1720:
1325:
200:
34:
1715:{\displaystyle \operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\text{if }}x>0,\\\arctan \left({\frac {y}{x}}\right)+\pi &{\text{if }}x<0{\text{ and }}y\geq 0,\\\arctan \left({\frac {y}{x}}\right)-\pi &{\text{if }}x<0{\text{ and }}y<0,\\+{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y>0,\\-{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y<0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}}
2960:
729:
2524:
2014:
2684:
2275:
1754:
2955:{\displaystyle {\begin{aligned}\operatorname {Arg} (z_{1}z_{2})&\equiv \operatorname {Arg} (z_{1})+\operatorname {Arg} (z_{2}){\pmod {\mathbb {R} /2\pi \mathbb {Z} }},\\\operatorname {Arg} \left({\frac {z_{1}}{z_{2}}}\right)&\equiv \operatorname {Arg} (z_{1})-\operatorname {Arg} (z_{2}){\pmod {\mathbb {R} /2\pi \mathbb {Z} }}.\end{aligned}}}
2519:{\displaystyle {\begin{cases}{\frac {1\pi }{4}}&{\text{if }}x=\infty {\text{ and }}y=\infty ,\\-{\frac {1\pi }{4}}&{\text{if }}x=\infty {\text{ and }}y=-\infty ,\\{\frac {3\pi }{4}}&{\text{if }}x=-\infty {\text{ and }}y=\infty ,\\-{\frac {3\pi }{4}}&{\text{if }}x=-\infty {\text{ and }}y=-\infty .\end{cases}}}
2009:{\displaystyle ={\begin{cases}{\text{undefined}}&{\text{if }}|x|=\infty {\text{ and }}|y|=\infty ,\\0&{\text{if }}x=0{\text{ and }}y=0,\\0&{\text{if }}x=\infty ,\\\pi &{\text{if }}x=-\infty ,\\\pm {\frac {\pi }{2}}&{\text{if }}y=\pm \infty ,\\\operatorname {Arg} (x+yi)&{\text{otherwise}}.\end{cases}}}
3242:
3076:
2115:
487:
Under both definitions, it can be seen that the argument of any non-zero complex number has many possible values: firstly, as a geometrical angle, it is clear that whole circle rotations do not change the point, so angles differing by an integer multiple of
3089:
3476:
616:
1019:
3659:
1113:
387:
3377:
2633:
1198:
2982:
2689:
2030:
3311:
2158:
466:
3237:{\displaystyle \operatorname {Arg} {\biggl (}{\frac {-1-i}{i}}{\biggr )}=\operatorname {Arg} (-1-i)-\operatorname {Arg} (i)=-{\frac {3\pi }{4}}-{\frac {\pi }{2}}=-{\frac {5\pi }{4}}}
3524:
2249:
3382:
1297:
1265:
835:
526:
769:
312:
281:
221:
138:
71:
3556:
718:
3581:
3576:
2221:
1233:
686:
2558:
656:
3709:
939:
1123:
function is available in the math libraries of many programming languages, sometimes under a different name, and usually returns a value in the range
317:
1052:
1766:
1386:
837:, where a vertical line (not shown in the figure) cuts the surface at heights representing all the possible choices of angle for that point.
3478:. As we are taking the imaginary part, any normalisation by a real scalar will not affect the result. This is useful when one has the
1305:
is not positive is relatively involved. Specifically, one may define the principal value of the argument separately on the half-plane
892:, excluding β180Β° itself). This represents an angle of up to half a complete circle from the positive real axis in either direction.
3316:
2582:
751:
Because a complete rotation around the origin leaves a complex number unchanged, there are many choices which could be made for
3071:{\displaystyle \operatorname {Arg} \left(z^{n}\right)\equiv n\operatorname {Arg} (z){\pmod {\mathbb {R} /2\pi \mathbb {Z} }}.}
1133:
747:. The red line here is the branch cut and corresponds to the two red lines in figure 4 seen vertically above each other).
418:
3869:
2284:
1035:
If a complex number is known in terms of its real and imaginary parts, then the function that calculates the principal value
2110:{\displaystyle ={\begin{cases}0&{\text{if }}x=0{\text{ and }}y=0,\\{\text{ArcTan}}&{\text{otherwise}}.\end{cases}}}
163:
of this function is single-valued, typically chosen to be the unique value of the argument that lies within the interval
3255:
3911:
3888:
3850:
631:
2125:
2042:
3493:
3471:{\displaystyle \operatorname {Arg} (z)=\operatorname {Im} (\ln {\frac {z}{|z|}})=\operatorname {Im} (\ln z)}
2234:
918:, especially when a general version of the argument is also being considered. Note that notation varies, so
3934:
1270:
1238:
109:
688:
isn't necessary for convergence to the correct value, but it does speed up convergence and ensures that
3960:
3812:
3770:
778:
3955:
144:
axis is drawn pointing upward, and complex numbers with positive real part are considered to have an
27:
771:
by circling the origin any number of times. This is shown in figure 2, a representation of the
2177:
473:
140:
in Figure 1. By convention the positive real axis is drawn pointing rightward, the positive
511:, the second definition also has this property. The argument of zero is usually left undefined.
851:
20:
2572:
is to be able to write complex numbers in modulus-argument form. Hence for any complex number
754:
297:
266:
206:
123:
56:
3950:
3843:
Complex
Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable
3529:
691:
495:(a complete circle) are the same, as reflected by figure 2 on the right. Similarly, from the
520:
152:
3561:
2206:
895:
Some authors define the range of the principal value as being in the closed-open interval
8:
1210:
661:
635:
3928:
3798:
3784:
2654:
2543:
2169:
641:
394:
199:
3746:
3907:
3884:
3865:
3846:
3685:
3479:
496:
33:
611:{\displaystyle \arg(z)=\lim _{n\to \infty }n\cdot \operatorname {Im} {\sqrt{z/|z|}}}
2188:
141:
82:
46:
1014:{\displaystyle \arg(z)=\{\operatorname {Arg} (z)+2\pi n\mid n\in \mathbb {Z} \}.}
846:
618:
This definition removes reliance on other difficult-to-compute functions such as
160:
3654:{\displaystyle {\overline {\arg }}(z)=\arg(z)+2k\pi ,\forall k\in \mathbb {Z} }
889:
403:
261:
156:
42:
38:
3944:
3903:
480:
257:
145:
117:
841:
772:
929:
The set of all possible values of the argument can be written in terms of
728:
106:
78:
1108:{\displaystyle \operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)}
911:
The principal value sometimes has the initial letter capitalized, as in
191:, but in some sources the capitalization of these symbols is exchanged.
619:
1732:
623:
627:
519:
The complex argument can also be defined algebraically in terms of
3881:
Complex
Analysis: The Argument Principle in Analysis and Topology
382:{\displaystyle z=r(\cos \varphi +i\sin \varphi )=re^{i\varphi }}
2259:
878:
506:
492:
288:
3372:{\displaystyle i\operatorname {Arg} (z)=\ln {\frac {z}{|z|}}}
2568:
One of the main motivations for defining the principal value
1726:
1040:
1030:
1024:
102:
175:. In this article the multi-valued function will be denoted
151:
When any real-valued angle is considered, the argument is a
2628:{\displaystyle z=\left|z\right|e^{i\operatorname {Arg} z}.}
2512:
2103:
2002:
1708:
500:
844:
function is required, then the usual choice, known as the
1193:{\displaystyle \operatorname {Arg} (x+iy)=\arctan(y/x),}
283:
from the positive real axis to the vector representing
1275:
1246:
3584:
3564:
3532:
3496:
3385:
3319:
3258:
3092:
2985:
2687:
2585:
2546:
2278:
2237:
2209:
2128:
2033:
1757:
1328:
1273:
1241:
1213:
1136:
1055:
942:
781:
757:
694:
664:
644:
529:
484:, for the argument, are sometimes used equivalently.
421:
320:
300:
269:
209:
126:
59:
1729:
for further detail and alternative implementations.
3653:
3570:
3550:
3518:
3470:
3371:
3305:
3236:
3070:
2954:
2627:
2552:
2518:
2243:
2215:
2152:
2109:
2008:
1733:Realizations of the function in computer languages
1714:
1291:
1259:
1227:
1192:
1107:
1013:
829:
763:
712:
680:
650:
610:
460:
381:
306:
275:
215:
132:
65:
3306:{\displaystyle z=|z|e^{i\operatorname {Arg} (z)}}
3129:
3101:
1737:
3942:
3490:The extended argument of a number z (denoted as
549:
3897:
2528:Unlike in Maple and Wolfram language, MATLAB's
291:, and is positive if measured counterclockwise.
3902:. Collins Dictionary (2nd ed.). Glasgow:
3898:Borowski, Ephraim; Borwein, Jonathan (2002) .
3845:(3rd ed.). New York;London: McGraw-Hill.
3799:"Four-quadrant inverse tangent - MATLAB atan2"
3526:) is the set of all real numbers congruent to
3247:
2642:is non-zero, but can be considered valid for
626:definition. Because it's defined in terms of
287:. The numeric value is given by the angle in
1005:
961:
2153:{\displaystyle \operatorname {atan2} (y,x)}
1319:, and then patch the definitions together:
1235:is well-defined and the angle lies between
1130:In some sources the argument is defined as
3864:(2nd ed.). New Delhi;Mumbai: Narosa.
3859:
1025:Computing from the real and imaginary part
3647:
3057:
3041:
2937:
2921:
2804:
2788:
1371:
1299:Extending this definition to cases where
1098:
1001:
514:
3813:"Algebraic Structure of Complex Numbers"
926:may be interchanged in different texts.
727:
622:as well as eliminating the need for the
461:{\displaystyle r={\sqrt {x^{2}+y^{2}}}.}
198:
183:and its principal value will be denoted
53:is the function which returns the angle
32:
3878:
3840:
2678:are two non-zero complex numbers, then
2270:in Wolfram language, except that it is
203:Figure 2. Two choices for the argument
16:Angle of complex number about real axis
3943:
3519:{\displaystyle {\overline {\arg }}(z)}
2244:{\displaystyle \operatorname {atan2} }
888:rad itself (equiv., from β180 to +180
3683:
252:, is defined in two equivalent ways:
3726:
3679:
3677:
3675:
3673:
3485:
2227:is the special floating-point value
294:Algebraically, as any real quantity
112:and the line joining the origin and
3039:
2919:
2786:
2660:Some further identities follow. If
2176:doesn't return anything (i.e. it's
1292:{\displaystyle {\tfrac {\pi }{2}}.}
1260:{\displaystyle -{\tfrac {\pi }{2}}}
13:
3732:Dictionary of Mathematics (2002).
3637:
2503:
2486:
2442:
2428:
2387:
2370:
2329:
2315:
2160:extended to work with infinities.
1955:
1916:
1887:
1821:
1797:
1200:however this is correct only when
1041:two-argument arctangent function,
850:, is the value in the open-closed
723:
559:
14:
3972:
3922:
3670:
2657:βrather than as being undefined.
2199:in Wolfram language, except that
830:{\displaystyle f(x,y)=\arg(x+iy)}
634:as its own principal branch. The
116:, represented as a point in the
3862:Foundations of Complex Analysis
3833:
3032:
2912:
2779:
632:principal branch of square root
49:. For each point on the plane,
3805:
3791:
3777:
3763:
3751:Wolfram Language Documentation
3739:
3702:
3619:
3613:
3601:
3595:
3545:
3539:
3513:
3507:
3465:
3453:
3441:
3434:
3426:
3410:
3398:
3392:
3362:
3354:
3335:
3329:
3298:
3292:
3274:
3266:
3176:
3170:
3158:
3143:
3061:
3033:
3028:
3022:
2941:
2913:
2908:
2895:
2883:
2870:
2808:
2780:
2775:
2762:
2750:
2737:
2721:
2698:
2147:
2135:
1986:
1971:
1814:
1806:
1790:
1782:
1738:Wolfram language (Mathematica)
1375:
1362:
1350:
1335:
1184:
1170:
1158:
1143:
1102:
1089:
1077:
1062:
976:
970:
955:
949:
824:
809:
797:
785:
732:Figure 3. The principal value
707:
701:
674:
666:
596:
588:
556:
542:
536:
357:
330:
1:
3663:
2638:This is only really valid if
2563:
1742:In Wolfram language, there's
194:
148:argument with positive sign.
3785:"Phase angle - MATLAB angle"
3590:
3502:
7:
3935:Encyclopedia of Mathematics
3248:Using the complex logarithm
2231:. Also, Maple doesn't have
1312:and the two quadrants with
906:
10:
3977:
3081:
1028:
25:
18:
2254:
155:operating on the nonzero
28:Argument (disambiguation)
2183:
2018:or using the language's
764:{\displaystyle \varphi }
307:{\displaystyle \varphi }
276:{\displaystyle \varphi }
216:{\displaystyle \varphi }
133:{\displaystyle \varphi }
66:{\displaystyle \varphi }
19:Not to be confused with
3860:Ponnuswamy, S. (2005).
3771:"Argument - Maple Help"
3551:{\displaystyle \arg(z)}
713:{\displaystyle \arg(0)}
630:, it also inherits the
407:(or absolute value) of
389:for some positive real
3879:Beardon, Alan (1979).
3841:Ahlfors, Lars (1979).
3655:
3572:
3552:
3520:
3472:
3373:
3307:
3238:
3072:
2956:
2629:
2554:
2520:
2245:
2217:
2154:
2111:
2010:
1716:
1293:
1261:
1229:
1194:
1109:
1015:
831:
775:(set-valued) function
765:
748:
714:
682:
652:
612:
515:Alternative definition
462:
383:
308:
277:
256:Geometrically, in the
230:of the complex number
223:
217:
134:
74:
67:
21:Argument of a function
3883:. Chichester: Wiley.
3690:mathworld.wolfram.com
3656:
3573:
3553:
3521:
3473:
3374:
3308:
3239:
3073:
2976:is any integer, then
2957:
2630:
2555:
2521:
2246:
2218:
2155:
2112:
2011:
1717:
1294:
1262:
1230:
1195:
1110:
1016:
832:
766:
736:of the blue point at
731:
715:
683:
653:
613:
478:for the modulus, and
463:
384:
309:
278:
218:
202:
135:
105:between the positive
68:
36:
3817:www.cut-the-knot.org
3582:
3571:{\displaystyle \pi }
3562:
3530:
3494:
3383:
3317:
3256:
3090:
2983:
2685:
2653:is considered as an
2583:
2544:
2276:
2266:behaves the same as
2235:
2216:{\displaystyle \pi }
2207:
2195:behaves the same as
2126:
2031:
1755:
1326:
1271:
1239:
1211:
1134:
1053:
940:
779:
755:
692:
662:
642:
527:
419:
318:
298:
267:
207:
153:multivalued function
124:
89:of a complex number
57:
26:For other uses, see
3684:Weisstein, Eric W.
1228:{\displaystyle y/x}
720:is left undefined.
681:{\displaystyle |z|}
3714:internal.ncl.ac.uk
3686:"Complex Argument"
3651:
3568:
3548:
3516:
3468:
3369:
3303:
3234:
3068:
2952:
2950:
2655:indeterminate form
2625:
2550:
2516:
2511:
2241:
2213:
2150:
2107:
2102:
2006:
2001:
1712:
1707:
1289:
1284:
1257:
1255:
1225:
1190:
1105:
1011:
827:
761:
749:
710:
678:
648:
608:
563:
458:
379:
304:
273:
224:
213:
130:
75:
63:
3961:Signal processing
3871:978-81-7319-629-4
3593:
3505:
3486:Extended argument
3480:complex logarithm
3439:
3367:
3232:
3211:
3198:
3125:
2851:
2553:{\displaystyle 0}
2532:is equivalent to
2492:
2475:
2468:
2434:
2417:
2410:
2376:
2362:
2355:
2321:
2307:
2300:
2095:
2088:
2067:
2053:
1994:
1944:
1937:
1905:
1879:
1853:
1839:
1803:
1779:
1772:
1694:
1680:
1673:
1652:
1638:
1631:
1602:
1588:
1581:
1552:
1538:
1521:
1485:
1471:
1454:
1418:
1407:
1283:
1254:
651:{\displaystyle z}
606:
548:
453:
81:(particularly in
3968:
3956:Complex analysis
3917:
3894:
3875:
3856:
3827:
3826:
3824:
3823:
3809:
3803:
3802:
3795:
3789:
3788:
3781:
3775:
3774:
3767:
3761:
3760:
3758:
3757:
3743:
3737:
3730:
3724:
3723:
3721:
3720:
3706:
3700:
3699:
3697:
3696:
3681:
3660:
3658:
3657:
3652:
3650:
3594:
3586:
3577:
3575:
3574:
3569:
3557:
3555:
3554:
3549:
3525:
3523:
3522:
3517:
3506:
3498:
3477:
3475:
3474:
3469:
3440:
3438:
3437:
3429:
3420:
3379:, alternatively
3378:
3376:
3375:
3370:
3368:
3366:
3365:
3357:
3348:
3312:
3310:
3309:
3304:
3302:
3301:
3277:
3269:
3243:
3241:
3240:
3235:
3233:
3228:
3220:
3212:
3204:
3199:
3194:
3186:
3133:
3132:
3126:
3121:
3107:
3105:
3104:
3077:
3075:
3074:
3069:
3064:
3060:
3049:
3044:
3009:
3005:
3004:
2975:
2971:
2961:
2959:
2958:
2953:
2951:
2944:
2940:
2929:
2924:
2907:
2906:
2882:
2881:
2856:
2852:
2850:
2849:
2840:
2839:
2830:
2811:
2807:
2796:
2791:
2774:
2773:
2749:
2748:
2720:
2719:
2710:
2709:
2677:
2668:
2652:
2648:
2641:
2634:
2632:
2631:
2626:
2621:
2620:
2602:
2575:
2571:
2559:
2557:
2556:
2551:
2539:
2535:
2531:
2525:
2523:
2522:
2517:
2515:
2514:
2493:
2490:
2476:
2473:
2469:
2464:
2456:
2435:
2432:
2418:
2415:
2411:
2406:
2398:
2377:
2374:
2363:
2360:
2356:
2351:
2343:
2322:
2319:
2308:
2305:
2301:
2296:
2288:
2269:
2265:
2250:
2248:
2247:
2242:
2230:
2226:
2222:
2220:
2219:
2214:
2202:
2198:
2194:
2175:
2172:defined), while
2167:
2163:
2159:
2157:
2156:
2151:
2121:
2116:
2114:
2113:
2108:
2106:
2105:
2096:
2093:
2089:
2086:
2068:
2065:
2054:
2051:
2027:
2021:
2015:
2013:
2012:
2007:
2005:
2004:
1995:
1992:
1945:
1942:
1938:
1930:
1906:
1903:
1880:
1877:
1854:
1851:
1840:
1837:
1817:
1809:
1804:
1801:
1793:
1785:
1780:
1777:
1773:
1770:
1751:
1745:
1721:
1719:
1718:
1713:
1711:
1710:
1695:
1692:
1681:
1678:
1674:
1671:
1653:
1650:
1639:
1636:
1632:
1624:
1603:
1600:
1589:
1586:
1582:
1574:
1553:
1550:
1539:
1536:
1526:
1522:
1514:
1486:
1483:
1472:
1469:
1459:
1455:
1447:
1419:
1416:
1412:
1408:
1400:
1318:
1311:
1304:
1298:
1296:
1295:
1290:
1285:
1276:
1266:
1264:
1263:
1258:
1256:
1247:
1234:
1232:
1231:
1226:
1221:
1206:
1199:
1197:
1196:
1191:
1180:
1126:
1122:
1114:
1112:
1111:
1106:
1044:
1038:
1020:
1018:
1017:
1012:
1004:
932:
925:
921:
917:
902:
887:
877:
871:
864:
836:
834:
833:
828:
770:
768:
767:
762:
746:
742:
735:
719:
717:
716:
711:
687:
685:
684:
679:
677:
669:
657:
655:
654:
649:
617:
615:
614:
609:
607:
605:
600:
599:
591:
586:
577:
562:
509:
503:
491:
467:
465:
464:
459:
454:
452:
451:
439:
438:
429:
414:
410:
400:
397:). The quantity
392:
388:
386:
385:
380:
378:
377:
313:
311:
310:
305:
286:
282:
280:
279:
274:
251:
243:
222:
220:
219:
214:
190:
182:
174:
139:
137:
136:
131:
115:
100:
92:
83:complex analysis
72:
70:
69:
64:
52:
3976:
3975:
3971:
3970:
3969:
3967:
3966:
3965:
3941:
3940:
3925:
3920:
3914:
3891:
3872:
3853:
3836:
3831:
3830:
3821:
3819:
3811:
3810:
3806:
3797:
3796:
3792:
3783:
3782:
3778:
3769:
3768:
3764:
3755:
3753:
3745:
3744:
3740:
3731:
3727:
3718:
3716:
3708:
3707:
3703:
3694:
3692:
3682:
3671:
3666:
3646:
3585:
3583:
3580:
3579:
3563:
3560:
3559:
3531:
3528:
3527:
3497:
3495:
3492:
3491:
3488:
3433:
3425:
3424:
3419:
3384:
3381:
3380:
3361:
3353:
3352:
3347:
3318:
3315:
3314:
3282:
3278:
3273:
3265:
3257:
3254:
3253:
3250:
3221:
3219:
3203:
3187:
3185:
3128:
3127:
3108:
3106:
3100:
3099:
3091:
3088:
3087:
3084:
3056:
3045:
3040:
3031:
3000:
2996:
2992:
2984:
2981:
2980:
2973:
2966:
2949:
2948:
2936:
2925:
2920:
2911:
2902:
2898:
2877:
2873:
2857:
2845:
2841:
2835:
2831:
2829:
2825:
2816:
2815:
2803:
2792:
2787:
2778:
2769:
2765:
2744:
2740:
2724:
2715:
2711:
2705:
2701:
2688:
2686:
2683:
2682:
2676:
2670:
2667:
2661:
2650:
2643:
2639:
2607:
2603:
2592:
2584:
2581:
2580:
2573:
2569:
2566:
2545:
2542:
2541:
2537:
2534:angle(x + y*1i)
2533:
2529:
2510:
2509:
2491: and
2489:
2472:
2470:
2457:
2455:
2449:
2448:
2433: and
2431:
2414:
2412:
2399:
2397:
2394:
2393:
2375: and
2373:
2359:
2357:
2344:
2342:
2336:
2335:
2320: and
2318:
2304:
2302:
2289:
2287:
2280:
2279:
2277:
2274:
2273:
2267:
2263:
2257:
2236:
2233:
2232:
2228:
2224:
2208:
2205:
2204:
2200:
2196:
2192:
2186:
2173:
2165:
2161:
2127:
2124:
2123:
2119:
2101:
2100:
2092:
2090:
2085:
2082:
2081:
2066: and
2064:
2050:
2048:
2038:
2037:
2032:
2029:
2028:
2025:
2019:
2000:
1999:
1991:
1989:
1962:
1961:
1941:
1939:
1929:
1923:
1922:
1902:
1900:
1894:
1893:
1876:
1874:
1868:
1867:
1852: and
1850:
1836:
1834:
1828:
1827:
1813:
1805:
1802: and
1800:
1789:
1781:
1776:
1774:
1769:
1762:
1761:
1756:
1753:
1752:
1749:
1743:
1740:
1735:
1706:
1705:
1693: and
1691:
1677:
1675:
1670:
1667:
1666:
1651: and
1649:
1635:
1633:
1623:
1617:
1616:
1601: and
1599:
1585:
1583:
1573:
1567:
1566:
1551: and
1549:
1535:
1533:
1513:
1509:
1500:
1499:
1484: and
1482:
1468:
1466:
1446:
1442:
1433:
1432:
1415:
1413:
1399:
1395:
1382:
1381:
1327:
1324:
1323:
1313:
1306:
1300:
1274:
1272:
1269:
1268:
1245:
1240:
1237:
1236:
1217:
1212:
1209:
1208:
1201:
1176:
1135:
1132:
1131:
1124:
1120:
1054:
1051:
1050:
1042:
1036:
1033:
1027:
1000:
941:
938:
937:
930:
923:
919:
912:
909:
896:
882:
873:
866:
865:, that is from
854:
847:principal value
780:
777:
776:
756:
753:
752:
744:
737:
733:
726:
724:Principal value
693:
690:
689:
673:
665:
663:
660:
659:
658:by dividing by
643:
640:
639:
601:
595:
587:
582:
578:
576:
552:
528:
525:
524:
517:
507:
501:
489:
447:
443:
434:
430:
428:
420:
417:
416:
412:
408:
398:
395:Euler's formula
390:
370:
366:
319:
316:
315:
299:
296:
295:
284:
268:
265:
264:
245:
231:
208:
205:
204:
197:
184:
176:
164:
161:principal value
157:complex numbers
125:
122:
121:
113:
94:
90:
58:
55:
54:
50:
41:represents the
37:Figure 1. This
31:
24:
17:
12:
11:
5:
3974:
3964:
3963:
3958:
3953:
3939:
3938:
3924:
3923:External links
3921:
3919:
3918:
3912:
3895:
3889:
3876:
3870:
3857:
3851:
3837:
3835:
3832:
3829:
3828:
3804:
3790:
3776:
3762:
3738:
3725:
3701:
3668:
3667:
3665:
3662:
3649:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3592:
3589:
3567:
3547:
3544:
3541:
3538:
3535:
3515:
3512:
3509:
3504:
3501:
3487:
3484:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3436:
3432:
3428:
3423:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3364:
3360:
3356:
3351:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3300:
3297:
3294:
3291:
3288:
3285:
3281:
3276:
3272:
3268:
3264:
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3249:
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3245:
3244:
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3227:
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3215:
3210:
3207:
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3145:
3142:
3139:
3136:
3131:
3124:
3120:
3117:
3114:
3111:
3103:
3098:
3095:
3083:
3080:
3079:
3078:
3067:
3063:
3059:
3055:
3052:
3048:
3043:
3038:
3035:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3008:
3003:
2999:
2995:
2991:
2988:
2963:
2962:
2947:
2943:
2939:
2935:
2932:
2928:
2923:
2918:
2915:
2910:
2905:
2901:
2897:
2894:
2891:
2888:
2885:
2880:
2876:
2872:
2869:
2866:
2863:
2860:
2858:
2855:
2848:
2844:
2838:
2834:
2828:
2824:
2821:
2818:
2817:
2814:
2810:
2806:
2802:
2799:
2795:
2790:
2785:
2782:
2777:
2772:
2768:
2764:
2761:
2758:
2755:
2752:
2747:
2743:
2739:
2736:
2733:
2730:
2727:
2725:
2723:
2718:
2714:
2708:
2704:
2700:
2697:
2694:
2691:
2690:
2674:
2665:
2636:
2635:
2624:
2619:
2616:
2613:
2610:
2606:
2601:
2598:
2595:
2591:
2588:
2565:
2562:
2549:
2513:
2508:
2505:
2502:
2499:
2496:
2488:
2485:
2482:
2479:
2471:
2467:
2463:
2460:
2454:
2451:
2450:
2447:
2444:
2441:
2438:
2430:
2427:
2424:
2421:
2413:
2409:
2405:
2402:
2396:
2395:
2392:
2389:
2386:
2383:
2380:
2372:
2369:
2366:
2358:
2354:
2350:
2347:
2341:
2338:
2337:
2334:
2331:
2328:
2325:
2317:
2314:
2311:
2303:
2299:
2295:
2292:
2286:
2285:
2283:
2256:
2253:
2240:
2212:
2185:
2182:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2104:
2099:
2091:
2084:
2083:
2080:
2077:
2074:
2071:
2063:
2060:
2057:
2049:
2047:
2044:
2043:
2041:
2036:
2003:
1998:
1990:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1963:
1960:
1957:
1954:
1951:
1948:
1940:
1936:
1933:
1928:
1925:
1924:
1921:
1918:
1915:
1912:
1909:
1901:
1899:
1896:
1895:
1892:
1889:
1886:
1883:
1875:
1873:
1870:
1869:
1866:
1863:
1860:
1857:
1849:
1846:
1843:
1835:
1833:
1830:
1829:
1826:
1823:
1820:
1816:
1812:
1808:
1799:
1796:
1792:
1788:
1784:
1775:
1768:
1767:
1765:
1760:
1739:
1736:
1734:
1731:
1723:
1722:
1709:
1704:
1701:
1698:
1690:
1687:
1684:
1676:
1669:
1668:
1665:
1662:
1659:
1656:
1648:
1645:
1642:
1634:
1630:
1627:
1622:
1619:
1618:
1615:
1612:
1609:
1606:
1598:
1595:
1592:
1584:
1580:
1577:
1572:
1569:
1568:
1565:
1562:
1559:
1556:
1548:
1545:
1542:
1534:
1532:
1529:
1525:
1520:
1517:
1512:
1508:
1505:
1502:
1501:
1498:
1495:
1492:
1489:
1481:
1478:
1475:
1467:
1465:
1462:
1458:
1453:
1450:
1445:
1441:
1438:
1435:
1434:
1431:
1428:
1425:
1422:
1414:
1411:
1406:
1403:
1398:
1394:
1391:
1388:
1387:
1385:
1380:
1377:
1374:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1288:
1282:
1279:
1253:
1250:
1244:
1224:
1220:
1216:
1189:
1186:
1183:
1179:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1117:
1116:
1104:
1101:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1039:is called the
1029:Main article:
1026:
1023:
1022:
1021:
1010:
1007:
1003:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
908:
905:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
760:
725:
722:
709:
706:
703:
700:
697:
676:
672:
668:
647:
604:
598:
594:
590:
585:
581:
575:
572:
569:
566:
561:
558:
555:
551:
547:
544:
541:
538:
535:
532:
516:
513:
469:
468:
457:
450:
446:
442:
437:
433:
427:
424:
376:
373:
369:
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
323:
303:
292:
272:
262:2D polar angle
212:
196:
193:
129:
62:
43:complex number
39:Argand diagram
15:
9:
6:
4:
3:
2:
3973:
3962:
3959:
3957:
3954:
3952:
3949:
3948:
3946:
3936:
3932:
3931:
3927:
3926:
3915:
3913:0-00-710295-X
3909:
3905:
3904:HarperCollins
3901:
3896:
3892:
3890:0-471-99671-8
3886:
3882:
3877:
3873:
3867:
3863:
3858:
3854:
3852:0-07-000657-1
3848:
3844:
3839:
3838:
3818:
3814:
3808:
3800:
3794:
3786:
3780:
3772:
3766:
3752:
3748:
3742:
3735:
3729:
3715:
3711:
3705:
3691:
3687:
3680:
3678:
3676:
3674:
3669:
3661:
3643:
3640:
3634:
3631:
3628:
3625:
3622:
3616:
3610:
3607:
3604:
3598:
3587:
3565:
3542:
3536:
3533:
3510:
3499:
3483:
3481:
3462:
3459:
3456:
3450:
3447:
3444:
3430:
3421:
3416:
3413:
3407:
3404:
3401:
3395:
3389:
3386:
3358:
3349:
3344:
3341:
3338:
3332:
3326:
3323:
3320:
3295:
3289:
3286:
3283:
3279:
3270:
3262:
3259:
3229:
3225:
3222:
3216:
3213:
3208:
3205:
3200:
3195:
3191:
3188:
3182:
3179:
3173:
3167:
3164:
3161:
3155:
3152:
3149:
3146:
3140:
3137:
3134:
3122:
3118:
3115:
3112:
3109:
3096:
3093:
3086:
3085:
3065:
3053:
3050:
3046:
3036:
3025:
3019:
3016:
3013:
3010:
3006:
3001:
2997:
2993:
2989:
2986:
2979:
2978:
2977:
2969:
2945:
2933:
2930:
2926:
2916:
2903:
2899:
2892:
2889:
2886:
2878:
2874:
2867:
2864:
2861:
2859:
2853:
2846:
2842:
2836:
2832:
2826:
2822:
2819:
2812:
2800:
2797:
2793:
2783:
2770:
2766:
2759:
2756:
2753:
2745:
2741:
2734:
2731:
2728:
2726:
2716:
2712:
2706:
2702:
2695:
2692:
2681:
2680:
2679:
2673:
2664:
2658:
2656:
2646:
2622:
2617:
2614:
2611:
2608:
2604:
2599:
2596:
2593:
2589:
2586:
2579:
2578:
2577:
2561:
2547:
2526:
2506:
2500:
2497:
2494:
2483:
2480:
2477:
2465:
2461:
2458:
2452:
2445:
2439:
2436:
2425:
2422:
2419:
2407:
2403:
2400:
2390:
2384:
2381:
2378:
2367:
2364:
2352:
2348:
2345:
2339:
2332:
2326:
2323:
2312:
2309:
2297:
2293:
2290:
2281:
2271:
2261:
2252:
2238:
2210:
2203:also returns
2190:
2181:
2179:
2171:
2166:Indeterminate
2144:
2141:
2138:
2132:
2129:
2117:
2097:
2078:
2075:
2072:
2069:
2061:
2058:
2055:
2045:
2039:
2034:
2023:
2016:
1996:
1983:
1980:
1977:
1974:
1968:
1965:
1958:
1952:
1949:
1946:
1934:
1931:
1926:
1919:
1913:
1910:
1907:
1897:
1890:
1884:
1881:
1871:
1864:
1861:
1858:
1855:
1847:
1844:
1841:
1831:
1824:
1818:
1810:
1794:
1786:
1763:
1758:
1747:
1730:
1728:
1702:
1699:
1696:
1688:
1685:
1682:
1663:
1660:
1657:
1654:
1646:
1643:
1640:
1628:
1625:
1620:
1613:
1610:
1607:
1604:
1596:
1593:
1590:
1578:
1575:
1570:
1563:
1560:
1557:
1554:
1546:
1543:
1540:
1530:
1527:
1523:
1518:
1515:
1510:
1506:
1503:
1496:
1493:
1490:
1487:
1479:
1476:
1473:
1463:
1460:
1456:
1451:
1448:
1443:
1439:
1436:
1429:
1426:
1423:
1420:
1409:
1404:
1401:
1396:
1392:
1389:
1383:
1378:
1372:
1368:
1365:
1359:
1356:
1353:
1347:
1344:
1341:
1338:
1332:
1329:
1322:
1321:
1320:
1316:
1309:
1303:
1286:
1280:
1277:
1251:
1248:
1242:
1222:
1218:
1214:
1204:
1187:
1181:
1177:
1173:
1167:
1164:
1161:
1155:
1152:
1149:
1146:
1140:
1137:
1128:
1099:
1095:
1092:
1086:
1083:
1080:
1074:
1071:
1068:
1065:
1059:
1056:
1049:
1048:
1047:
1045:
1032:
1008:
997:
994:
991:
988:
985:
982:
979:
973:
967:
964:
958:
952:
946:
943:
936:
935:
934:
927:
916:
904:
900:
893:
891:
886:
880:
876:
870:
862:
858:
853:
849:
848:
843:
838:
821:
818:
815:
812:
806:
803:
800:
794:
791:
788:
782:
774:
758:
741:
730:
721:
704:
698:
695:
670:
645:
637:
636:normalization
633:
629:
625:
621:
602:
592:
583:
579:
573:
570:
567:
564:
553:
545:
539:
533:
530:
522:
521:complex roots
512:
510:
504:
498:
494:
485:
483:
482:
477:
475:
455:
448:
444:
440:
435:
431:
425:
422:
406:
405:
396:
374:
371:
367:
363:
360:
354:
351:
348:
345:
342:
339:
336:
333:
327:
324:
321:
301:
293:
290:
270:
263:
259:
258:complex plane
255:
254:
253:
249:
242:
238:
234:
229:
210:
201:
192:
188:
180:
172:
168:
162:
158:
154:
149:
147:
146:anticlockwise
143:
127:
119:
118:complex plane
111:
108:
104:
98:
88:
84:
80:
60:
48:
44:
40:
35:
29:
22:
3951:Trigonometry
3929:
3899:
3880:
3861:
3842:
3834:Bibliography
3820:. Retrieved
3816:
3807:
3793:
3779:
3765:
3754:. Retrieved
3750:
3741:
3733:
3728:
3717:. Retrieved
3713:
3710:"Pure Maths"
3704:
3693:. Retrieved
3689:
3489:
3251:
2967:
2964:
2671:
2662:
2659:
2644:
2637:
2567:
2527:
2272:
2258:
2187:
2118:
2024:
2017:
1748:
1741:
1724:
1314:
1307:
1301:
1202:
1129:
1118:
1034:
928:
914:
910:
898:
894:
884:
881:, excluding
874:
868:
860:
856:
845:
842:well-defined
839:
773:multi-valued
750:
739:
518:
486:
479:
472:
470:
402:
247:
240:
236:
232:
227:
225:
186:
178:
170:
166:
150:
96:
86:
76:
3900:Mathematics
3482:available.
2538:atan2(0, 0)
2536:. That is,
2530:atan2(y, x)
2201:argument(z)
2193:argument(z)
2168:(i.e. it's
1125:(βΟ, Ο]
497:periodicity
411:, denoted |
120:, shown as
79:mathematics
45:lying on a
3945:Categories
3822:2021-08-29
3756:2024-08-30
3719:2020-08-31
3695:2020-08-31
3664:References
2564:Identities
620:arctangent
471:The names
314:such that
244:, denoted
195:Definition
93:, denoted
3644:∈
3638:∀
3632:π
3611:
3591:¯
3566:π
3537:
3503:¯
3460:
3451:
3417:
3408:
3390:
3345:
3327:
3313:, we get
3290:
3226:π
3217:−
3206:π
3201:−
3192:π
3183:−
3168:
3162:−
3153:−
3147:−
3141:
3116:−
3110:−
3097:
3054:π
3020:
3011:≡
2990:
2934:π
2893:
2887:−
2868:
2862:≡
2823:
2801:π
2760:
2735:
2729:≡
2696:
2615:
2504:∞
2501:−
2487:∞
2484:−
2462:π
2453:−
2443:∞
2429:∞
2426:−
2404:π
2388:∞
2385:−
2371:∞
2349:π
2340:−
2330:∞
2316:∞
2294:π
2211:π
2178:undefined
2133:
2094:otherwise
1993:otherwise
1969:
1956:∞
1953:±
1932:π
1927:±
1917:∞
1914:−
1898:π
1888:∞
1822:∞
1798:∞
1771:undefined
1672:undefined
1626:π
1621:−
1576:π
1531:π
1528:−
1507:
1491:≥
1464:π
1440:
1393:
1360:
1333:
1278:π
1249:π
1243:−
1168:
1141:
1087:
1060:
998:∈
992:∣
986:π
968:
947:
897:[0, 2
807:
759:φ
699:
624:piecewise
574:
568:⋅
560:∞
557:→
534:
474:magnitude
375:φ
355:φ
352:
340:φ
337:
302:φ
271:φ
260:, as the
211:φ
142:imaginary
128:φ
101:, is the
61:φ
3930:Argument
3558:modulo 2
2474:if
2416:if
2361:if
2306:if
2264:angle(z)
2052:if
1943:if
1904:if
1878:if
1838:if
1778:if
1679:if
1637:if
1587:if
1537:if
1470:if
1417:if
1207:, where
907:Notation
863:rad]
852:interval
228:argument
87:argument
3082:Example
890:degrees
879:radians
840:When a
493:radians
404:modulus
401:is the
289:radians
85:), the
3910:
3887:
3868:
3849:
2651:Arg(0)
2260:MATLAB
2255:MATLAB
2174:ArcTan
2162:ArcTan
2120:ArcTan
2087:ArcTan
2020:ArcTan
1504:arctan
1437:arctan
1390:arctan
1317:< 0
1310:> 0
1205:> 0
1165:arctan
159:. The
3747:"Arg"
3734:phase
3252:From
2239:atan2
2189:Maple
2184:Maple
2170:still
2130:atan2
1727:atan2
1357:atan2
1121:atan2
1084:atan2
1043:atan2
1031:atan2
859:rad,
628:roots
481:phase
393:(see
173:]
103:angle
47:plane
3908:ISBN
3885:ISBN
3866:ISBN
3847:ISBN
2972:and
2669:and
1725:See
1658:<
1608:>
1558:<
1544:<
1477:<
1424:>
1267:and
1119:The
933:as:
922:and
913:Arg
738:1 +
523:as:
505:and
246:arg(
185:Arg(
177:arg(
110:axis
107:real
95:arg(
3933:at
3608:arg
3588:arg
3534:arg
3500:arg
3387:Arg
3324:Arg
3287:Arg
3165:Arg
3138:Arg
3094:Arg
3037:mod
3017:Arg
2987:Arg
2970:β 0
2965:If
2917:mod
2890:Arg
2865:Arg
2820:Arg
2784:mod
2757:Arg
2732:Arg
2693:Arg
2649:if
2647:= 0
2612:Arg
2570:Arg
2540:is
2268:Arg
2262:'s
2229:β0.
2223:if
2197:Arg
2191:'s
2180:).
2164:is
2122:is
2026:Arg
1966:Arg
1750:Arg
1744:Arg
1330:Arg
1138:Arg
1057:Arg
1037:Arg
965:Arg
944:arg
931:Arg
924:Arg
920:arg
872:to
804:arg
745:Ο/4
743:is
734:Arg
696:arg
638:of
550:lim
531:arg
508:cos
502:sin
499:of
415:|:
349:sin
334:cos
226:An
77:In
51:arg
3947::
3906:.
3815:.
3749:.
3712:.
3688:.
3672:^
3457:ln
3448:Im
3414:ln
3405:Im
3342:ln
2576:,
2560:.
2251:.
2022::
1746::
1703:0.
1127:.
1046::
903:.
855:(β
571:Im
490:2Ο
241:iy
239:+
235:=
169:,
165:(β
3937:.
3916:.
3893:.
3874:.
3855:.
3825:.
3801:.
3787:.
3773:.
3759:.
3736:.
3722:.
3698:.
3648:Z
3641:k
3635:,
3629:k
3626:2
3623:+
3620:)
3617:z
3614:(
3605:=
3602:)
3599:z
3596:(
3578:.
3546:)
3543:z
3540:(
3514:)
3511:z
3508:(
3466:)
3463:z
3454:(
3445:=
3442:)
3435:|
3431:z
3427:|
3422:z
3411:(
3402:=
3399:)
3396:z
3393:(
3363:|
3359:z
3355:|
3350:z
3339:=
3336:)
3333:z
3330:(
3321:i
3299:)
3296:z
3293:(
3284:i
3280:e
3275:|
3271:z
3267:|
3263:=
3260:z
3230:4
3223:5
3214:=
3209:2
3196:4
3189:3
3180:=
3177:)
3174:i
3171:(
3159:)
3156:i
3150:1
3144:(
3135:=
3130:)
3123:i
3119:i
3113:1
3102:(
3066:.
3062:)
3058:Z
3051:2
3047:/
3042:R
3034:(
3029:)
3026:z
3023:(
3014:n
3007:)
3002:n
2998:z
2994:(
2974:n
2968:z
2946:.
2942:)
2938:Z
2931:2
2927:/
2922:R
2914:(
2909:)
2904:2
2900:z
2896:(
2884:)
2879:1
2875:z
2871:(
2854:)
2847:2
2843:z
2837:1
2833:z
2827:(
2813:,
2809:)
2805:Z
2798:2
2794:/
2789:R
2781:(
2776:)
2771:2
2767:z
2763:(
2754:+
2751:)
2746:1
2742:z
2738:(
2722:)
2717:2
2713:z
2707:1
2703:z
2699:(
2675:2
2672:z
2666:1
2663:z
2645:z
2640:z
2623:.
2618:z
2609:i
2605:e
2600:|
2597:z
2594:|
2590:=
2587:z
2574:z
2548:0
2507:.
2498:=
2495:y
2481:=
2478:x
2466:4
2459:3
2446:,
2440:=
2437:y
2423:=
2420:x
2408:4
2401:3
2391:,
2382:=
2379:y
2368:=
2365:x
2353:4
2346:1
2333:,
2327:=
2324:y
2313:=
2310:x
2298:4
2291:1
2282:{
2225:z
2148:)
2145:x
2142:,
2139:y
2136:(
2098:.
2079:,
2076:0
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2070:y
2062:0
2059:=
2056:x
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2040:{
2035:=
1997:.
1987:)
1984:i
1981:y
1978:+
1975:x
1972:(
1959:,
1950:=
1947:y
1935:2
1920:,
1911:=
1908:x
1891:,
1885:=
1882:x
1872:0
1865:,
1862:0
1859:=
1856:y
1848:0
1845:=
1842:x
1832:0
1825:,
1819:=
1815:|
1811:y
1807:|
1795:=
1791:|
1787:x
1783:|
1764:{
1759:=
1700:=
1697:y
1689:0
1686:=
1683:x
1664:,
1661:0
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1647:0
1644:=
1641:x
1629:2
1614:,
1611:0
1605:y
1597:0
1594:=
1591:x
1579:2
1571:+
1564:,
1561:0
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1547:0
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1524:)
1519:x
1516:y
1511:(
1497:,
1494:0
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1457:)
1452:x
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1444:(
1430:,
1427:0
1421:x
1410:)
1405:x
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1397:(
1384:{
1379:=
1376:)
1373:x
1369:,
1366:y
1363:(
1354:=
1351:)
1348:y
1345:i
1342:+
1339:x
1336:(
1315:x
1308:x
1302:x
1287:.
1281:2
1252:2
1223:x
1219:/
1215:y
1203:x
1188:,
1185:)
1182:x
1178:/
1174:y
1171:(
1162:=
1159:)
1156:y
1153:i
1150:+
1147:x
1144:(
1115:.
1103:)
1100:x
1096:,
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1090:(
1081:=
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1075:y
1072:i
1069:+
1066:x
1063:(
1009:.
1006:}
1002:Z
995:n
989:n
983:2
980:+
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974:z
971:(
962:{
959:=
956:)
953:z
950:(
915:z
901:)
899:Ο
885:Ο
883:β
875:Ο
869:Ο
867:β
861:Ο
857:Ο
825:)
822:y
819:i
816:+
813:x
810:(
801:=
798:)
795:y
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786:(
783:f
740:i
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705:0
702:(
675:|
671:z
667:|
646:z
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546:=
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540:z
537:(
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456:.
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426:=
423:r
413:z
409:z
399:r
391:r
372:i
368:e
364:r
361:=
358:)
346:i
343:+
331:(
328:r
325:=
322:z
285:z
250:)
248:z
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189:)
187:z
181:)
179:z
171:Ο
167:Ο
114:z
99:)
97:z
91:z
73:.
30:.
23:.
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