309:
1251:
1639:
1744:
419:
540:
170:
836:
150:
1472:
1362:
1126:
1094:
1038:
950:
768:
584:
112:
1394:
1423:
727:
669:
71:
1526:
1518:
997:
909:
624:
1284:
1313:
869:
698:
451:
1653:
1118:
973:
889:
604:
320:
1829:
1945:
1794:
1899:
1868:
463:
304:{\displaystyle f(x)=\{{\mathcal {M}}^{-1}\varphi \}={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }x^{-s}\varphi (s)\,ds}
1940:
1950:
33:
1878:
631:
773:
117:
1246:{\displaystyle \|f\|=\left(\int _{0}^{\infty }|x^{\nu }f(x)|^{p}\,{\frac {dx}{x}}\right)^{1/p}<\infty }
457:, taking a value halfway between the limit values at any jump discontinuities, and suppose the integral
454:
1428:
1318:
1050:
1002:
914:
732:
548:
76:
1763:
976:
1634:{\displaystyle f(x)={\frac {1}{2\pi i}}\int _{\nu -i\infty }^{\nu +i\infty }x^{-s}\varphi (s)\,ds.}
1367:
1399:
703:
645:
47:
871:
as defined by the inversion integral exists and is continuous; moreover the Mellin transform of
1477:
1895:
1864:
1817:
1800:
1790:
982:
894:
627:
609:
1838:
1758:
1263:
29:
1739:{\displaystyle \left\{{\mathcal {B}}f\right\}(s)=\left\{{\mathcal {M}}f(-\ln x)\right\}(s)}
1289:
845:
674:
427:
1924:
1821:
1103:
999:
to simply make it of polynomial growth in any closed strip contained in the open strip
958:
874:
589:
25:
1644:
Here functions, identical everywhere except on a set of measure zero, are identified.
1934:
1842:
1044:
414:{\displaystyle \varphi (s)=\{{\mathcal {M}}f\}=\int _{0}^{\infty }x^{s-1}f(x)\,dx.}
17:
1804:
606:
is recoverable via the inverse Mellin transform from its Mellin transform
1097:
630:
by a change of variables and then applying an appropriate version of the
626:. These results can be obtained by relating the Mellin transform to the
1784:
164:, with its integral along such a line converging absolutely, then if
535:{\displaystyle \varphi (s)=\int _{0}^{\infty }x^{s-1}f(x)\,dx}
1696:
1664:
344:
195:
955:
On the other hand, if we are willing to accept an original
1647:
Since the two-sided
Laplace transform can be defined as
1749:
these theorems can be immediately applied to it also.
1656:
1529:
1480:
1431:
1402:
1370:
1321:
1292:
1266:
1129:
1106:
1053:
1005:
985:
961:
917:
897:
877:
848:
776:
735:
706:
677:
648:
612:
592:
551:
466:
430:
323:
173:
120:
79:
50:
36:, are defined and recover the transformed function.
1822:"Mellin transforms and asymptotics: Harmonic sums"
1738:
1633:
1512:
1466:
1417:
1388:
1356:
1307:
1278:
1245:
1112:
1088:
1032:
991:
967:
944:
903:
883:
863:
830:
762:
721:
692:
663:
618:
598:
578:
534:
445:
413:
303:
144:
106:
65:
1927:at EqWorld: The World of Mathematical Equations.
1883:Introduction to the Theory of Fourier Integrals
1852:Complex Variable Theory and Transform Calculus
28:) tells us conditions under which the inverse
979:, we may relax the boundedness condition on
8:
1136:
1130:
352:
339:
212:
189:
1885:(Second ed.). Oxford University Press.
1859:Polyanin, A. D.; Manzhirov, A. V. (1998).
1786:Integral transforms and their applications
831:{\displaystyle |\varphi (s)|<K|s|^{-2}}
1695:
1694:
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1662:
1655:
1621:
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1567:
1545:
1528:
1490:
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1401:
1369:
1345:
1326:
1320:
1291:
1265:
1227:
1223:
1203:
1202:
1196:
1191:
1172:
1163:
1157:
1152:
1128:
1105:
1077:
1058:
1052:
1004:
984:
960:
916:
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819:
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805:
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734:
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273:
254:
240:
218:
200:
194:
193:
172:
119:
78:
49:
114:, and if it tends to zero uniformly as
1775:
1047:version of this theorem. If we call by
145:{\displaystyle \Im (s)\to \pm \infty }
7:
1591:
1577:
1240:
1158:
1012:
924:
742:
558:
492:
368:
264:
250:
139:
121:
86:
14:
1820:; Gourdon, X.; Dumas, P. (1995).
1467:{\displaystyle L_{\nu ,q}(R^{+})}
1357:{\displaystyle L_{\nu ,p}(R^{+})}
1089:{\displaystyle L_{\nu ,p}(R^{+})}
1033:{\displaystyle a<\Re (s)<b}
945:{\displaystyle a<\Re (s)<b}
763:{\displaystyle a<\Re (s)<b}
579:{\displaystyle a<\Re (s)<b}
107:{\displaystyle a<\Re (s)<b}
1120:on the positive reals such that
1911:Generalized Integral Transforms
453:is piecewise continuous on the
1861:Handbook of Integral Equations
1733:
1727:
1719:
1704:
1683:
1677:
1618:
1612:
1539:
1533:
1507:
1495:
1461:
1448:
1412:
1406:
1351:
1338:
1302:
1296:
1192:
1187:
1181:
1164:
1083:
1070:
1021:
1015:
933:
927:
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815:
806:
795:
791:
785:
778:
751:
745:
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710:
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652:
567:
561:
545:is absolutely convergent when
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516:
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440:
434:
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392:
333:
327:
291:
285:
183:
177:
133:
130:
124:
95:
89:
60:
54:
32:, or equivalently the inverse
1:
1925:Tables of Integral Transforms
1854:. Cambridge University Press.
842:is a positive constant, then
642:The boundedness condition on
1946:Theorems in complex analysis
1843:10.1016/0304-3975(95)00002-E
1830:Theoretical Computer Science
1389:{\displaystyle 1<p\leq 2}
1260:are fixed real numbers with
1100:of complex valued functions
1418:{\displaystyle \varphi (s)}
722:{\displaystyle \varphi (s)}
664:{\displaystyle \varphi (s)}
66:{\displaystyle \varphi (s)}
34:two-sided Laplace transform
1967:
1890:Yakubovich, S. B. (1996).
1783:Debnath, Lokenath (2015).
1863:. Boca Raton: CRC Press.
1850:McLachlan, N. W. (1953).
1513:{\displaystyle q=p/(p-1)}
729:is analytic in the strip
632:Fourier inversion theorem
73:is analytic in the strip
1913:. John Wiley & Sons.
1909:Zemanian, A. H. (1968).
992:{\displaystyle \varphi }
904:{\displaystyle \varphi }
671:can be strengthened if
619:{\displaystyle \varphi }
22:Mellin inversion formula
1740:
1635:
1514:
1468:
1419:
1390:
1358:
1309:
1280:
1279:{\displaystyle p>1}
1247:
1114:
1090:
1034:
993:
969:
946:
905:
885:
865:
832:
764:
723:
694:
665:
620:
600:
580:
536:
447:
415:
305:
146:
108:
67:
1741:
1636:
1515:
1469:
1420:
1391:
1359:
1310:
1281:
1248:
1115:
1091:
1043:We may also define a
1035:
994:
970:
947:
906:
886:
866:
833:
765:
724:
695:
666:
638:Boundedness condition
621:
601:
581:
537:
455:positive real numbers
448:
416:
306:
147:
109:
68:
1894:. World Scientific.
1654:
1527:
1478:
1429:
1400:
1368:
1319:
1308:{\displaystyle f(x)}
1290:
1264:
1127:
1104:
1051:
1003:
983:
977:generalized function
959:
915:
895:
875:
864:{\displaystyle f(x)}
846:
774:
733:
704:
693:{\displaystyle f(x)}
675:
646:
610:
590:
549:
464:
446:{\displaystyle f(x)}
428:
424:Conversely, suppose
321:
171:
118:
77:
48:
1941:Integral transforms
1595:
1162:
496:
372:
268:
152:for any real value
1951:Laplace transforms
1736:
1631:
1563:
1510:
1464:
1415:
1386:
1354:
1305:
1276:
1243:
1148:
1110:
1086:
1030:
989:
965:
942:
901:
881:
861:
828:
760:
719:
700:is continuous. If
690:
661:
616:
596:
576:
532:
482:
443:
411:
358:
301:
236:
142:
104:
63:
1879:Titchmarsh, E. C.
1796:978-1-4822-2357-6
1764:Nachbin's theorem
1561:
1216:
1113:{\displaystyle f}
968:{\displaystyle f}
884:{\displaystyle f}
628:Fourier transform
599:{\displaystyle f}
234:
1958:
1914:
1905:
1892:Index Transforms
1886:
1874:
1855:
1846:
1826:
1809:
1808:
1780:
1759:Mellin transform
1745:
1743:
1742:
1737:
1726:
1722:
1700:
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1668:
1667:
1640:
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1111:
1095:
1093:
1092:
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1081:
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1068:
1039:
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996:
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30:Mellin transform
1966:
1965:
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1955:
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1921:
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1919:External links
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26:Hjalmar Mellin
13:
10:
9:
6:
4:
3:
2:
1963:
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1949:
1947:
1944:
1942:
1939:
1938:
1936:
1926:
1923:
1922:
1918:
1912:
1907:
1903:
1901:981-02-2216-5
1897:
1893:
1888:
1884:
1880:
1876:
1872:
1870:0-8493-2876-4
1866:
1862:
1857:
1853:
1848:
1844:
1840:
1837:(1–2): 3–58.
1836:
1832:
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1798:
1792:
1789:. CRC Press.
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1173:
1169:
1153:
1149:
1144:
1139:
1133:
1123:
1122:
1121:
1107:
1099:
1096:the weighted
1078:
1074:
1065:
1062:
1059:
1055:
1046:
1041:
1027:
1024:
1018:
1009:
1006:
986:
978:
962:
953:
939:
936:
930:
921:
918:
911:for at least
898:
878:
855:
849:
841:
823:
820:
810:
802:
799:
788:
782:
757:
754:
748:
739:
736:
713:
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655:
649:
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479:
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467:
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459:
458:
456:
437:
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408:
405:
402:
395:
389:
384:
381:
378:
374:
363:
359:
355:
349:
336:
330:
324:
317:
316:
315:
314:we have that
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282:
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51:
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35:
31:
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