2258:
2445:
2769:
One feature that all of the examples listed above have in common is that, in each case, two or more known functions (sometimes multiplication by a constant, sometimes addition of two variables, sometimes the
2751:
2638:
2018:
1272:
2523:
1636:
2131:
1927:
1830:
1729:
1169:
3113:
1552:
148:
3044:
1474:
1172:
2340:
375:
2946:
822:
737:
1269:
1068:
892:
1336:
962:
2140:
1400:
304:
can be seen as functional equations in functions over the integers or natural numbers, in which the differences between terms' indexes can be seen as an application of the
662:
239:
2997:
602:
494:
548:
2867:
2698:
441:
408:
274:
3497:
3485:
3223:
2346:
1833:
1732:
3477:
2543:
184:
is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the
1955:
3506:
3453:
2703:
2793:
are the 'reasonable' ones, while other solutions that are not likely to have practical application can be constructed (by using a
2045:
2265:
3525:
3473:
1555:
3553:
17:
2451:
3288:
3237:
1563:
3427:
3203:
3183:
1841:
1744:
1643:
1079:
3276:
740:
281:
3051:
3445:
79:
3569:
70:
3003:
1411:
3304:
3253:
2286:
1485:
315:
2879:
760:
675:
1180:
976:
833:
276:
There are many functions that satisfy these conditions, but the gamma function is the unique one that is
1280:
906:
3458:
3399:
3229:
1347:
2810:
828:. Like Cauchy's additive functional equation, this too may have pathological, discontinuous solutions
289:
1477:
1403:
3280:
2822:
614:
191:
2957:
969:
557:
449:
1944:
are functional equations. In its familiar form, the associative law is expressed by writing the
506:
3547:
3153:
2828:
665:
161:
3356:
Házy, Attila (2004-03-01). "Solving linear two variable functional equations with computer".
3171:
2782:
2665:
2261:
413:
380:
244:
54:
50:
3268:
1930:
1736:
825:
277:
153:
2253:{\displaystyle f(s)=2^{s}\pi ^{s-1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)f(1-s)}
8:
3412:
3269:
3193:
3163:
2790:
751:, also have other pathological nonlinear solutions, whose existence can be proven with a
301:
165:
2440:{\displaystyle f(y)f\left(y+{\frac {1}{2}}\right)={\frac {\sqrt {\pi }}{2^{2y-1}}}f(2y)}
3381:
3338:
3298:
3247:
3134:
2532:
66:
2281:
The gamma function is the unique solution of the following system of three equations:
3521:
3469:
3441:
3385:
3373:
3284:
3233:
3198:
2771:
1071:
899:
497:
309:
58:
3159:
Some classes of functional equations can be solved by computer-assisted techniques.
61:
are functional equations. However, a more restricted meaning is often used, where a
3365:
3330:
3188:
3167:
1945:
177:
31:
65:
is an equation that relates several values of the same function. For example, the
3515:
3432:
2870:
2806:
1941:
1937:
748:
181:
164:, and, in this case, a functional equation (in the narrower meaning) is called a
3541:
3437:
2529:
2273:
1949:
965:
305:
185:
157:
3369:
3563:
3377:
3142:
2042:
then the associative law looks more like a conventional functional equation,
1339:
551:
173:
3492:
3138:
2802:
2758:
605:
3318:
2798:
2794:
752:
38:
3411:
Sniedovich, M. (2010). Dynamic
Programming: Foundations and Principles,
3342:
3130:
3126:
One method of solving elementary functional equations is substitution.
744:
3321:(1916). "On Certain Real Solutions of Babbage's Functional Equation".
2774:) are inside the argument of the unknown functions to be solved for.
895:
3334:
3498:
Introduction to the Theory of
Functional Equations and Inequalities
46:
2659:
3166:
a variety of successive approximation methods are used to solve
3149:
3118:
which includes the previous three as special cases or limits.
2746:{\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}}
2633:{\displaystyle f\left({az+b \over cz+d}\right)=(cz+d)^{k}f(z)}
2526:
3148:
Some functional equations have been solved with the use of
3271:
Functional
Equations and Inequalities in Several Variables
2951:
Other involutions, and solutions of the equation, include
3486:
Functional
Equations and Inequalities with Applications
3433:
Lectures on
Functional Equations and Their Applications
3129:
Some solutions to functional equations have exploited
2712:
3054:
3006:
2960:
2882:
2831:
2706:
2668:
2546:
2454:
2349:
2289:
2143:
2048:
2013:{\displaystyle (a\circ b)\circ c=a\circ (b\circ c)~,}
1958:
1844:
1747:
1646:
1566:
1488:
1414:
1350:
1283:
1183:
1082:
979:
909:
836:
763:
678:
617:
560:
509:
452:
416:
383:
318:
247:
194:
188:
is a function that satisfies the functional equation
82:
3513:
308:. For example, the recurrence relation defining the
3510:, first edition, World Scientific Publishing, 2013.
2785:should be applied; for example, in the case of the
2781:solutions, it may be the case that conditions from
2518:{\displaystyle f(z)f(1-z)={\pi \over \sin(\pi z)}}
3107:
3038:
2991:
2940:
2861:
2745:
2692:
2632:
2517:
2439:
2334:
2252:
2125:
2012:
1921:
1824:
1723:
1630:
1546:
1468:
1394:
1330:
1263:
1163:
1062:
956:
886:
816:
731:
656:
596:
542:
488:
435:
402:
369:
268:
233:
142:
2122:
1918:
1821:
1720:
1627:
1543:
1465:
1391:
1327:
1260:
1160:
1059:
953:
883:
813:
728:
3561:
3550:at EqWorld: The World of Mathematical Equations.
3544:at EqWorld: The World of Mathematical Equations.
1631:{\displaystyle f(xy)=\sum g_{l}(x)h_{l}(y)\,\!}
898:functions and, over coprime integer arguments,
3275:. P O Box 128, Farrer Road, Singapore 912805:
156:of the unknown function is supposed to be the
2825:are characterized by the functional equation
2126:{\displaystyle f(f(a,b),c)=f(a,f(b,c)).\,\!}
1922:{\displaystyle g(x+y)=g(x)g(y)+f(y)f(x)\,\!}
1825:{\displaystyle g(x+y)=g(x)g(y)-f(y)f(x)\,\!}
1724:{\displaystyle f(x+y)=f(x)g(y)+f(y)g(x)\,\!}
1164:{\displaystyle f((x+y)/2)=(f(x)+f(y))/2\,\!}
49:in which one or several functions appear as
3556:on functional equations in problem solving.
3517:Functional Equations and How to Solve Them
3108:{\displaystyle f(x)={\frac {b-x}{1+cx}}~,}
3520:. Springer Science & Business Media.
3454:Functional Equations in Several Variables
3440:, 1966, reprinted by Dover Publications,
3398:Bellman, R. (1957). Dynamic Programming,
3032:
2985:
2934:
2121:
1917:
1820:
1719:
1626:
1542:
1464:
1390:
1326:
1259:
1159:
1058:
952:
882:
812:
727:
143:{\displaystyle \log(xy)=\log(x)+\log(y).}
2789:mentioned above, the solutions that are
160:, the function is generally viewed as a
3266:
3260:
3221:
14:
3562:
3228:. 3300 AA Dordrecht, The Netherlands:
3215:
3039:{\displaystyle f(x)={\frac {a}{x}}\,,}
1469:{\displaystyle f(h(x))=(f(x))^{c}\,\!}
747:. The equation may, contingent on the
3542:Functional Equations: Exact Solutions
3514:Christopher G. Small (3 April 2007).
3225:Functional Equations and Inequalities
2335:{\displaystyle f(x)={f(x+1) \over x}}
1547:{\displaystyle f(h(x))=h'(x)f(x)\,\!}
968:and, over coprime integer arguments,
370:{\displaystyle F_{n}=F_{n-1}+F_{n-2}}
3466:Introduction to Functional Equations
3355:
3317:
2941:{\displaystyle f(f(x))=1-(1-x)=x\,.}
817:{\displaystyle f(x+y)=f(x)f(y),\,\!}
732:{\displaystyle f(x+y)=f(x)+f(y)\,\!}
27:Equation whose unknown is a function
3501:, second edition, Birkhäuser, 2009.
1264:{\displaystyle g(x+y)+g(x-y)=2\,\!}
1063:{\displaystyle f(x+y)+f(x-y)=2\,\!}
887:{\displaystyle f(xy)=f(x)+f(y)\,\!}
24:
2214:
1931:hyperbolic cosine addition formula
1331:{\displaystyle f(h(x))=h(x+1)\,\!}
957:{\displaystyle f(xy)=f(x)f(y)\,\!}
25:
3581:
3535:
3222:Rassias, Themistocles M. (2000).
1395:{\displaystyle f(h(x))=cf(x)\,\!}
3204:Functional differential equation
3184:Functional equation (L-function)
1737:hyperbolic sine addition formula
1273:d'Alembert's functional equation
280:in the whole complex plane, and
45:is, in the broadest meaning, an
3451:János Aczél & J. Dhombres,
3277:World Scientific Publishing Co.
2813:is another well-known example.
75:logarithmic functional equation
3554:IMO Compendium text (archived)
3507:Functional Equations on Groups
3405:
3392:
3349:
3311:
3064:
3058:
3016:
3010:
2970:
2964:
2925:
2913:
2901:
2898:
2892:
2886:
2850:
2847:
2841:
2835:
2816:
2627:
2621:
2609:
2593:
2509:
2500:
2482:
2470:
2464:
2458:
2434:
2425:
2359:
2353:
2323:
2311:
2299:
2293:
2247:
2235:
2229:
2217:
2153:
2147:
2115:
2112:
2100:
2088:
2079:
2070:
2058:
2052:
2001:
1989:
1971:
1959:
1914:
1908:
1902:
1896:
1887:
1881:
1875:
1869:
1860:
1848:
1817:
1811:
1805:
1799:
1790:
1784:
1778:
1772:
1763:
1751:
1716:
1710:
1704:
1698:
1689:
1683:
1677:
1671:
1662:
1650:
1623:
1617:
1604:
1598:
1579:
1570:
1539:
1533:
1527:
1521:
1507:
1504:
1498:
1492:
1455:
1451:
1445:
1439:
1433:
1430:
1424:
1418:
1387:
1381:
1369:
1366:
1360:
1354:
1323:
1311:
1302:
1299:
1293:
1287:
1256:
1253:
1247:
1241:
1235:
1229:
1220:
1208:
1199:
1187:
1148:
1145:
1139:
1130:
1124:
1118:
1112:
1101:
1089:
1086:
1055:
1052:
1046:
1037:
1031:
1025:
1016:
1004:
995:
983:
949:
943:
937:
931:
922:
913:
879:
873:
864:
858:
849:
840:
806:
800:
794:
788:
779:
767:
724:
718:
709:
703:
694:
682:
651:
645:
636:
633:
627:
621:
591:
582:
570:
564:
537:
528:
519:
513:
483:
477:
468:
456:
257:
251:
228:
222:
210:
198:
134:
128:
116:
110:
98:
89:
13:
1:
3421:
3170:, including methods based on
3168:Bellman's functional equation
2873:functional equation (1820),
2777:When it comes to asking for
1173:Jensen's functional equation
741:Cauchy's functional equation
657:{\displaystyle f(f(x))=g(x)}
234:{\displaystyle f(x+1)=xf(x)}
7:
3548:Functional Equations: Index
3177:
3121:
2992:{\displaystyle f(x)=a-x\,,}
597:{\displaystyle f(x)=-f(-x)}
489:{\displaystyle f(x+P)=f(x)}
295:
10:
3586:
3459:Cambridge University Press
3400:Princeton University Press
3230:Kluwer Academic Publishers
664:, which characterizes the
604:, which characterizes the
550:, which characterizes the
543:{\displaystyle f(x)=f(-x)}
496:, which characterizes the
29:
3370:10.1007/s00010-003-2703-9
3323:The Annals of Mathematics
3267:Czerwik, Stephan (2002).
2862:{\displaystyle f(f(x))=x}
71:essentially characterized
3358:Aequationes Mathematicae
3303:: CS1 maint: location (
3252:: CS1 maint: location (
3209:
2540:The functional equation
2137:The functional equation
970:multiplicative functions
30:Not to be confused with
2693:{\displaystyle ad-bc=1}
1834:cosine addition formula
1070:(quadratic equation or
666:functional square roots
436:{\displaystyle F_{1}=1}
403:{\displaystyle F_{0}=0}
269:{\displaystyle f(1)=1.}
3172:fixed point iterations
3154:mathematical induction
3109:
3040:
2993:
2942:
2863:
2747:
2694:
2634:
2519:
2441:
2336:
2254:
2127:
2014:
1923:
1826:
1725:
1632:
1548:
1470:
1396:
1332:
1265:
1165:
1064:
958:
888:
818:
733:
658:
598:
544:
490:
437:
404:
371:
282:logarithmically convex
270:
241:and the initial value
235:
144:
55:differential equations
3110:
3041:
2994:
2943:
2864:
2811:Bohr–Mollerup theorem
2783:mathematical analysis
2748:
2695:
2635:
2520:
2442:
2337:
2262:Riemann zeta function
2255:
2128:
2015:
1924:
1827:
1733:sine addition formula
1726:
1633:
1549:
1471:
1397:
1333:
1266:
1166:
1065:
959:
889:
826:exponential functions
819:
734:
659:
599:
545:
491:
438:
405:
372:
290:Bohr–Mollerup theorem
271:
236:
145:
3570:Functional equations
3413:Taylor & Francis
3052:
3004:
2958:
2880:
2829:
2791:continuous functions
2704:
2666:
2544:
2452:
2347:
2287:
2260:is satisfied by the
2141:
2046:
1956:
1842:
1745:
1644:
1564:
1486:
1412:
1348:
1281:
1181:
1080:
977:
907:
834:
761:
755:for the real numbers
676:
615:
558:
507:
450:
414:
381:
316:
302:Recurrence relations
245:
192:
182:smoothness condition
80:
18:Functional equations
3194:Dynamic programming
3164:dynamic programming
1478:Böttcher's equation
1404:Schröder's equation
964:, satisfied by all
894:, satisfied by all
288:real and positive (
172:is used mainly for
170:functional equation
166:recurrence relation
67:logarithm functions
63:functional equation
43:functional equation
3105:
3036:
2989:
2938:
2869:. These appear in
2859:
2743:
2737:
2690:
2630:
2533:reflection formula
2515:
2437:
2332:
2250:
2123:
2010:
1919:
1822:
1721:
1628:
1544:
1466:
1392:
1328:
1261:
1161:
1060:
954:
900:additive functions
884:
814:
729:
654:
594:
540:
498:periodic functions
486:
433:
400:
367:
266:
231:
140:
59:integral equations
3527:978-0-387-48901-8
3489:, Springer, 2009.
3474:978-0-8218-5314-6
3199:Implicit function
3101:
3097:
3030:
2772:identity function
2584:
2513:
2420:
2400:
2384:
2330:
2208:
2006:
1072:parallelogram law
824:satisfied by all
668:of the function g
310:Fibonacci numbers
178:complex functions
16:(Redirected from
3577:
3531:
3504:Henrik Stetkær,
3416:
3409:
3403:
3396:
3390:
3389:
3353:
3347:
3346:
3315:
3309:
3308:
3302:
3294:
3274:
3264:
3258:
3257:
3251:
3243:
3219:
3189:Bellman equation
3114:
3112:
3111:
3106:
3099:
3098:
3096:
3082:
3071:
3045:
3043:
3042:
3037:
3031:
3023:
2998:
2996:
2995:
2990:
2947:
2945:
2944:
2939:
2868:
2866:
2865:
2860:
2807:rational numbers
2764:
2756:
2752:
2750:
2749:
2744:
2742:
2741:
2699:
2697:
2696:
2691:
2657:
2639:
2637:
2636:
2631:
2617:
2616:
2589:
2585:
2583:
2569:
2555:
2527:
2524:
2522:
2521:
2516:
2514:
2512:
2489:
2446:
2444:
2443:
2438:
2421:
2419:
2418:
2396:
2395:
2390:
2386:
2385:
2377:
2341:
2339:
2338:
2333:
2331:
2326:
2306:
2271:
2259:
2257:
2256:
2251:
2213:
2209:
2204:
2196:
2184:
2183:
2168:
2167:
2132:
2130:
2129:
2124:
2041:
2020:but if we write
2019:
2017:
2016:
2011:
2004:
1946:binary operation
1942:associative laws
1928:
1926:
1925:
1920:
1831:
1829:
1828:
1823:
1730:
1728:
1727:
1722:
1637:
1635:
1634:
1629:
1616:
1615:
1597:
1596:
1556:Julia's equation
1553:
1551:
1550:
1545:
1520:
1475:
1473:
1472:
1467:
1463:
1462:
1401:
1399:
1398:
1393:
1337:
1335:
1334:
1329:
1270:
1268:
1267:
1262:
1170:
1168:
1167:
1162:
1155:
1108:
1069:
1067:
1066:
1061:
963:
961:
960:
955:
893:
891:
890:
885:
823:
821:
820:
815:
743:), satisfied by
738:
736:
735:
730:
663:
661:
660:
655:
603:
601:
600:
595:
549:
547:
546:
541:
495:
493:
492:
487:
442:
440:
439:
434:
426:
425:
409:
407:
406:
401:
393:
392:
376:
374:
373:
368:
366:
365:
347:
346:
328:
327:
287:
275:
273:
272:
267:
240:
238:
237:
232:
168:. Thus the term
149:
147:
146:
141:
32:Functional model
21:
3585:
3584:
3580:
3579:
3578:
3576:
3575:
3574:
3560:
3559:
3538:
3528:
3483:Pl. Kannappan,
3424:
3419:
3410:
3406:
3397:
3393:
3354:
3350:
3335:10.2307/2007270
3316:
3312:
3296:
3295:
3291:
3265:
3261:
3245:
3244:
3240:
3232:. p. 335.
3220:
3216:
3212:
3180:
3124:
3083:
3072:
3070:
3053:
3050:
3049:
3022:
3005:
3002:
3001:
2959:
2956:
2955:
2881:
2878:
2877:
2830:
2827:
2826:
2819:
2787:Cauchy equation
2762:
2754:
2736:
2735:
2730:
2724:
2723:
2718:
2708:
2707:
2705:
2702:
2701:
2667:
2664:
2663:
2641:
2612:
2608:
2570:
2556:
2554:
2550:
2545:
2542:
2541:
2525:
2493:
2488:
2453:
2450:
2449:
2405:
2401:
2394:
2376:
2369:
2365:
2348:
2345:
2344:
2307:
2305:
2288:
2285:
2284:
2269:
2197:
2195:
2191:
2173:
2169:
2163:
2159:
2142:
2139:
2138:
2047:
2044:
2043:
2033:
1957:
1954:
1953:
1843:
1840:
1839:
1746:
1743:
1742:
1645:
1642:
1641:
1611:
1607:
1592:
1588:
1565:
1562:
1561:
1513:
1487:
1484:
1483:
1458:
1454:
1413:
1410:
1409:
1349:
1346:
1345:
1282:
1279:
1278:
1182:
1179:
1178:
1151:
1104:
1081:
1078:
1077:
978:
975:
974:
966:power functions
908:
905:
904:
835:
832:
831:
762:
759:
758:
749:axiom of choice
677:
674:
673:
616:
613:
612:
559:
556:
555:
554:, and likewise
508:
505:
504:
451:
448:
447:
421:
417:
415:
412:
411:
388:
384:
382:
379:
378:
355:
351:
336:
332:
323:
319:
317:
314:
313:
298:
285:
246:
243:
242:
193:
190:
189:
158:natural numbers
81:
78:
77:
35:
28:
23:
22:
15:
12:
11:
5:
3583:
3573:
3572:
3558:
3557:
3551:
3545:
3537:
3536:External links
3534:
3533:
3532:
3526:
3511:
3502:
3490:
3481:
3464:C. Efthimiou,
3462:
3449:
3438:Academic Press
3423:
3420:
3418:
3417:
3404:
3391:
3348:
3329:(3): 113–122.
3310:
3289:
3259:
3238:
3213:
3211:
3208:
3207:
3206:
3201:
3196:
3191:
3186:
3179:
3176:
3123:
3120:
3116:
3115:
3104:
3095:
3092:
3089:
3086:
3081:
3078:
3075:
3069:
3066:
3063:
3060:
3057:
3047:
3035:
3029:
3026:
3021:
3018:
3015:
3012:
3009:
2999:
2988:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2949:
2948:
2937:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2818:
2815:
2767:
2766:
2740:
2734:
2731:
2729:
2726:
2725:
2722:
2719:
2717:
2714:
2713:
2711:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2629:
2626:
2623:
2620:
2615:
2611:
2607:
2604:
2601:
2598:
2595:
2592:
2588:
2582:
2579:
2576:
2573:
2568:
2565:
2562:
2559:
2553:
2549:
2538:
2537:
2536:
2511:
2508:
2505:
2502:
2499:
2496:
2492:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2447:
2436:
2433:
2430:
2427:
2424:
2417:
2414:
2411:
2408:
2404:
2399:
2393:
2389:
2383:
2380:
2375:
2372:
2368:
2364:
2361:
2358:
2355:
2352:
2342:
2329:
2325:
2322:
2319:
2316:
2313:
2310:
2304:
2301:
2298:
2295:
2292:
2278:
2277:
2274:gamma function
2268:. The capital
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2212:
2207:
2203:
2200:
2194:
2190:
2187:
2182:
2179:
2176:
2172:
2166:
2162:
2158:
2155:
2152:
2149:
2146:
2134:
2133:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2009:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1950:infix notation
1934:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1837:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1740:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1639:
1638:(Levi-Civita),
1625:
1622:
1619:
1614:
1610:
1606:
1603:
1600:
1595:
1591:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1559:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1519:
1516:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1481:
1461:
1457:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1407:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1343:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1276:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1176:
1158:
1154:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1107:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1075:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
972:
951:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
902:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
829:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
756:
726:
723:
720:
717:
714:
711:
708:
705:
702:
699:
696:
693:
690:
687:
684:
681:
670:
669:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
620:
609:
608:
593:
590:
587:
584:
581:
578:
575:
572:
569:
566:
563:
552:even functions
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
501:
500:
485:
482:
479:
476:
473:
470:
467:
464:
461:
458:
455:
444:
443:
432:
429:
424:
420:
399:
396:
391:
387:
364:
361:
358:
354:
350:
345:
342:
339:
335:
331:
326:
322:
306:shift operator
297:
294:
265:
262:
259:
256:
253:
250:
230:
227:
224:
221:
218:
215:
212:
209:
206:
203:
200:
197:
186:gamma function
174:real functions
139:
136:
133:
130:
127:
124:
121:
118:
115:
112:
109:
106:
103:
100:
97:
94:
91:
88:
85:
26:
9:
6:
4:
3:
2:
3582:
3571:
3568:
3567:
3565:
3555:
3552:
3549:
3546:
3543:
3540:
3539:
3529:
3523:
3519:
3518:
3512:
3509:
3508:
3503:
3500:
3499:
3494:
3491:
3488:
3487:
3482:
3479:
3475:
3471:
3468:, AMS, 2011,
3467:
3463:
3460:
3456:
3455:
3450:
3447:
3443:
3439:
3435:
3434:
3429:
3426:
3425:
3414:
3408:
3401:
3395:
3387:
3383:
3379:
3375:
3371:
3367:
3363:
3359:
3352:
3344:
3340:
3336:
3332:
3328:
3324:
3320:
3314:
3306:
3300:
3292:
3290:981-02-4837-7
3286:
3282:
3278:
3273:
3272:
3263:
3255:
3249:
3241:
3239:0-7923-6484-8
3235:
3231:
3227:
3226:
3218:
3214:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3181:
3175:
3173:
3169:
3165:
3160:
3157:
3155:
3151:
3146:
3144:
3140:
3136:
3132:
3127:
3119:
3102:
3093:
3090:
3087:
3084:
3079:
3076:
3073:
3067:
3061:
3055:
3048:
3033:
3027:
3024:
3019:
3013:
3007:
3000:
2986:
2982:
2979:
2976:
2973:
2967:
2961:
2954:
2953:
2952:
2935:
2931:
2928:
2922:
2919:
2916:
2910:
2907:
2904:
2895:
2889:
2883:
2876:
2875:
2874:
2872:
2856:
2853:
2844:
2838:
2832:
2824:
2814:
2812:
2808:
2804:
2800:
2796:
2792:
2788:
2784:
2780:
2775:
2773:
2760:
2753:= 1, defines
2738:
2732:
2727:
2720:
2715:
2709:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2661:
2656:
2652:
2648:
2644:
2624:
2618:
2613:
2605:
2602:
2599:
2596:
2590:
2586:
2580:
2577:
2574:
2571:
2566:
2563:
2560:
2557:
2551:
2547:
2539:
2534:
2531:
2506:
2503:
2497:
2494:
2490:
2485:
2479:
2476:
2473:
2467:
2461:
2455:
2448:
2431:
2428:
2422:
2415:
2412:
2409:
2406:
2402:
2397:
2391:
2387:
2381:
2378:
2373:
2370:
2366:
2362:
2356:
2350:
2343:
2327:
2320:
2317:
2314:
2308:
2302:
2296:
2290:
2283:
2282:
2280:
2279:
2275:
2267:
2263:
2244:
2241:
2238:
2232:
2226:
2223:
2220:
2210:
2205:
2201:
2198:
2192:
2188:
2185:
2180:
2177:
2174:
2170:
2164:
2160:
2156:
2150:
2144:
2136:
2135:
2118:
2109:
2106:
2103:
2097:
2094:
2091:
2085:
2082:
2076:
2073:
2067:
2064:
2061:
2055:
2049:
2040:
2036:
2032:) instead of
2031:
2027:
2023:
2007:
1998:
1995:
1992:
1986:
1983:
1980:
1977:
1974:
1968:
1965:
1962:
1951:
1947:
1943:
1939:
1935:
1932:
1911:
1905:
1899:
1893:
1890:
1884:
1878:
1872:
1866:
1863:
1857:
1854:
1851:
1845:
1838:
1835:
1814:
1808:
1802:
1796:
1793:
1787:
1781:
1775:
1769:
1766:
1760:
1757:
1754:
1748:
1741:
1738:
1734:
1713:
1707:
1701:
1695:
1692:
1686:
1680:
1674:
1668:
1665:
1659:
1656:
1653:
1647:
1640:
1620:
1612:
1608:
1601:
1593:
1589:
1585:
1582:
1576:
1573:
1567:
1560:
1557:
1536:
1530:
1524:
1517:
1514:
1510:
1501:
1495:
1489:
1482:
1479:
1459:
1448:
1442:
1436:
1427:
1421:
1415:
1408:
1405:
1384:
1378:
1375:
1372:
1363:
1357:
1351:
1344:
1341:
1340:Abel equation
1320:
1317:
1314:
1308:
1305:
1296:
1290:
1284:
1277:
1274:
1250:
1244:
1238:
1232:
1226:
1223:
1217:
1214:
1211:
1205:
1202:
1196:
1193:
1190:
1184:
1177:
1174:
1156:
1152:
1142:
1136:
1133:
1127:
1121:
1115:
1109:
1105:
1098:
1095:
1092:
1083:
1076:
1073:
1049:
1043:
1040:
1034:
1028:
1022:
1019:
1013:
1010:
1007:
1001:
998:
992:
989:
986:
980:
973:
971:
967:
946:
940:
934:
928:
925:
919:
916:
910:
903:
901:
897:
876:
870:
867:
861:
855:
852:
846:
843:
837:
830:
827:
809:
803:
797:
791:
785:
782:
776:
773:
770:
764:
757:
754:
750:
746:
742:
721:
715:
712:
706:
700:
697:
691:
688:
685:
679:
672:
671:
667:
648:
642:
639:
630:
624:
618:
611:
610:
607:
606:odd functions
588:
585:
579:
576:
573:
567:
561:
553:
534:
531:
525:
522:
516:
510:
503:
502:
499:
480:
474:
471:
465:
462:
459:
453:
446:
445:
430:
427:
422:
418:
397:
394:
389:
385:
362:
359:
356:
352:
348:
343:
340:
337:
333:
329:
324:
320:
311:
307:
303:
300:
299:
293:
291:
283:
279:
263:
260:
254:
248:
225:
219:
216:
213:
207:
204:
201:
195:
187:
183:
180:. Moreover a
179:
175:
171:
167:
163:
159:
155:
150:
137:
131:
125:
122:
119:
113:
107:
104:
101:
95:
92:
86:
83:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
3516:
3505:
3496:
3493:Marek Kuczma
3484:
3465:
3452:
3431:
3407:
3394:
3364:(1): 47–62.
3361:
3357:
3351:
3326:
3322:
3313:
3270:
3262:
3224:
3217:
3161:
3158:
3147:
3131:surjectivity
3128:
3125:
3117:
2950:
2820:
2803:vector space
2799:real numbers
2786:
2778:
2776:
2768:
2759:modular form
2654:
2650:
2646:
2642:
2272:denotes the
2264:, as proved
2038:
2034:
2029:
2025:
2021:
169:
151:
74:
62:
42:
36:
3428:János Aczél
3319:Ritt, J. F.
3135:injectivity
2823:involutions
2817:Involutions
2795:Hamel basis
2662:satisfying
1938:commutative
896:logarithmic
753:Hamel basis
745:linear maps
278:meromorphic
39:mathematics
3446:0486445232
3422:References
3386:118563768
3378:1420-8903
3299:cite book
3248:cite book
3077:−
2980:−
2920:−
2911:−
2871:Babbage's
2805:over the
2761:of order
2676:−
2504:π
2498:
2491:π
2477:−
2413:−
2398:π
2242:−
2224:−
2215:Γ
2199:π
2189:
2178:−
2171:π
1996:∘
1987:∘
1975:∘
1966:∘
1794:−
1586:∑
1215:−
1011:−
586:−
577:−
532:−
360:−
341:−
126:
108:
87:
3564:Category
3476: ;
3279:p.
3178:See also
3150:ansatzes
3143:evenness
3122:Solution
2797:for the
2757:to be a
2660:integers
2028:,
1518:′
377:, where
296:Examples
162:sequence
51:unknowns
47:equation
3461:, 1989.
3343:2007270
3139:oddness
2809:). The
2700:, i.e.
2530:Euler's
152:If the
73:by the
3524:
3478:online
3472:
3444:
3384:
3376:
3341:
3287:
3236:
3141:, and
3100:
2640:where
2005:
154:domain
53:. So,
3382:S2CID
3339:JSTOR
3210:Notes
3522:ISBN
3470:ISBN
3442:ISBN
3374:ISSN
3305:link
3285:ISBN
3254:link
3234:ISBN
2821:The
2658:are
2266:here
1940:and
1936:The
1735:and
410:and
284:for
176:and
69:are
57:and
41:, a
3366:doi
3331:doi
3281:410
3162:In
3046:and
2801:as
2779:all
2495:sin
2186:sin
1948:in
292:).
123:log
105:log
84:log
37:In
3566::
3495:,
3457:,
3436:,
3430:,
3380:.
3372:.
3362:67
3360:.
3337:.
3327:17
3325:.
3301:}}
3297:{{
3283:.
3250:}}
3246:{{
3174:.
3156:.
3152:,
3145:.
3137:,
3133:,
2653:,
2649:,
2645:,
2037:â—‹
1952:,
1933:).
1836:),
1739:),
1558:).
1480:).
1406:).
312:,
264:1.
3530:.
3480:.
3448:.
3415:.
3402:.
3388:.
3368::
3345:.
3333::
3307:)
3293:.
3256:)
3242:.
3103:,
3094:x
3091:c
3088:+
3085:1
3080:x
3074:b
3068:=
3065:)
3062:x
3059:(
3056:f
3034:,
3028:x
3025:a
3020:=
3017:)
3014:x
3011:(
3008:f
2987:,
2983:x
2977:a
2974:=
2971:)
2968:x
2965:(
2962:f
2936:.
2932:x
2929:=
2926:)
2923:x
2917:1
2914:(
2908:1
2905:=
2902:)
2899:)
2896:x
2893:(
2890:f
2887:(
2884:f
2857:x
2854:=
2851:)
2848:)
2845:x
2842:(
2839:f
2836:(
2833:f
2765:.
2763:k
2755:f
2739:|
2733:d
2728:c
2721:b
2716:a
2710:|
2688:1
2685:=
2682:c
2679:b
2673:d
2670:a
2655:d
2651:c
2647:b
2643:a
2628:)
2625:z
2622:(
2619:f
2614:k
2610:)
2606:d
2603:+
2600:z
2597:c
2594:(
2591:=
2587:)
2581:d
2578:+
2575:z
2572:c
2567:b
2564:+
2561:z
2558:a
2552:(
2548:f
2535:)
2528:(
2510:)
2507:z
2501:(
2486:=
2483:)
2480:z
2474:1
2471:(
2468:f
2465:)
2462:z
2459:(
2456:f
2435:)
2432:y
2429:2
2426:(
2423:f
2416:1
2410:y
2407:2
2403:2
2392:=
2388:)
2382:2
2379:1
2374:+
2371:y
2367:(
2363:f
2360:)
2357:y
2354:(
2351:f
2328:x
2324:)
2321:1
2318:+
2315:x
2312:(
2309:f
2303:=
2300:)
2297:x
2294:(
2291:f
2276:.
2270:Γ
2248:)
2245:s
2239:1
2236:(
2233:f
2230:)
2227:s
2221:1
2218:(
2211:)
2206:2
2202:s
2193:(
2181:1
2175:s
2165:s
2161:2
2157:=
2154:)
2151:s
2148:(
2145:f
2119:.
2116:)
2113:)
2110:c
2107:,
2104:b
2101:(
2098:f
2095:,
2092:a
2089:(
2086:f
2083:=
2080:)
2077:c
2074:,
2071:)
2068:b
2065:,
2062:a
2059:(
2056:f
2053:(
2050:f
2039:b
2035:a
2030:b
2026:a
2024:(
2022:f
2008:,
2002:)
1999:c
1993:b
1990:(
1984:a
1981:=
1978:c
1972:)
1969:b
1963:a
1960:(
1929:(
1915:)
1912:x
1909:(
1906:f
1903:)
1900:y
1897:(
1894:f
1891:+
1888:)
1885:y
1882:(
1879:g
1876:)
1873:x
1870:(
1867:g
1864:=
1861:)
1858:y
1855:+
1852:x
1849:(
1846:g
1832:(
1818:)
1815:x
1812:(
1809:f
1806:)
1803:y
1800:(
1797:f
1791:)
1788:y
1785:(
1782:g
1779:)
1776:x
1773:(
1770:g
1767:=
1764:)
1761:y
1758:+
1755:x
1752:(
1749:g
1731:(
1717:)
1714:x
1711:(
1708:g
1705:)
1702:y
1699:(
1696:f
1693:+
1690:)
1687:y
1684:(
1681:g
1678:)
1675:x
1672:(
1669:f
1666:=
1663:)
1660:y
1657:+
1654:x
1651:(
1648:f
1624:)
1621:y
1618:(
1613:l
1609:h
1605:)
1602:x
1599:(
1594:l
1590:g
1583:=
1580:)
1577:y
1574:x
1571:(
1568:f
1554:(
1540:)
1537:x
1534:(
1531:f
1528:)
1525:x
1522:(
1515:h
1511:=
1508:)
1505:)
1502:x
1499:(
1496:h
1493:(
1490:f
1476:(
1460:c
1456:)
1452:)
1449:x
1446:(
1443:f
1440:(
1437:=
1434:)
1431:)
1428:x
1425:(
1422:h
1419:(
1416:f
1402:(
1388:)
1385:x
1382:(
1379:f
1376:c
1373:=
1370:)
1367:)
1364:x
1361:(
1358:h
1355:(
1352:f
1342:)
1338:(
1324:)
1321:1
1318:+
1315:x
1312:(
1309:h
1306:=
1303:)
1300:)
1297:x
1294:(
1291:h
1288:(
1285:f
1275:)
1271:(
1257:]
1254:)
1251:y
1248:(
1245:g
1242:)
1239:x
1236:(
1233:g
1230:[
1227:2
1224:=
1221:)
1218:y
1212:x
1209:(
1206:g
1203:+
1200:)
1197:y
1194:+
1191:x
1188:(
1185:g
1175:)
1171:(
1157:2
1153:/
1149:)
1146:)
1143:y
1140:(
1137:f
1134:+
1131:)
1128:x
1125:(
1122:f
1119:(
1116:=
1113:)
1110:2
1106:/
1102:)
1099:y
1096:+
1093:x
1090:(
1087:(
1084:f
1074:)
1056:]
1053:)
1050:y
1047:(
1044:f
1041:+
1038:)
1035:x
1032:(
1029:f
1026:[
1023:2
1020:=
1017:)
1014:y
1008:x
1005:(
1002:f
999:+
996:)
993:y
990:+
987:x
984:(
981:f
950:)
947:y
944:(
941:f
938:)
935:x
932:(
929:f
926:=
923:)
920:y
917:x
914:(
911:f
880:)
877:y
874:(
871:f
868:+
865:)
862:x
859:(
856:f
853:=
850:)
847:y
844:x
841:(
838:f
810:,
807:)
804:y
801:(
798:f
795:)
792:x
789:(
786:f
783:=
780:)
777:y
774:+
771:x
768:(
765:f
739:(
725:)
722:y
719:(
716:f
713:+
710:)
707:x
704:(
701:f
698:=
695:)
692:y
689:+
686:x
683:(
680:f
652:)
649:x
646:(
643:g
640:=
637:)
634:)
631:x
628:(
625:f
622:(
619:f
592:)
589:x
583:(
580:f
574:=
571:)
568:x
565:(
562:f
538:)
535:x
529:(
526:f
523:=
520:)
517:x
514:(
511:f
484:)
481:x
478:(
475:f
472:=
469:)
466:P
463:+
460:x
457:(
454:f
431:1
428:=
423:1
419:F
398:0
395:=
390:0
386:F
363:2
357:n
353:F
349:+
344:1
338:n
334:F
330:=
325:n
321:F
286:x
261:=
258:)
255:1
252:(
249:f
229:)
226:x
223:(
220:f
217:x
214:=
211:)
208:1
205:+
202:x
199:(
196:f
138:.
135:)
132:y
129:(
120:+
117:)
114:x
111:(
102:=
99:)
96:y
93:x
90:(
34:.
20:)
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