33:
1509:
2395:
3039:
1028:
2227:
2511:
1340:
1769:
2852:
2688:
1931:
667:
324:
1378:
2242:
2089:
895:
2870:
2591:
1848:
899:
516:
has less to say in general, and the solutions are harder to study, except in terms of their asymptotic expansions. The irregularity of an irregular singularity is measured by the
2096:
2413:
1217:
150:
128:
471:
1193:
1153:
1113:
1077:
1624:
1619:
822:
1957:
1798:
2707:
2598:
1853:
207:
1198:
Listed below are several examples from ordinary differential equations from mathematical physics that have singular points and known solutions.
62:
1982:
827:
17:
2528:
559:
3237:
360:
may be applied to move ∞ into the finite part of the complex plane if required, see example on Bessel differential equation below.
3203:
1804:
180:, where the full solution set requires functions with higher growth rates. This distinction occurs, for example, between the
1504:{\displaystyle {\frac {d^{2}f}{dx^{2}}}+{\frac {1}{x}}{\frac {df}{dx}}+\left(1-{\frac {\alpha ^{2}}{x^{2}}}\right)f=0.}
84:
2390:{\displaystyle {\frac {d^{2}f}{dx^{2}}}-{\frac {2x}{1-x^{2}}}{\frac {df}{dx}}+{\frac {\ell (\ell +1)}{1-x^{2}}}f=0.}
55:
2407:
One encounters this ordinary second order differential equation in solving the one-dimensional time independent
434:
3071:
539:
102:
3034:{\displaystyle {\frac {d^{2}f}{dz^{2}}}+{\frac {c-(a+b+1)z}{z(1-z)}}{\frac {df}{dz}}-{\frac {ab}{z(1-z)}}f=0.}
3195:
3165:
3119:
3155:
3220:
3160:
3114:
2514:
189:
3109:
3242:
542:
whose only singular points, including the point at infinity, are regular singular points is called a
165:
45:
3191:
49:
41:
1584:
357:
133:
111:
3044:
1211:
474:
181:
2408:
790:
We can check whether there is an irregular singular point at infinity by using the substitution
1156:
66:
1162:
1122:
1082:
1046:
3061:
1976:
1972:
1207:
1023:{\displaystyle {\frac {d^{2}f}{dx^{2}}}=w^{4}{\frac {d^{2}f}{dw^{2}}}+2w^{3}{\frac {df}{dw}}}
513:
342:
2222:{\displaystyle \left(1-x^{2}\right){d^{2}f \over dx^{2}}-2x{df \over dx}+\ell (\ell +1)f=0.}
1590:
793:
1971:
This is an ordinary differential equation of second order. It is found in the solution of
1206:
This is an ordinary differential equation of second order. It is found in the solution to
8:
3173:
3127:
2693:
1936:
1777:
1115:
are quotients of polynomials, then there will be an irregular singular point at infinite
3043:
This differential equation has regular singular points at 0, 1 and ∞. A solution is the
3083:
173:
3134:
2506:{\displaystyle E\psi =-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}+V(x)\psi }
1335:{\displaystyle x^{2}{\frac {d^{2}f}{dx^{2}}}+x{\frac {df}{dx}}+(x^{2}-\alpha ^{2})f=0}
3199:
3185:
3067:
531:
condition, in the sense that the allowed poles are in a region, when plotted against
368:
353:
157:
517:
486:
364:
3212:
1354:
544:
509:
404:
185:
2692:
This differential equation has an irregular singularity at ∞. Its solutions are
1933:
has a pole of fourth order. Thus, this equation has an irregular singularity at
1764:{\displaystyle {\frac {d^{2}f}{dw^{2}}}+{\frac {1}{w}}{\frac {df}{dw}}+\leftf=0}
1343:
528:
408:
349:
3139:
3231:
3181:
3140:
A Course in
Mathematical Analysis, Volume II, Part II: Differential Equations
3057:
420:
395:
need not be an integer; this function may exist, therefore, only thanks to a
106:
3216:
3097:
2847:{\displaystyle z(1-z){\frac {d^{2}f}{dz^{2}}}+\left{\frac {df}{dz}}-abf=0.}
372:
168:. Then amongst singular points, an important distinction is made between a
1159:
at least one more than the degree of its numerator and the denominator of
3090:
2595:
This leads to the following ordinary second order differential equation:
98:
2683:{\displaystyle {\frac {d^{2}f}{dx^{2}}}-2x{\frac {df}{dx}}+\lambda f=0.}
1926:{\displaystyle p_{0}(w)={\frac {1}{w^{4}}}-{\frac {\alpha ^{2}}{w^{2}}}}
396:
172:, where the growth of solutions is bounded (in any small sector) by an
200:
More precisely, consider an ordinary linear differential equation of
2399:
This differential equation has regular singular points at ±1 and ∞.
3128:
Theory of
Differential Equations Vol. IV: Ordinary Linear Equations
551:
1362:
1195:
is of degree at least two more than the degree of its numerator.
192:, but where the analytic properties are substantially different.
3102:
An
Introduction to the Theory of Functions of a Complex Variable
319:{\displaystyle f^{(n)}(z)+\sum _{i=0}^{n-1}p_{i}(z)f^{(i)}(z)=0}
2084:{\displaystyle {\frac {d}{dx}}\left+\ell (\ell +1)f=0.}
1357:). The most common and important special case is where
890:{\displaystyle {\frac {df}{dx}}=-w^{2}{\frac {df}{dw}}}
3187:
2873:
2710:
2601:
2586:{\displaystyle V(x)={\frac {1}{2}}m\omega ^{2}x^{2}.}
2531:
2416:
2245:
2099:
1985:
1939:
1856:
1807:
1780:
1627:
1593:
1573:. Thus this equation has a regular singularity at 0.
1381:
1220:
1165:
1125:
1085:
1049:
1032:
We can thus transform the equation to an equation in
902:
830:
796:
562:
437:
210:
136:
114:
662:{\displaystyle f''(x)+p_{1}(x)f'(x)+p_{0}(x)f(x)=0.}
3224:
371:may be applied to find possible solutions that are
3082:
3080:
3033:
2846:
2682:
2585:
2505:
2389:
2221:
2083:
1951:
1925:
1842:
1792:
1763:
1613:
1503:
1334:
1187:
1147:
1107:
1071:
1022:
889:
816:
661:
465:
318:
144:
122:
3060:(1995). "Heun's Equation". In A. Ronveaux (ed.).
1966:
3229:
2402:
552:Examples for second order differential equations
54:but its sources remain unclear because it lacks
3153:
3143:pp. 128−ff. (Ginn & co., Boston, 1917)
1201:
556:In this case the equation above is reduced to:
3081:Coddington, Earl A.; Levinson, Norman (1955).
184:, with three regular singular points, and the
1119:unless the polynomial in the denominator of
156:, at which the equation's coefficients are
2699:
3085:Theory of Ordinary Differential Equations
138:
116:
85:Learn how and when to remove this message
3107:
535:, bounded by a line at 45° to the axes.
3066:. Oxford University Press. p. 74.
3056:
1843:{\displaystyle p_{1}(w)={\frac {1}{w}}}
671:One distinguishes the following cases:
521:
14:
3230:
3180:
527:The regularity condition is a kind of
348:The equation should be studied on the
3177:p. 243 (MacMillan, London, 1917)
489:also can be made to work and provide
195:
2517:. In this case the potential energy
26:
24:
3131:(Cambridge University Press, 1906)
2093:Opening the square bracket gives:
415:. This presents no difficulty for
164:, at which some coefficient has a
25:
3254:
431:, which by definition means that
1621:. After performing the algebra:
548:ordinary differential equation.
356:as a possible singular point. A
31:
3238:Ordinary differential equations
3174:Functions of a Complex Variable
3148:Ordinary Differential Equations
2704:The equation may be defined as
1850:has a pole of first order, but
765:has a pole of order up to 2 at
103:ordinary differential equations
3016:
3004:
2961:
2949:
2938:
2920:
2795:
2777:
2726:
2714:
2541:
2535:
2497:
2491:
2354:
2342:
2207:
2195:
2069:
2057:
2025:
2006:
1967:Legendre differential equation
1873:
1867:
1824:
1818:
1566:has a pole of second order at
1320:
1294:
1182:
1176:
1142:
1136:
1102:
1096:
1066:
1060:
650:
644:
638:
632:
616:
610:
599:
593:
577:
571:
540:ordinary differential equation
460:
454:
307:
301:
296:
290:
282:
276:
233:
227:
222:
216:
13:
1:
3196:American Mathematical Society
3154:Il'yashenko, Yu. S. (2001) ,
3063:Heun's Differential Equations
3050:
2403:Hermite differential equation
1530:has a pole of first order at
18:Regular differential equation
1202:Bessel differential equation
1036:, and check what happens at
746:has a pole up to order 1 at
145:{\displaystyle \mathbb {C} }
123:{\displaystyle \mathbb {C} }
7:
3221:A Course of Modern Analysis
3161:Encyclopedia of Mathematics
3150:, Dover Publications (1944)
3115:Encyclopedia of Mathematics
493:independent solutions near
391:in the complex plane where
10:
3259:
1371:Dividing this equation by
466:{\displaystyle p_{n-i}(z)}
3108:Fedoryuk, M. V. (2001) ,
1576:To see what happens when
1342:for an arbitrary real or
3156:"Regular singular point"
1188:{\displaystyle p_{2}(x)}
1148:{\displaystyle p_{1}(x)}
1108:{\displaystyle p_{2}(x)}
1072:{\displaystyle p_{1}(x)}
784:irregular singular point
178:irregular singular point
40:This article includes a
3045:hypergeometric function
2856:Dividing both sides by
2700:Hypergeometric equation
1212:cylindrical coordinates
182:hypergeometric equation
69:more precise citations.
3035:
2848:
2684:
2587:
2507:
2391:
2223:
2085:
1953:
1927:
1844:
1794:
1765:
1615:
1505:
1336:
1189:
1149:
1109:
1073:
1024:
891:
818:
730:regular singular point
663:
512:relating solutions by
467:
429:regular singular point
320:
265:
188:which is in a sense a
170:regular singular point
146:
124:
3036:
2849:
2685:
2588:
2508:
2392:
2224:
2086:
1977:spherical coordinates
1954:
1928:
1845:
1795:
1766:
1616:
1614:{\displaystyle x=1/w}
1585:Möbius transformation
1506:
1337:
1190:
1150:
1110:
1074:
1025:
892:
819:
817:{\displaystyle w=1/x}
664:
514:analytic continuation
506:irregular singularity
468:
375:times complex powers
358:Möbius transformation
343:meromorphic functions
321:
239:
147:
125:
2871:
2708:
2599:
2529:
2414:
2409:Schrödinger equation
2243:
2097:
1983:
1937:
1854:
1805:
1778:
1625:
1591:
1379:
1218:
1163:
1123:
1083:
1047:
900:
828:
794:
560:
500:Otherwise the point
435:
208:
152:are classified into
134:
112:
3110:"Fuchsian equation"
2694:Hermite polynomials
2515:harmonic oscillator
1952:{\displaystyle w=0}
1793:{\displaystyle w=0}
824:and the relations:
508:. In that case the
419:an ordinary point (
399:extending out from
101:, in the theory of
3031:
2844:
2680:
2583:
2503:
2387:
2219:
2081:
1973:Laplace's equation
1949:
1923:
1840:
1790:
1761:
1611:
1501:
1332:
1208:Laplace's equation
1185:
1145:
1105:
1069:
1020:
887:
814:
659:
463:
316:
196:Formal definitions
174:algebraic function
158:analytic functions
142:
120:
42:list of references
3205:978-0-8218-8328-0
3020:
2985:
2965:
2906:
2824:
2761:
2663:
2634:
2555:
2483:
2449:
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2331:
2311:
2278:
2187:
2158:
2041:
1999:
1959:corresponding to
1921:
1894:
1838:
1745:
1718:
1693:
1673:
1660:
1583:one has to use a
1485:
1447:
1427:
1414:
1289:
1263:
1018:
982:
935:
885:
849:
477:of order at most
369:indicial equation
354:point at infinity
95:
94:
87:
16:(Redirected from
3250:
3243:Complex analysis
3209:
3171:T. M. MacRobert
3168:
3122:
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3021:
3019:
2999:
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2984:
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2905:
2904:
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2890:
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2845:
2825:
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2292:
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2279:
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2276:
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2262:
2258:
2257:
2247:
2238:
2231:And dividing by
2228:
2226:
2225:
2220:
2188:
2186:
2178:
2170:
2159:
2157:
2156:
2155:
2142:
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2121:
2120:
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2090:
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1998:
1987:
1958:
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778:Otherwise point
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711:are analytic at
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487:Frobenius method
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365:Frobenius method
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130:, the points of
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65:this article by
56:inline citations
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3213:E. T. Whittaker
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1355:Bessel function
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684:
683:when functions
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583:
563:
561:
558:
557:
554:
534:
510:monodromy group
501:
494:
490:
482:
478:
442:
438:
436:
433:
432:
424:
416:
412:
405:Riemann surface
400:
392:
388:
387:near any given
376:
352:to include the
335:
327:
289:
285:
270:
266:
254:
243:
215:
211:
209:
206:
205:
201:
198:
186:Bessel equation
162:singular points
154:ordinary points
137:
135:
132:
131:
115:
113:
110:
109:
91:
80:
74:
71:
60:
46:related reading
36:
32:
23:
22:
15:
12:
11:
5:
3256:
3246:
3245:
3240:
3226:
3225:
3210:
3204:
3182:Teschl, Gerald
3178:
3169:
3151:
3144:
3132:
3125:A. R. Forsyth
3123:
3105:
3095:
3078:
3072:
3052:
3049:
3030:
3027:
3024:
3018:
3015:
3012:
3009:
3006:
3003:
2998:
2995:
2989:
2983:
2980:
2975:
2972:
2963:
2960:
2957:
2954:
2951:
2948:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2910:
2902:
2898:
2894:
2889:
2884:
2880:
2843:
2840:
2837:
2834:
2831:
2828:
2822:
2819:
2814:
2811:
2804:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2769:
2765:
2757:
2753:
2749:
2744:
2739:
2735:
2728:
2725:
2722:
2719:
2716:
2713:
2701:
2698:
2679:
2676:
2673:
2670:
2667:
2661:
2658:
2653:
2650:
2644:
2641:
2638:
2630:
2626:
2622:
2617:
2612:
2608:
2582:
2577:
2573:
2567:
2563:
2559:
2554:
2551:
2546:
2543:
2540:
2537:
2534:
2502:
2499:
2496:
2493:
2490:
2487:
2479:
2475:
2471:
2466:
2461:
2457:
2447:
2444:
2438:
2434:
2428:
2425:
2422:
2419:
2404:
2401:
2386:
2383:
2380:
2372:
2368:
2364:
2361:
2356:
2353:
2350:
2347:
2344:
2341:
2335:
2329:
2326:
2321:
2318:
2307:
2303:
2299:
2296:
2291:
2288:
2282:
2274:
2270:
2266:
2261:
2256:
2252:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2185:
2182:
2177:
2174:
2168:
2165:
2162:
2154:
2150:
2146:
2141:
2136:
2132:
2124:
2118:
2114:
2110:
2107:
2103:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2049:
2045:
2039:
2036:
2032:
2027:
2022:
2018:
2014:
2011:
2008:
2004:
1997:
1994:
1990:
1968:
1965:
1948:
1945:
1942:
1918:
1914:
1908:
1904:
1898:
1891:
1887:
1883:
1878:
1875:
1872:
1869:
1864:
1860:
1837:
1834:
1829:
1826:
1823:
1820:
1815:
1811:
1789:
1786:
1783:
1760:
1757:
1754:
1750:
1742:
1738:
1732:
1728:
1722:
1715:
1711:
1707:
1701:
1697:
1691:
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1656:
1652:
1648:
1643:
1638:
1634:
1610:
1606:
1602:
1599:
1596:
1587:, for example
1549:
1518:
1500:
1497:
1494:
1490:
1482:
1478:
1472:
1468:
1462:
1459:
1455:
1451:
1445:
1442:
1437:
1434:
1426:
1423:
1418:
1410:
1406:
1402:
1397:
1392:
1388:
1344:complex number
1331:
1328:
1325:
1322:
1317:
1313:
1309:
1304:
1300:
1296:
1293:
1287:
1284:
1279:
1276:
1270:
1267:
1259:
1255:
1251:
1246:
1241:
1237:
1228:
1224:
1203:
1200:
1184:
1181:
1178:
1173:
1169:
1144:
1141:
1138:
1133:
1129:
1104:
1101:
1098:
1093:
1089:
1068:
1065:
1062:
1057:
1053:
1016:
1013:
1008:
1005:
997:
993:
989:
986:
978:
974:
970:
965:
960:
956:
947:
943:
939:
931:
927:
923:
918:
913:
909:
883:
880:
875:
872:
864:
860:
856:
853:
847:
844:
839:
836:
813:
809:
805:
802:
799:
788:
787:
776:
761:
737:
722:
702:
688:
681:ordinary point
658:
655:
652:
649:
646:
643:
640:
637:
634:
629:
625:
621:
618:
615:
612:
608:
605:
601:
598:
595:
590:
586:
582:
579:
576:
573:
569:
566:
553:
550:
532:
529:Newton polygon
522:Arscott (1995)
462:
459:
456:
451:
448:
445:
441:
409:punctured disc
350:Riemann sphere
331:
315:
312:
309:
306:
303:
298:
295:
292:
288:
284:
281:
278:
273:
269:
263:
260:
257:
252:
249:
246:
242:
238:
235:
232:
229:
224:
221:
218:
214:
197:
194:
140:
118:
93:
92:
50:external links
39:
37:
30:
9:
6:
4:
3:
2:
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3179:
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3162:
3157:
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3141:
3136:
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3129:
3124:
3121:
3117:
3116:
3111:
3106:
3103:
3099:
3096:
3092:
3087:
3086:
3079:
3075:
3069:
3065:
3064:
3059:
3058:Arscott, F.M.
3055:
3054:
3048:
3046:
3041:
3028:
3025:
3022:
3013:
3010:
3007:
3001:
2996:
2993:
2987:
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2973:
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2737:
2733:
2723:
2720:
2717:
2711:
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2677:
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2665:
2659:
2656:
2651:
2648:
2642:
2639:
2636:
2628:
2624:
2620:
2615:
2610:
2606:
2593:
2580:
2575:
2571:
2565:
2561:
2557:
2552:
2549:
2544:
2538:
2532:
2524:
2520:
2516:
2500:
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2488:
2485:
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2473:
2469:
2464:
2459:
2455:
2445:
2442:
2436:
2432:
2426:
2423:
2420:
2417:
2410:
2400:
2397:
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2381:
2378:
2370:
2366:
2362:
2359:
2351:
2348:
2345:
2339:
2333:
2327:
2324:
2319:
2316:
2305:
2301:
2297:
2294:
2289:
2286:
2280:
2272:
2268:
2264:
2259:
2254:
2250:
2236:
2229:
2216:
2213:
2210:
2204:
2201:
2198:
2192:
2189:
2183:
2180:
2175:
2172:
2166:
2163:
2160:
2152:
2148:
2144:
2139:
2134:
2130:
2122:
2116:
2112:
2108:
2105:
2101:
2091:
2078:
2075:
2072:
2066:
2063:
2060:
2054:
2051:
2047:
2043:
2037:
2034:
2030:
2020:
2016:
2012:
2009:
2002:
1995:
1992:
1988:
1978:
1974:
1964:
1962:
1946:
1943:
1940:
1916:
1912:
1906:
1902:
1896:
1889:
1885:
1881:
1876:
1870:
1862:
1858:
1835:
1832:
1827:
1821:
1813:
1809:
1787:
1784:
1781:
1771:
1758:
1755:
1752:
1748:
1740:
1736:
1730:
1726:
1720:
1713:
1709:
1705:
1699:
1695:
1689:
1686:
1681:
1678:
1670:
1667:
1662:
1654:
1650:
1646:
1641:
1636:
1632:
1608:
1604:
1600:
1597:
1594:
1586:
1580:
1574:
1570:
1563:
1559:
1555:
1548:
1541:
1534:
1528:
1524:
1517:
1513:In this case
1511:
1498:
1495:
1492:
1488:
1480:
1476:
1470:
1466:
1460:
1457:
1453:
1449:
1443:
1440:
1435:
1432:
1424:
1421:
1416:
1408:
1404:
1400:
1395:
1390:
1386:
1374:
1369:
1364:
1356:
1352:
1345:
1329:
1326:
1323:
1315:
1311:
1307:
1302:
1298:
1291:
1285:
1282:
1277:
1274:
1268:
1265:
1257:
1253:
1249:
1244:
1239:
1235:
1226:
1222:
1213:
1209:
1199:
1196:
1179:
1171:
1167:
1158:
1139:
1131:
1127:
1118:
1099:
1091:
1087:
1063:
1055:
1051:
1040:
1030:
1014:
1011:
1006:
1003:
995:
991:
987:
984:
976:
972:
968:
963:
958:
954:
945:
941:
937:
929:
925:
921:
916:
911:
907:
881:
878:
873:
870:
862:
858:
854:
851:
845:
842:
837:
834:
811:
807:
803:
800:
797:
785:
777:
773:
769:
760:
754:
750:
743:
736:
731:
723:
719:
715:
708:
701:
694:
687:
682:
674:
673:
672:
669:
656:
653:
647:
641:
635:
627:
623:
619:
613:
606:
603:
596:
588:
584:
580:
574:
567:
564:
549:
547:
546:
541:
536:
530:
525:
523:
519:
515:
511:
507:
498:
488:
476:
457:
449:
446:
443:
439:
430:
422:
421:Lazarus Fuchs
410:
406:
398:
384:
380:
374:
370:
367:based on the
366:
361:
359:
355:
351:
346:
344:
339:
334:
330:
313:
310:
304:
293:
286:
279:
271:
267:
261:
258:
255:
250:
247:
244:
240:
236:
230:
219:
212:
193:
191:
190:limiting case
187:
183:
179:
175:
171:
167:
163:
159:
155:
108:
107:complex plane
104:
100:
89:
86:
78:
68:
64:
58:
57:
51:
47:
43:
38:
29:
28:
19:
3219:
3217:G. N. Watson
3186:
3172:
3159:
3147:
3146:E. L. Ince,
3138:
3126:
3113:
3101:
3098:E. T. Copson
3089:. New York:
3084:
3062:
3042:
2862:
2858:
2855:
2703:
2691:
2594:
2522:
2518:
2406:
2398:
2234:
2230:
2092:
1970:
1960:
1772:
1578:
1575:
1568:
1561:
1557:
1553:
1546:
1539:
1532:
1526:
1522:
1515:
1512:
1372:
1370:
1350:
1205:
1197:
1116:
1038:
1031:
789:
783:
771:
767:
758:
752:
748:
741:
734:
729:
717:
713:
706:
699:
692:
685:
680:
670:
555:
543:
537:
526:
505:
499:
428:
423:1866). When
382:
378:
373:power series
362:
347:
337:
332:
328:
199:
177:
169:
161:
153:
96:
81:
72:
61:Please help
53:
3091:McGraw-Hill
166:singularity
99:mathematics
67:introducing
3232:Categories
3192:Providence
3073:0198596952
3051:References
403:, or on a
397:branch cut
204:-th order
3166:EMS Press
3120:EMS Press
3011:−
2988:−
2956:−
2918:−
2827:−
2775:−
2721:−
2669:λ
2637:−
2562:ω
2501:ψ
2465:ψ
2433:ℏ
2427:−
2421:ψ
2363:−
2346:ℓ
2340:ℓ
2298:−
2281:−
2199:ℓ
2193:ℓ
2161:−
2109:−
2061:ℓ
2055:ℓ
2013:−
1903:α
1897:−
1727:α
1721:−
1556:) = (1 −
1467:α
1461:−
1312:α
1308:−
855:−
447:−
363:Then the
259:−
241:∑
176:, and an
75:June 2017
3184:(2012).
607:′
568:″
545:Fuchsian
518:Poincaré
407:of some
2867:gives:
1773:Now at
1537:. When
1375:gives:
1363:integer
1353:of the
679:is an
411:around
105:in the
63:improve
3202:
3104:(1935)
3070:
2525:) is:
2513:for a
1963:at ∞.
1525:) = 1/
1361:is an
1157:degree
1155:is of
782:is an
724:Point
675:Point
520:rank (
504:is an
485:, the
473:has a
160:, and
2861:(1 −
2233:(1 −
1351:order
1349:(the
1043:. If
728:is a
427:is a
326:with
48:, or
3215:and
3200:ISBN
3068:ISBN
1079:and
756:and
697:and
475:pole
1975:in
1581:→ ∞
1571:= 0
1542:≠ 0
1535:= 0
1210:in
1041:= 0
732:if
538:An
524:).
481:at
97:In
3234::
3198:.
3194::
3190:.
3164:,
3158:,
3137:,
3118:,
3112:,
3100:,
3047:.
3029:0.
2842:0.
2696:.
2678:0.
2385:0.
2239::
2217:0.
2079:0.
1979::
1544:,
1499:0.
1368:.
1214::
770:=
751:=
716:=
657:0.
497:.
381:−
345:.
52:,
44:,
3208:.
3093:.
3076:.
3026:=
3023:f
3017:)
3014:z
3008:1
3005:(
3002:z
2997:b
2994:a
2982:z
2979:d
2974:f
2971:d
2962:)
2959:z
2953:1
2950:(
2947:z
2942:z
2939:)
2936:1
2933:+
2930:b
2927:+
2924:a
2921:(
2915:c
2909:+
2901:2
2897:z
2893:d
2888:f
2883:2
2879:d
2865:)
2863:z
2859:z
2839:=
2836:f
2833:b
2830:a
2821:z
2818:d
2813:f
2810:d
2803:]
2799:z
2796:)
2793:1
2790:+
2787:b
2784:+
2781:a
2778:(
2772:c
2768:[
2764:+
2756:2
2752:z
2748:d
2743:f
2738:2
2734:d
2727:)
2724:z
2718:1
2715:(
2712:z
2675:=
2672:f
2666:+
2660:x
2657:d
2652:f
2649:d
2643:x
2640:2
2629:2
2625:x
2621:d
2616:f
2611:2
2607:d
2581:.
2576:2
2572:x
2566:2
2558:m
2553:2
2550:1
2545:=
2542:)
2539:x
2536:(
2533:V
2523:x
2521:(
2519:V
2498:)
2495:x
2492:(
2489:V
2486:+
2478:2
2474:x
2470:d
2460:2
2456:d
2446:m
2443:2
2437:2
2424:=
2418:E
2382:=
2379:f
2371:2
2367:x
2360:1
2355:)
2352:1
2349:+
2343:(
2334:+
2328:x
2325:d
2320:f
2317:d
2306:2
2302:x
2295:1
2290:x
2287:2
2273:2
2269:x
2265:d
2260:f
2255:2
2251:d
2237:)
2235:x
2214:=
2211:f
2208:)
2205:1
2202:+
2196:(
2190:+
2184:x
2181:d
2176:f
2173:d
2167:x
2164:2
2153:2
2149:x
2145:d
2140:f
2135:2
2131:d
2123:)
2117:2
2113:x
2106:1
2102:(
2076:=
2073:f
2070:)
2067:1
2064:+
2058:(
2052:+
2048:]
2044:f
2038:x
2035:d
2031:d
2026:)
2021:2
2017:x
2010:1
2007:(
2003:[
1996:x
1993:d
1989:d
1961:x
1947:0
1944:=
1941:w
1917:2
1913:w
1907:2
1890:4
1886:w
1882:1
1877:=
1874:)
1871:w
1868:(
1863:0
1859:p
1836:w
1833:1
1828:=
1825:)
1822:w
1819:(
1814:1
1810:p
1800:,
1788:0
1785:=
1782:w
1759:0
1756:=
1753:f
1749:]
1741:2
1737:w
1731:2
1714:4
1710:w
1706:1
1700:[
1696:+
1690:w
1687:d
1682:f
1679:d
1671:w
1668:1
1663:+
1655:2
1651:w
1647:d
1642:f
1637:2
1633:d
1609:w
1605:/
1601:1
1598:=
1595:x
1579:x
1569:x
1564:)
1562:x
1560:/
1558:α
1554:x
1552:(
1550:0
1547:p
1540:α
1533:x
1527:x
1523:x
1521:(
1519:1
1516:p
1496:=
1493:f
1489:)
1481:2
1477:x
1471:2
1458:1
1454:(
1450:+
1444:x
1441:d
1436:f
1433:d
1425:x
1422:1
1417:+
1409:2
1405:x
1401:d
1396:f
1391:2
1387:d
1373:x
1366:n
1359:α
1347:α
1330:0
1327:=
1324:f
1321:)
1316:2
1303:2
1299:x
1295:(
1292:+
1286:x
1283:d
1278:f
1275:d
1269:x
1266:+
1258:2
1254:x
1250:d
1245:f
1240:2
1236:d
1227:2
1223:x
1183:)
1180:x
1177:(
1172:2
1168:p
1143:)
1140:x
1137:(
1132:1
1128:p
1117:x
1103:)
1100:x
1097:(
1092:2
1088:p
1067:)
1064:x
1061:(
1056:1
1052:p
1039:w
1034:w
1015:w
1012:d
1007:f
1004:d
996:3
992:w
988:2
985:+
977:2
973:w
969:d
964:f
959:2
955:d
946:4
942:w
938:=
930:2
926:x
922:d
917:f
912:2
908:d
882:w
879:d
874:f
871:d
863:2
859:w
852:=
846:x
843:d
838:f
835:d
812:x
808:/
804:1
801:=
798:w
786:.
780:a
775:.
772:a
768:x
762:0
759:p
753:a
749:x
744:)
742:x
740:(
738:1
735:p
726:a
721:.
718:a
714:x
709:)
707:x
705:(
703:0
700:p
695:)
693:x
691:(
689:1
686:p
677:a
654:=
651:)
648:x
645:(
642:f
639:)
636:x
633:(
628:0
624:p
620:+
617:)
614:x
611:(
604:f
600:)
597:x
594:(
589:1
585:p
581:+
578:)
575:x
572:(
565:f
533:i
502:a
495:a
491:n
483:a
479:i
461:)
458:z
455:(
450:i
444:n
440:p
425:a
417:a
413:a
401:a
393:r
389:a
385:)
383:a
379:z
377:(
340:)
338:z
336:(
333:i
329:p
314:0
311:=
308:)
305:z
302:(
297:)
294:i
291:(
287:f
283:)
280:z
277:(
272:i
268:p
262:1
256:n
251:0
248:=
245:i
237:+
234:)
231:z
228:(
223:)
220:n
217:(
213:f
202:n
139:C
117:C
88:)
82:(
77:)
73:(
59:.
20:)
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