4610:x
4607:(
4604:G
4594:V
4590:S
4550:)
4548:r
4532:.
4523:2
4518:)
4515:r
4512:(
4497:=
4494:u
4452:.
4446:r
4440:4
4436:1
4431:=
4428:u
4418:r
4404:,
4396:2
4392:r
4385:4
4381:1
4373:=
4367:r
4364:d
4359:u
4356:d
4331:.
4326:a
4323:=
4320:r
4315:|
4308:r
4305:d
4300:u
4297:d
4289:2
4285:a
4278:4
4270:=
4267:S
4264:d
4257:r
4254:d
4249:u
4246:d
4238:S
4230:=
4227:V
4224:d
4220:u
4206:V
4198:=
4195:1
4178:a
4174:r
4152:=
4149:V
4146:d
4142:u
4128:V
4112:u
4093:z
4089:y
4085:x
4083:(
4078:δ
4060:,
4057:)
4050:z
4043:z
4040:,
4033:y
4026:y
4023:,
4016:x
4009:x
4006:(
3997:=
3992:z
3989:z
3985:u
3981:+
3976:y
3973:y
3969:u
3965:+
3960:x
3957:x
3953:u
3949:=
3946:u
3902:,
3893:=
3888:y
3885:y
3877:+
3872:x
3869:x
3844:.
3841:v
3835:=
3830:y
3821:,
3818:u
3812:=
3807:x
3791:φ
3776:,
3773:y
3770:d
3766:v
3760:x
3757:d
3753:u
3747:=
3741:d
3730:ρ
3715:,
3709:=
3706:)
3703:v
3700:,
3697:u
3694:(
3668:,
3664:0
3660:=
3650:k
3643:)
3638:y
3634:u
3625:x
3621:v
3617:(
3614:=
3611:)
3608:0
3605:,
3602:v
3599:,
3596:u
3593:(
3577:)
3575:v
3571:u
3569:(
3542:.
3539:v
3533:=
3528:y
3519:,
3516:u
3510:=
3505:x
3485:ψ
3479:φ
3473:ψ
3458:,
3455:u
3449:=
3444:y
3435:,
3432:v
3429:=
3424:x
3408:ψ
3385:,
3382:y
3379:d
3375:u
3369:x
3366:d
3362:v
3359:=
3353:d
3342:ψ
3324:=
3319:y
3315:u
3306:x
3302:v
3298:=
3294:V
3267:,
3264:0
3261:=
3256:y
3252:v
3248:+
3243:x
3239:u
3227:v
3221:u
3197:f
3182:,
3178:]
3171:n
3160:n
3156:r
3150:n
3146:b
3142:+
3136:n
3125:n
3121:r
3115:n
3111:a
3106:[
3095:1
3092:=
3089:n
3081:i
3078:+
3074:]
3067:n
3056:n
3052:r
3046:n
3042:b
3032:n
3021:n
3017:r
3011:n
3007:a
3002:[
2991:0
2988:=
2985:n
2977:=
2974:)
2971:z
2968:(
2965:f
2945:.
2940:n
2936:b
2932:i
2929:+
2924:n
2920:a
2916:=
2911:n
2907:c
2886:,
2881:n
2877:z
2871:n
2867:c
2856:0
2853:=
2850:n
2842:=
2839:)
2836:z
2833:(
2830:f
2820:R
2815:f
2792:θ
2776:.
2770:i
2767:+
2764:r
2755:=
2752:z
2743:=
2740:)
2737:z
2734:(
2731:f
2711:,
2708:r
2699:=
2686:θ
2682:r
2669:ψ
2654:,
2649:x
2646:y
2638:=
2633:y
2630:x
2614:ψ
2608:φ
2593:.
2590:y
2587:d
2581:x
2573:+
2570:x
2567:d
2561:y
2550:=
2544:d
2533:ψ
2518:.
2513:x
2505:=
2500:y
2491:,
2486:y
2475:=
2470:x
2445:,
2442:)
2439:y
2436:,
2433:x
2430:(
2424:i
2421:+
2418:)
2415:y
2412:,
2409:x
2406:(
2400:=
2397:)
2394:z
2391:(
2388:f
2378:)
2376:z
2374:(
2372:f
2366:v
2360:u
2345:.
2340:x
2336:)
2330:x
2326:u
2322:(
2316:=
2311:x
2307:)
2301:y
2297:v
2293:(
2287:=
2282:y
2278:)
2272:x
2268:v
2261:(
2258:=
2253:y
2250:y
2246:u
2235:x
2230:u
2223:x
2221:u
2206:.
2201:y
2197:u
2190:=
2185:x
2181:v
2176:,
2171:y
2167:v
2163:=
2158:x
2154:u
2138:v
2132:u
2127:)
2125:z
2123:(
2121:f
2106:,
2103:)
2100:y
2097:,
2094:x
2091:(
2088:v
2085:i
2082:+
2079:)
2076:y
2073:,
2070:x
2067:(
2064:u
2061:=
2058:)
2055:z
2052:(
2049:f
2034:x
2030:z
2001:=
1996:y
1993:y
1985:+
1980:x
1977:x
1961:2
1957:y
1943:2
1932:+
1924:2
1920:x
1906:2
1863:φ
1857:D
1848:D
1843:φ
1827:D
1822:φ
1817:D
1812:φ
1794:)
1792:θ
1788:R
1786:(
1784:u
1777:r
1775:(
1773:u
1766:R
1759:r
1720:g
1705:)
1702:}
1697:j
1694:i
1690:g
1686:{
1680:=
1677:g
1674:(
1670:,
1667:0
1664:=
1660:)
1651:j
1638:f
1627:j
1624:i
1620:g
1613:|
1609:g
1605:|
1598:(
1588:i
1567:|
1563:g
1559:|
1554:1
1549:=
1546:f
1541:2
1516:,
1513:0
1510:=
1505:n
1500:n
1497:m
1487:m
1484:j
1480:g
1471:j
1458:f
1449:+
1445:)
1439:j
1436:k
1432:g
1423:k
1410:f
1400:(
1391:j
1374:=
1371:f
1366:2
1327:=
1319:2
1306:f
1301:2
1279:2
1269:2
1265:r
1260:1
1255:+
1251:)
1236:f
1217:(
1184:2
1180:r
1175:1
1170:+
1166:)
1159:r
1151:f
1140:2
1136:r
1131:(
1124:r
1108:2
1104:r
1100:1
1095:=
1092:f
1087:2
1062:)
1056:,
1050:,
1047:r
1044:(
1013:=
1005:2
1001:z
992:f
987:2
976:+
968:2
955:f
950:2
935:2
931:r
927:1
922:+
918:)
911:r
903:f
894:r
890:(
883:r
869:r
866:1
861:=
858:f
853:2
817:=
809:2
805:z
796:f
791:2
780:+
772:2
768:y
759:f
754:2
743:+
735:2
731:x
722:f
717:2
706:=
703:f
698:2
684:,
619:h
616:=
613:f
590:)
587:z
584:,
581:y
578:,
575:x
572:(
569:h
546:)
543:z
540:,
537:y
534:,
531:x
528:(
525:f
448:2
440:=
428:=
405:,
402:0
399:=
396:f
373:0
370:=
367:f
361:2
317:e
310:t
303:v
34:.
20:)
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