5660:
2327:
1856:
5079:
4905:
5655:{\displaystyle {\begin{aligned}{\frac {\partial }{\partial z_{i}}}\left(f\circ g\right)&=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial z_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}\left(f\circ g\right)&=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial {\bar {z}}_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial {\bar {z}}_{i}}}\end{aligned}}}
2322:{\displaystyle {\begin{cases}{\frac {\partial }{\partial z_{1}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{1}}}-i{\frac {\partial }{\partial y_{1}}}\right)\\\qquad \vdots \\{\frac {\partial }{\partial z_{n}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{n}}}-i{\frac {\partial }{\partial y_{n}}}\right)\\\end{cases}},\qquad {\begin{cases}{\frac {\partial }{\partial {\bar {z}}_{1}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{1}}}+i{\frac {\partial }{\partial y_{1}}}\right)\\\qquad \vdots \\{\frac {\partial }{\partial {\bar {z}}_{n}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{n}}}+i{\frac {\partial }{\partial y_{n}}}\right)\\\end{cases}}.}
4514:
2914:
4900:{\displaystyle {\begin{aligned}{\frac {\partial }{\partial z}}(f\circ g)&=\left({\frac {\partial f}{\partial z}}\circ g\right){\frac {\partial g}{\partial z}}+\left({\frac {\partial f}{\partial {\bar {z}}}}\circ g\right){\frac {\partial {\bar {g}}}{\partial z}}\\{\frac {\partial }{\partial {\bar {z}}}}(f\circ g)&=\left({\frac {\partial f}{\partial z}}\circ g\right){\frac {\partial g}{\partial {\bar {z}}}}+\left({\frac {\partial f}{\partial {\bar {z}}}}\circ g\right){\frac {\partial {\bar {g}}}{\partial {\bar {z}}}}\end{aligned}}}
3814:
2597:
4203:
3529:
1537:
5974:
2909:{\displaystyle {\begin{aligned}{\frac {\partial f}{\partial z}}&={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}-i{\frac {\partial f}{\partial y}}\right)\\&={\frac {1}{2}}\left({\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}-i{\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial y}}\right)\\&={\frac {\partial u}{\partial z}}+i{\frac {\partial v}{\partial z}}={\frac {df}{dz}}\end{aligned}}}
3926:
3809:{\displaystyle {\begin{aligned}{\frac {\partial }{\partial z_{i}}}\left(\alpha f+\beta g\right)&=\alpha {\frac {\partial f}{\partial z_{i}}}+\beta {\frac {\partial g}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}\left(\alpha f+\beta g\right)&=\alpha {\frac {\partial f}{\partial {\bar {z}}_{i}}}+\beta {\frac {\partial g}{\partial {\bar {z}}_{i}}}\end{aligned}}}
1348:
5762:
4198:{\displaystyle {\begin{aligned}{\frac {\partial }{\partial z_{i}}}(f\cdot g)&={\frac {\partial f}{\partial z_{i}}}\cdot g+f\cdot {\frac {\partial g}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}(f\cdot g)&={\frac {\partial f}{\partial {\bar {z}}_{i}}}\cdot g+f\cdot {\frac {\partial g}{\partial {\bar {z}}_{i}}}\end{aligned}}}
1844:
876:
1532:{\displaystyle {\begin{aligned}{\frac {\partial }{\partial z}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right)\\{\frac {\partial }{\partial {\bar {z}}}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right)\end{aligned}}}
5969:{\displaystyle {\begin{aligned}{\overline {\left({\frac {\partial f}{\partial z_{i}}}\right)}}&={\frac {\partial {\bar {f}}}{\partial {\bar {z}}_{i}}}\\{\overline {\left({\frac {\partial f}{\partial {\bar {z}}_{i}}}\right)}}&={\frac {\partial {\bar {f}}}{\partial z_{i}}}\end{aligned}}}
1658:
3026:
700:
283:
1258:
2922:
3236:
1839:{\displaystyle \mathbb {C} ^{n}=\mathbb {R} ^{2n}=\left\{\left(\mathbf {x} ,\mathbf {y} \right)=\left(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}\right)\mid \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{n}\right\}.}
4321:
3081:
5029:
4281:
1137:, is considerably simplified and consequently easier to handle. The paper is deliberately written from a formal point of view, i.e. without giving a rigorous derivation of the properties deduced.
952:
5767:
5084:
4519:
3931:
3534:
2602:
1353:
1615:
480:
4973:
2407:
4398:
4502:
871:{\displaystyle {{\frac {\partial g}{\partial {\bar {z}}}}(z_{0})}\mathrel {\overset {\mathrm {def} }{=}} \lim _{r\to 0}{\frac {1}{2\pi ir^{2}}}\oint _{\Gamma (z_{0},r)}g(z)\mathrm {d} z,}
4361:
6313:
4464:
3876:
685:
155:
6451:
3453:
3124:
2489:
5716:
6992:"Über die Krümmung von Niveaukurven bei der konformen Abbildung einfachzusammenhängender Gebiete auf das Innere eines Kreises. Eine Verallgemeinerung eines Satzes von E. Study."
7291:
3374:
3335:
143:
2589:
1145:
Despite their ubiquitous use, it seems that there is no text listing all the properties of
Wirtinger derivatives: however, fairly complete references are the short course on
7651:, Graduate Texts in Mathematics, vol. 122 (Fourth corrected 1998 printing ed.), New York–Berlin–Heidelberg–Barcelona–Hong Kong–London–Milan–Paris–Singapore–Tokyo:
6349:
3479:
5754:
5067:
3914:
3521:
360:
6517:
aspects: however, in the introductory sections, Wirtinger derivatives and a few other analytical tools are introduced and their application to the theory is described.
1071:
3293:
6856:(1911), "Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse",
1293:
6603:
6551:
2523:
2361:
1572:
1016:
6379:
640:
3262:
386:
6403:
6143:
2451:
1333:
1313:
980:
406:
306:
7136:
7094:
6655:
1182:
7605:
7265:
6692:" (free translation of the title) is the first paper where a set of (fairly complicate) necessary and sufficient conditions for the solvability of the
6893:
On the hypersurfaces of the 4-dimensional space that can be the boundary of the domain of existence of an analytic function of two complex variables
3021:{\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}},{\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}}
6556:: in this single aspect, their approach is different from the one adopted by the other authors cited in this section, and perhaps more complete.
6858:
6796:
7711:
7686:
7525:
6719:
3133:
6750:
7720:
Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the
Istituto Nazionale di Alta Matematica in Rome
7660:
7578:
7484:
7442:
7398:
7356:
7699:
Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto
Nazionale di Alta Matematica in Roma
6035:
Some of the basic properties of
Wirtinger derivatives are the same ones as the properties characterizing the ordinary (or partial)
7607:
Introduzione elementare alla teoria delle funzioni di variabili complesse con particolare riguardo alle rappresentazioni integrali
3034:
7741:
7610:, Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni (in Italian), vol. 67, Rome:
7270:, Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni (in Italian), vol. 24, Rome:
7746:
7631:
7611:
7382:
7334:
7271:
6574:
6219:
4241:
3296:
2366:
1577:
68:
7200:
7628:
Elementary introduction to the theory of functions of complex variables with particular regard to integral representations
7562:
7424:
4209:
1850:
1339:
884:
516:
44:
7247:
6189:
4286:
1134:
6095:
4246:
1582:
1098:
414:
2371:
643:
76:
52:
7305:(1986), "Unification of global and local existence theorems for holomorphic functions of several complex variables",
6935:" (free English translation of the title) is the first paper where a sufficient condition for the solvability of the
1102:
4978:
6911:
4925:
4469:
7736:
7243:
6940:
6896:
6834:
6697:
6510:
6223:
6171:
4229:
1146:
1122:
548:
496:
40:
24:
6278:
3389:
2526:
278:{\displaystyle {\begin{cases}x_{k}+iy_{k}=z_{k}\\x_{k}-iy_{k}=u_{k}\end{cases}}\qquad 1\leqslant k\leqslant n.}
60:
7475:, North–Holland Mathematical Library, vol. 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo:
6526:
In this work, the authors prove some of the properties of
Wirtinger derivatives also for the general case of
7615:
7275:
7172:, International Series of Monographs in Pure and Applied Mathematics, vol. 25, London–Paris–Frankfurt:
6497:
6109:
4418:
4366:
3830:
1034:
652:
603:
560:
64:
6419:
4329:
3410:
3094:
2460:
6553:
6410:
6150:
6091:
6000:
5676:
1086:
1026:
320:
7714:(which at present bears his name), containing appendices of Enzo Martinelli, Giovanni Battista Rizza and
7290:
is a short course in the theory of functions of several complex variables, held in
February 1972 at the
7205:
6996:
6780:
6485:
6231:
6170:) of Osgood's 1913 paper contains much important historical information on the early development of the
6040:
2414:
1622:
1126:
1078:
688:
72:
6991:
3347:
3308:
119:
7715:
7566:
7434:
7134:(1913), "Sur une classe de fonctions d'une variable complexe et sur certaines équations intégrales",
6846:
6620:
6493:
6489:
2532:
2422:
2332:
1630:
1543:
983:
56:
6318:
3458:
2086:
1865:
164:
7390:
6947:
6842:
1130:
955:
692:
647:
564:
556:
540:
520:
6916:
Rendiconti della
Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
6792:"Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse"
5721:
5034:
3881:
3488:
7230:
7153:
7119:
7029:
6955:
6907:
6883:
6821:
6680:
6163:
6146:
5995:
5985:
3393:
2454:
959:
576:
325:
313:
7307:
Memorie della
Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
7682:
7656:
7574:
7521:
7517:
7480:
7468:
7438:
7394:
7352:
7196:
7053:
6853:
6787:
6715:
6693:
6464:
6382:
6057:
5990:
4324:
1114:
1040:
1023:
572:
552:
146:
109:
36:
7570:
7556:
7433:, Wiley Classics Library (Reprint ed.), New York–Chichester–Brisbane–Toronto–Singapore:
3278:
7702:
7690:
7674:
7642:
7592:
7539:
7498:
7456:
7412:
7370:
7344:
7322:
7222:
7214:
7185:
7145:
7131:
7111:
7103:
7089:
7077:
7067:
7058:
7037:
7013:
7005:
6975:
6959:
6923:
6875:
6867:
6813:
6805:
6767:
6733:
6672:
6664:
6406:
6207:
6065:
4213:
1263:
611:
592:
544:
532:
113:
93:
20:
7670:
7588:
7535:
7494:
7452:
7408:
7366:
7318:
7181:
7025:
6971:
6831:
Studies on essential singular points of analytic functions of two or more complex variables
6779:" (free English translation of the title) is an important reference paper in the theory of
6763:
6729:
6581:
6529:
2494:
2339:
1550:
989:
7706:
7678:
7666:
7652:
7596:
7584:
7543:
7531:
7502:
7490:
7460:
7448:
7416:
7404:
7374:
7362:
7326:
7314:
7302:
7226:
7189:
7177:
7115:
7081:
7041:
7021:
7017:
6979:
6967:
6963:
6927:
6879:
6817:
6771:
6759:
6745:
6737:
6725:
6711:
6676:
6578:
6355:
4401:
3303:
2336:
1847:
1649:
1547:
1336:
1082:
616:
7565:
Series of
Monographs and Advanced Texts, New York–Chichester–Brisbane–Toronto–Singapore:
3241:
365:
1165:, p. 2,4) which are used as general references in this and the following sections.
75:
for such functions that is entirely analogous to the ordinary differential calculus for
7552:
7261:
7173:
6936:
6481:
6414:
6388:
6128:
3482:
3127:
2436:
2418:
1626:
1318:
1298:
965:
600:
528:
391:
291:
6707:
Nevanlinna's theory of value distribution: the second main theorem and its error terms
7730:
7387:
Introduction to
Holomorphic Functions of Several Variables. Volume I: Function Theory
7348:
7234:
7157:
7123:
7033:
6887:
6825:
6684:
6646:
3341:
1653:
1253:{\displaystyle \mathbb {C} \equiv \mathbb {R} ^{2}=\{(x,y)\mid x,y\in \mathbb {R} \}}
1177:
1022:. This is evidently an alternative definition of Wirtinger derivative respect to the
607:
568:
7701:(in Italian), Padova: CEDAM – Casa Editrice Dott. Antonio Milani, pp. XIV+255,
7630:" (English translation of the title) are the notes form a course, published by the
6987:
6838:
6514:
3917:
1129:, the form of all the differential operators commonly used in the theory, like the
495:, p. 112). Apparently, this paper was not noticed by early researchers in the
7242:. In this important paper, Wirtinger introduces several important concepts in the
7646:
7511:
7428:
7338:
6705:
3338:
3300:
3265:
2331:
As for Wirtinger derivatives for functions of one complex variable, the natural
317:
7558:
Methods of Complex Analysis in Partial Differential Equations and Applications
7165:
6036:
5070:
4505:
4237:
4228:
This property takes two different forms respectively for functions of one and
4217:
3031:
The second Wirtinger derivative is also related with complex differentiation;
1090:
48:
7292:
Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni "
6103:
2335:
of definition of these partial differential operators is again the space of
1154:
1097:
uses this newly defined concept in order to introduce his generalization of
524:
7056:(1899), "Sur les propriétés du potentiel et sur les fonctions Abéliennes",
6627: = 1: however, the resulting formulas are formally very similar.
7476:
6315:
6125:, p. 112), equation 2': note that, throughout the paper, the symbol
591:, p. 294), a new step in the definition of the concept was taken by
47:
of the first order which behave in a very similar manner to the ordinary
1113:
The first systematic introduction of Wirtinger derivatives seems due to
7218:
7201:"Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen"
7149:
7107:
7072:
7009:
6895:" (English translation of the title) is another important paper in the
6871:
6809:
6668:
3231:{\displaystyle z\equiv (x,y)=(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})}
1546:
of definition of these partial differential operators is the space of
491:
below: to see this it is sufficient to compare equations 2 and 2' of (
7267:
Introduzione all'analisi complessa (Lezioni tenute nel febbraio 1972)
6227:
3273:
2410:
1618:
1121:
in order to simplify the calculations of quantities occurring in the
1019:
6791:
6650:
3091:
In the present section and in the following ones it is assumed that
1033:, p. 294), the limit may exist for functions that are not even
7516:, de Gruyter Studies in Mathematics, vol. 3, Berlin–New York:
7239:
7046:
6484:
uses the properties of Wirtinger derivatives in order to prove the
6833:" (English translation of the title) is an important paper in the
6500:: this purpose is common to all references cited in this section.
3083:
is equivalent to the Cauchy-Riemann equations in a complex form.
583:
The work of Dimitrie Pompeiu in 1912 and 1913: a new formulation
39:
who introduced them in 1927 in the course of his studies on the
7092:(1912), "Sur une classe de fonctions d'une variable complexe",
6952:
Topics in the theory of functions of several complex variables
6748:(1969), "Derivata areolare e funzioni a variazione limitata",
108:, pp. 66–67). In the third paragraph of his 1899 paper,
3076:{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}=0}
2919:
where the third equality uses the Cauchy-Riemann equations
2312:
2073:
252:
1335:). The Wirtinger derivatives are defined as the following
7473:
An Introduction to Complex Analysis in Several Variables
4240:
in its full generality it is necessary to consider two
1846:
The Wirtinger derivatives are defined as the following
7710:. Notes from a course held by Francesco Severi at the
6777:
Areolar derivative and functions of bounded variation
6584:
6532:
6422:
6391:
6358:
6321:
6281:
6131:
6068:
does not cite any reference to support his assertion.
5765:
5724:
5679:
5082:
5037:
4981:
4928:
4517:
4472:
4421:
4369:
4332:
4289:
4249:
4208:
This property implies that Wirtinger derivatives are
3929:
3884:
3833:
3532:
3491:
3461:
3413:
3350:
3311:
3281:
3244:
3136:
3097:
3037:
2925:
2600:
2535:
2497:
2463:
2439:
2374:
2342:
1859:
1661:
1585:
1553:
1351:
1321:
1301:
1266:
1185:
1043:
992:
968:
887:
703:
655:
619:
417:
394:
368:
328:
294:
158:
122:
947:{\displaystyle \Gamma (z_{0},r)=\partial D(z_{0},r)}
7430:
Applied and Computational Complex Analysis Volume 3
6751:
Revue Roumaine de Mathématiques Pures et Appliquées
6619:, p. 6, as already pointed out, consider only
6098:
defining them, i.e. the operator in equation 2 of (
4316:{\displaystyle \Omega ''\subseteq \mathbb {C} ^{p}}
288:Then he writes the equation defining the functions
7389:, Wadsworth & Brooks/Cole Mathematics Series,
6941:holomorphic functions of several complex variables
6597:
6545:
6463:With or without the attribution of the concept to
6445:
6397:
6373:
6343:
6307:
6214:, p. 27) remarks, the original definition of
6137:
5968:
5748:
5710:
5654:
5061:
5023:
4967:
4899:
4496:
4458:
4392:
4355:
4315:
4276:{\displaystyle \Omega '\subseteq \mathbb {C} ^{m}}
4275:
4197:
3908:
3870:
3808:
3515:
3473:
3447:
3368:
3329:
3287:
3256:
3230:
3118:
3075:
3020:
2908:
2583:
2517:
2483:
2445:
2401:
2355:
2321:
1838:
1610:{\displaystyle \Omega \subseteq \mathbb {R} ^{2},}
1609:
1566:
1531:
1327:
1307:
1287:
1252:
1065:
1010:
974:
946:
870:
679:
634:
475:{\displaystyle {\frac {d^{2}V}{dz_{k}\,du_{q}}}=0}
474:
400:
380:
354:
300:
277:
137:
88:Early days (1899–1911): the work of Henri Poincaré
7718:. An English translation of the title reads as:-"
6933:On the functions of two or more complex variables
6912:"Sulle funzioni di due o più variabili complesse"
6260:, p. 294 for one example of such a function.
2402:{\displaystyle \Omega \subset \mathbb {R} ^{2n},}
1029:: it is a more general one, since, as noted a by
549:holomorphic function of several complex variables
7393:: Wadsworth & Brooks/Cole, pp. xx+203,
7244:theory of functions of several complex variables
6897:theory of functions of several complex variables
6837:, where the problem of determining what kind of
6835:theory of functions of several complex variables
6511:theory of functions of several complex variables
6172:theory of functions of several complex variables
1123:theory of functions of several complex variables
770:
497:theory of functions of several complex variables
41:theory of functions of several complex variables
7693:including many historical notes on the subject.
7340:Analytic Functions of Several Complex Variables
6954:(unabridged and corrected ed.), New York:
6899:, investigating further the theory started in (
6153:, instead of the now commonplace symbol ∂.
71:. These operators permit the construction of a
6710:, Springer Monographs in Mathematics, Berlin:
6023:
5024:{\displaystyle f\in C^{1}(\Omega ,\Omega ''),}
519:of the theory are expressed directly by using
6467:: see, for example, the well known monograph
8:
7137:Rendiconti del Circolo Matematico di Palermo
7095:Rendiconti del Circolo Matematico di Palermo
6656:Rendiconti del Circolo Matematico di Palermo
6616:
4968:{\displaystyle g\in C^{1}(\Omega ',\Omega )}
4497:{\displaystyle g(\Omega )\subseteq \Omega ,}
3272: ≥ 1: also it is assumed that the
1247:
1209:
1158:
7343:, Prentice-Hall series in Modern Analysis,
6039:and used for the construction of the usual
3388:and of the corresponding properties of the
1260:(in a sense of expressing a complex number
1125:: as a result of the introduction of these
500:
7513:Holomorphic functions of several variables
6698:holomorphic functions of several variables
7246:, namely Wirtinger's derivatives and the
7071:
6612:
6589:
6583:
6537:
6531:
6468:
6428:
6427:
6421:
6390:
6357:
6326:
6320:
6308:{\displaystyle \partial _{\bar {z}}w\in }
6287:
6286:
6280:
6130:
5953:
5933:
5932:
5926:
5901:
5890:
5889:
5874:
5868:
5855:
5844:
5843:
5826:
5825:
5819:
5794:
5776:
5770:
5766:
5764:
5723:
5690:
5678:
5639:
5628:
5627:
5615:
5604:
5603:
5596:
5576:
5565:
5564:
5549:
5538:
5527:
5511:
5500:
5499:
5487:
5477:
5457:
5439:
5428:
5417:
5378:
5367:
5366:
5356:
5343:
5328:
5317:
5316:
5309:
5289:
5278:
5277:
5262:
5251:
5240:
5224:
5209:
5199:
5179:
5161:
5150:
5139:
5100:
5087:
5083:
5081:
5036:
4992:
4980:
4939:
4927:
4879:
4878:
4862:
4861:
4855:
4830:
4829:
4815:
4793:
4792:
4778:
4747:
4706:
4705:
4696:
4670:
4669:
4663:
4638:
4637:
4623:
4595:
4564:
4522:
4518:
4516:
4471:
4438:
4420:
4368:
4331:
4307:
4303:
4302:
4288:
4267:
4263:
4262:
4248:
4182:
4171:
4170:
4155:
4131:
4120:
4119:
4104:
4073:
4062:
4061:
4051:
4038:
4020:
3996:
3978:
3947:
3934:
3930:
3928:
3883:
3850:
3832:
3793:
3782:
3781:
3766:
3751:
3740:
3739:
3724:
3680:
3669:
3668:
3658:
3645:
3627:
3612:
3594:
3550:
3537:
3533:
3531:
3490:
3460:
3430:
3412:
3357:
3353:
3352:
3349:
3318:
3314:
3313:
3310:
3280:
3243:
3219:
3200:
3187:
3168:
3135:
3110:
3106:
3105:
3096:
3053:
3052:
3038:
3036:
2998:
2972:
2949:
2926:
2924:
2882:
2859:
2833:
2798:
2775:
2749:
2723:
2708:
2673:
2647:
2632:
2605:
2601:
2599:
2534:
2504:
2496:
2470:
2462:
2438:
2387:
2383:
2382:
2373:
2347:
2341:
2295:
2282:
2267:
2254:
2239:
2227:
2216:
2215:
2205:
2179:
2166:
2151:
2138:
2123:
2111:
2100:
2099:
2089:
2081:
2056:
2043:
2028:
2015:
2000:
1988:
1975:
1949:
1936:
1921:
1908:
1893:
1881:
1868:
1860:
1858:
1822:
1818:
1817:
1808:
1800:
1786:
1767:
1754:
1735:
1713:
1705:
1683:
1679:
1678:
1668:
1664:
1663:
1660:
1598:
1594:
1593:
1584:
1558:
1552:
1505:
1484:
1469:
1448:
1447:
1438:
1414:
1393:
1378:
1356:
1352:
1350:
1320:
1300:
1265:
1243:
1242:
1200:
1196:
1195:
1187:
1186:
1184:
1150:
1118:
1077:, p. 28), the first to identify the
1054:
1042:
991:
967:
929:
898:
886:
857:
828:
817:
804:
785:
773:
756:
751:
741:
720:
719:
705:
704:
702:
670:
669:
660:
654:
618:
567:, he follows the established practice of
457:
449:
443:
425:
418:
416:
393:
367:
346:
333:
327:
293:
243:
230:
214:
200:
187:
171:
159:
157:
129:
125:
124:
121:
6566:
6122:
6099:
6078:
6061:
6053:
3376:All the proofs are easy consequences of
2417:, they can be readily extended to every
1625:, they can be readily extended to every
1162:
492:
101:
97:
6570:
6257:
6215:
6211:
6203:
6187:, pp. 23–24): curiously, he calls
6019:
6011:
4236: > 1 case, to express the
1094:
1074:
1030:
596:
588:
512:
105:
6859:Annali di Matematica Pura ed Applicata
6797:Annali di Matematica Pura ed Applicata
6244:
6184:
6167:
4459:{\displaystyle f,g\in C^{1}(\Omega ),}
4393:{\displaystyle f:\Omega \to \Omega ''}
3871:{\displaystyle f,g\in C^{1}(\Omega ),}
1161:, pp. 3–6), and the monograph of
680:{\displaystyle z_{0}\in \mathbb {C} ,}
536:
535:involved. In the long survey paper by
7712:Istituto Nazionale di Alta Matematica
7510:Kaup, Ludger; Kaup, Burchard (1983),
6446:{\displaystyle \partial _{\bar {z}}w}
6270:
4356:{\displaystyle g:\Omega '\to \Omega }
4216:point of view, exactly like ordinary
3448:{\displaystyle f,g\in C^{1}(\Omega )}
3119:{\displaystyle z\in \mathbb {C} ^{n}}
2484:{\displaystyle \partial f/\partial z}
2457:at a point, the Wirtinger derivative
2429:Relation with complex differentiation
2409:and again, since these operators are
485:This implies that he implicitly used
7:
6900:
5711:{\displaystyle f\in C^{1}(\Omega ),}
508:
504:
7248:tangential Cauchy-Riemann condition
6174:, and is therefore a useful source.
2491:agrees with the complex derivative
96:at least as early as in the paper (
92:Wirtinger derivatives were used in
7634:, held by Martinelli when he was "
6424:
6335:
6283:
6206:, p. 294) in his approach to
5946:
5929:
5885:
5877:
5839:
5822:
5787:
5779:
5699:
5623:
5599:
5560:
5552:
5495:
5480:
5450:
5442:
5362:
5358:
5336:
5312:
5273:
5265:
5217:
5202:
5172:
5164:
5093:
5089:
5008:
5001:
4959:
4949:
4875:
4858:
4826:
4818:
4789:
4781:
4758:
4750:
4702:
4698:
4683:
4666:
4634:
4626:
4606:
4598:
4575:
4567:
4528:
4524:
4488:
4479:
4447:
4383:
4376:
4350:
4340:
4291:
4251:
4166:
4158:
4115:
4107:
4057:
4053:
4031:
4023:
3989:
3981:
3940:
3936:
3859:
3777:
3769:
3735:
3727:
3664:
3660:
3638:
3630:
3605:
3597:
3543:
3539:
3439:
3282:
3049:
3041:
3009:
3001:
2983:
2975:
2960:
2952:
2937:
2929:
2870:
2862:
2844:
2836:
2809:
2801:
2786:
2778:
2760:
2752:
2734:
2726:
2684:
2676:
2658:
2650:
2616:
2608:
2475:
2464:
2375:
2288:
2284:
2260:
2256:
2211:
2207:
2172:
2168:
2144:
2140:
2095:
2091:
2049:
2045:
2021:
2017:
1981:
1977:
1942:
1938:
1914:
1910:
1874:
1870:
1586:
1511:
1507:
1490:
1486:
1444:
1440:
1420:
1416:
1399:
1395:
1362:
1358:
916:
888:
858:
818:
763:
760:
757:
716:
708:
14:
7614:, pp. 236+II, archived from
4408:Functions of one complex variable
3369:{\displaystyle \mathbb {C} ^{n}.}
3330:{\displaystyle \mathbb {R} ^{2n}}
1169:Functions of one complex variable
1147:multidimensional complex analysis
7288:Introduction to complex analysis
6509:This is a classical work on the
6409:, the latter being equal to the
6202:This is the definition given by
2591:which is complex differentiable
1809:
1801:
1714:
1706:
138:{\displaystyle \mathbb {C} ^{n}}
6651:"Sopra un problema al contorno"
6269:See also the excellent book by
6102:, p. 112), is exactly the
2584:{\displaystyle f(z)=u(z)+iv(z)}
2197:
2080:
1967:
1617:but, since these operators are
408:, exactly in the following way
256:
7632:Accademia Nazionale dei Lincei
7612:Accademia Nazionale dei Lincei
7272:Accademia Nazionale dei Lincei
7170:Generalized Analytic Functions
6577:considers the general case of
6433:
6368:
6362:
6352:, p > 1, then the function
6344:{\displaystyle L_{p}(\Omega )}
6338:
6332:
6292:
6275:If the generalized derivative
6090:These functions are precisely
5938:
5895:
5849:
5831:
5756:the following equalities hold
5702:
5696:
5633:
5609:
5570:
5505:
5372:
5322:
5283:
5015:
4998:
4962:
4945:
4884:
4867:
4835:
4798:
4732:
4720:
4711:
4675:
4643:
4549:
4537:
4482:
4476:
4450:
4444:
4379:
4347:
4176:
4125:
4094:
4082:
4067:
3968:
3956:
3862:
3856:
3787:
3745:
3674:
3523:the following equalities hold
3474:{\displaystyle \alpha ,\beta }
3442:
3436:
3225:
3161:
3155:
3143:
3058:
2578:
2572:
2560:
2554:
2545:
2539:
2221:
2105:
1851:partial differential operators
1453:
1340:partial differential operators
1224:
1212:
1002:
996:
941:
922:
910:
891:
854:
848:
840:
821:
777:
747:
734:
725:
629:
623:
517:partial differential operators
45:partial differential operators
1:
7563:Canadian Mathematical Society
6405:a derivative in the sense of
6273:, p. 55), Theorem 1.31:
1109:The work of Wilhelm Wirtinger
7274:, p. 34, archived from
6096:linear differential operator
5914:
5807:
5749:{\displaystyle i=1,\dots ,n}
5062:{\displaystyle i=1,\dots ,m}
3909:{\displaystyle i=1,\dots ,n}
3516:{\displaystyle i=1,\dots ,n}
2529:. For the complex function
7648:Theory of Complex Functions
6704:Cherry, W.; Ye, Z. (2001),
6690:On a boundary value problem
6611: = 1. References
3384:
3378:
555:: as a matter of fact when
539:(first published in 1913),
487:
355:{\displaystyle x_{k},y_{q}}
312:, previously written using
77:functions of real variables
16:Concept in complex analysis
7763:
7697:Severi, Francesco (1958),
6024:Kracht & Kreyszig 1988
5069:the following form of the
1093:. In his following paper,
982:entirely contained in the
7604:Martinelli, Enzo (1984),
6480:In this course lectures,
4230:several complex variables
2525:. This follows from the
1159:Gunning & Rossi (1965
1099:Cauchy's integral formula
61:antiholomorphic functions
25:several complex variables
6617:Gunning & Rossi 1965
6513:dealing mainly with its
6234:for further information.
6190:Cauchy–Riemann equations
4915:> 1 complex variables
2527:Cauchy-Riemann equations
1641:> 1 complex variables
1066:{\displaystyle z=z_{0}.}
65:differentiable functions
6862:, s. III (in Italian),
6800:, s. III (in Italian),
6147:partial differentiation
6092:pluriharmonic functions
6060:is precisely stated by
3295:can be thought of as a
3288:{\displaystyle \Omega }
1135:Cauchy–Riemann operator
604:differentiable function
100:), as briefly noted by
31:(sometimes also called
21:complex analysis of one
7742:Differential operators
7337:; Rossi, Hugo (1965),
6599:
6547:
6447:
6411:Generalized derivative
6399:
6375:
6345:
6309:
6193:this set of equations.
6139:
6110:pluriharmonic operator
6062:Cherry & Ye (2001)
6052:Reference to the work
6001:Pluriharmonic function
5970:
5750:
5712:
5656:
5543:
5433:
5256:
5155:
5063:
5025:
4969:
4901:
4498:
4460:
4394:
4357:
4317:
4277:
4199:
3910:
3872:
3810:
3517:
3475:
3449:
3370:
3331:
3289:
3258:
3232:
3120:
3077:
3022:
2910:
2585:
2519:
2485:
2455:complex differentiable
2447:
2403:
2357:
2323:
1840:
1611:
1568:
1533:
1329:
1309:
1289:
1288:{\displaystyle z=x+iy}
1254:
1127:differential operators
1103:Cauchy–Pompeiu formula
1067:
1012:
976:
948:
872:
681:
636:
561:pluriharmonic operator
476:
402:
382:
356:
302:
279:
139:
7747:Mathematical analysis
7567:John Wiley & Sons
7435:John Wiley & Sons
7255:Scientific references
7206:Mathematische Annalen
6997:Mathematische Annalen
6640:Historical references
6600:
6598:{\displaystyle C^{1}}
6569:, p. 4 and also
6548:
6546:{\displaystyle C^{1}}
6494:holomorphic functions
6448:
6400:
6376:
6346:
6310:
6218:does not require the
6140:
6041:differential calculus
5971:
5751:
5713:
5657:
5523:
5413:
5236:
5135:
5064:
5026:
4970:
4902:
4499:
4461:
4395:
4358:
4318:
4278:
4200:
3911:
3873:
3811:
3518:
3476:
3450:
3371:
3332:
3290:
3259:
3233:
3121:
3078:
3023:
2911:
2586:
2520:
2518:{\displaystyle df/dz}
2486:
2448:
2423:generalized functions
2415:constant coefficients
2404:
2358:
2356:{\displaystyle C^{1}}
2324:
1841:
1631:generalized functions
1623:constant coefficients
1612:
1569:
1567:{\displaystyle C^{1}}
1542:Clearly, the natural
1534:
1330:
1310:
1290:
1255:
1163:Kaup & Kaup (1983
1153:, pp. 3–5), the
1068:
1013:
1011:{\displaystyle g(z),}
977:
949:
873:
682:
637:
543:with respect to each
477:
403:
383:
357:
303:
280:
140:
104:, p. 31) and by
102:Cherry & Ye (2001
73:differential calculus
57:holomorphic functions
29:Wirtinger derivatives
7351:, pp. xiv+317,
7176:, pp. xxx+668,
6948:Osgood, William Fogg
6847:domain of holomorphy
6714:, pp. XII+202,
6582:
6567:Kaup & Kaup 1983
6530:
6420:
6389:
6374:{\displaystyle w(z)}
6356:
6319:
6279:
6129:
5763:
5722:
5677:
5080:
5035:
4979:
4926:
4515:
4470:
4419:
4367:
4330:
4287:
4247:
3927:
3882:
3831:
3530:
3489:
3459:
3411:
3348:
3309:
3279:
3242:
3134:
3095:
3035:
2923:
2598:
2533:
2495:
2461:
2437:
2372:
2340:
1857:
1659:
1583:
1551:
1349:
1319:
1299:
1264:
1183:
1041:
990:
984:domain of definition
966:
885:
701:
653:
635:{\displaystyle g(z)}
617:
551:seem to be meant as
415:
392:
366:
326:
316:with respect to the
292:
156:
120:
51:with respect to one
7655:, pp. xx+453,
7520:, pp. XV+349,
7391:Belmont, California
6958:, pp. IV+120,
6908:Levi-Civita, Tullio
6781:areolar derivatives
6149:respect to a given
6145:is used to signify
6081:, pp. 111–114)
3257:{\displaystyle x,y}
541:partial derivatives
521:partial derivatives
499:: in the papers of
381:{\displaystyle k,q}
314:partial derivatives
33:Wirtinger operators
7437:, pp. X+637,
7383:Gunning, Robert C.
7335:Gunning, Robert C.
7219:10.1007/BF01447872
7197:Wirtinger, Wilhelm
7150:10.1007/BF03015607
7108:10.1007/BF03015292
7073:10.1007/BF02417872
7010:10.1007/BF01455902
6918:, 5 (in Italian),
6872:10.1007/BF02420535
6854:Levi, Eugenio Elia
6810:10.1007/BF02419336
6788:Levi, Eugenio Elia
6669:10.1007/BF03015289
6595:
6543:
6443:
6395:
6371:
6341:
6305:
6232:areolar derivative
6135:
5996:Dolbeault operator
5966:
5964:
5746:
5708:
5652:
5650:
5059:
5021:
4965:
4897:
4895:
4494:
4456:
4390:
4353:
4313:
4273:
4195:
4193:
3906:
3868:
3806:
3804:
3513:
3471:
3445:
3366:
3327:
3285:
3254:
3228:
3116:
3073:
3018:
2906:
2904:
2581:
2515:
2481:
2443:
2399:
2353:
2319:
2311:
2072:
1836:
1607:
1564:
1529:
1527:
1325:
1305:
1285:
1250:
1079:areolar derivative
1063:
1018:i.e. his bounding
1008:
972:
944:
868:
784:
689:areolar derivative
677:
632:
553:formal derivatives
501:Levi-Civita (1905)
472:
398:
388:ranging from 1 to
378:
352:
298:
275:
251:
135:
112:first defines the
55:, when applied to
7687:978-0-387-97195-7
7643:Remmert, Reinhold
7636:Professore Linceo
7551:Kracht, Manfred;
7527:978-3-11-004150-7
7518:Walter de Gruyter
7240:DigiZeitschriften
7047:DigiZeitschriften
6721:978-3-540-66416-1
6694:Dirichlet problem
6465:Wilhelm Wirtinger
6436:
6398:{\displaystyle G}
6383:almost everywhere
6295:
6256:See problem 2 in
6245:Formal definition
6243:See the section "
6138:{\displaystyle d}
6022:, p. 62 and
5991:Dolbeault complex
5960:
5941:
5917:
5908:
5898:
5862:
5852:
5834:
5810:
5801:
5646:
5636:
5612:
5583:
5573:
5518:
5508:
5464:
5385:
5375:
5350:
5325:
5296:
5286:
5231:
5186:
5107:
4891:
4887:
4870:
4842:
4838:
4805:
4801:
4765:
4718:
4714:
4690:
4678:
4650:
4646:
4613:
4582:
4535:
4189:
4179:
4138:
4128:
4080:
4070:
4045:
4003:
3954:
3800:
3790:
3758:
3748:
3687:
3677:
3652:
3619:
3557:
3065:
3061:
3016:
2990:
2967:
2944:
2900:
2877:
2851:
2816:
2793:
2767:
2741:
2716:
2691:
2665:
2640:
2623:
2446:{\displaystyle f}
2302:
2274:
2247:
2234:
2224:
2186:
2158:
2131:
2118:
2108:
2063:
2035:
2008:
1995:
1956:
1928:
1901:
1888:
1518:
1497:
1477:
1460:
1456:
1427:
1406:
1386:
1369:
1328:{\displaystyle y}
1308:{\displaystyle x}
1295:for real numbers
1141:Formal definition
1115:Wilhelm Wirtinger
1101:, the now called
1024:complex conjugate
975:{\displaystyle r}
811:
769:
767:
732:
728:
691:as the following
606:(in the sense of
533:complex variables
464:
401:{\displaystyle n}
301:{\displaystyle V}
147:complex conjugate
37:Wilhelm Wirtinger
7754:
7737:Complex analysis
7709:
7691:complex analysis
7689:. A textbook on
7681:
7625:
7624:
7623:
7599:
7546:
7505:
7463:
7419:
7377:
7345:Englewood Cliffs
7329:
7303:Fichera, Gaetano
7285:
7284:
7283:
7237:
7192:
7160:
7126:
7084:
7075:
7059:Acta Mathematica
7044:
6982:
6930:
6890:
6828:
6774:
6746:Fichera, Gaetano
6740:
6687:
6628:
6621:holomorphic maps
6615:, p. 5 and
6604:
6602:
6601:
6596:
6594:
6593:
6563:
6557:
6552:
6550:
6549:
6544:
6542:
6541:
6524:
6518:
6507:
6501:
6478:
6472:
6461:
6455:
6452:
6450:
6449:
6444:
6439:
6438:
6437:
6429:
6413:in the sense of
6404:
6402:
6401:
6396:
6380:
6378:
6377:
6372:
6350:
6348:
6347:
6342:
6331:
6330:
6314:
6312:
6311:
6306:
6298:
6297:
6296:
6288:
6267:
6261:
6254:
6248:
6247:" of this entry.
6241:
6235:
6230:. See the entry
6200:
6194:
6181:
6175:
6160:
6154:
6144:
6142:
6141:
6136:
6119:
6113:
6088:
6082:
6075:
6069:
6066:Reinhold Remmert
6050:
6044:
6033:
6027:
6016:
5975:
5973:
5972:
5967:
5965:
5961:
5959:
5958:
5957:
5944:
5943:
5942:
5934:
5927:
5918:
5913:
5909:
5907:
5906:
5905:
5900:
5899:
5891:
5883:
5875:
5869:
5863:
5861:
5860:
5859:
5854:
5853:
5845:
5837:
5836:
5835:
5827:
5820:
5811:
5806:
5802:
5800:
5799:
5798:
5785:
5777:
5771:
5755:
5753:
5752:
5747:
5717:
5715:
5714:
5709:
5695:
5694:
5672:
5661:
5659:
5658:
5653:
5651:
5647:
5645:
5644:
5643:
5638:
5637:
5629:
5621:
5620:
5619:
5614:
5613:
5605:
5597:
5595:
5591:
5584:
5582:
5581:
5580:
5575:
5574:
5566:
5558:
5550:
5542:
5537:
5519:
5517:
5516:
5515:
5510:
5509:
5501:
5493:
5492:
5491:
5478:
5476:
5472:
5465:
5463:
5462:
5461:
5448:
5440:
5432:
5427:
5405:
5401:
5386:
5384:
5383:
5382:
5377:
5376:
5368:
5357:
5351:
5349:
5348:
5347:
5334:
5333:
5332:
5327:
5326:
5318:
5310:
5308:
5304:
5297:
5295:
5294:
5293:
5288:
5287:
5279:
5271:
5263:
5255:
5250:
5232:
5230:
5229:
5228:
5215:
5214:
5213:
5200:
5198:
5194:
5187:
5185:
5184:
5183:
5170:
5162:
5154:
5149:
5127:
5123:
5108:
5106:
5105:
5104:
5088:
5068:
5066:
5065:
5060:
5030:
5028:
5027:
5022:
5014:
4997:
4996:
4974:
4972:
4971:
4966:
4955:
4944:
4943:
4921:
4906:
4904:
4903:
4898:
4896:
4892:
4890:
4889:
4888:
4880:
4873:
4872:
4871:
4863:
4856:
4854:
4850:
4843:
4841:
4840:
4839:
4831:
4824:
4816:
4806:
4804:
4803:
4802:
4794:
4787:
4779:
4777:
4773:
4766:
4764:
4756:
4748:
4719:
4717:
4716:
4715:
4707:
4697:
4691:
4689:
4681:
4680:
4679:
4671:
4664:
4662:
4658:
4651:
4649:
4648:
4647:
4639:
4632:
4624:
4614:
4612:
4604:
4596:
4594:
4590:
4583:
4581:
4573:
4565:
4536:
4534:
4523:
4503:
4501:
4500:
4495:
4465:
4463:
4462:
4457:
4443:
4442:
4414:
4399:
4397:
4396:
4391:
4389:
4362:
4360:
4359:
4354:
4346:
4322:
4320:
4319:
4314:
4312:
4311:
4306:
4297:
4282:
4280:
4279:
4274:
4272:
4271:
4266:
4257:
4214:abstract algebra
4204:
4202:
4201:
4196:
4194:
4190:
4188:
4187:
4186:
4181:
4180:
4172:
4164:
4156:
4139:
4137:
4136:
4135:
4130:
4129:
4121:
4113:
4105:
4081:
4079:
4078:
4077:
4072:
4071:
4063:
4052:
4046:
4044:
4043:
4042:
4029:
4021:
4004:
4002:
4001:
4000:
3987:
3979:
3955:
3953:
3952:
3951:
3935:
3915:
3913:
3912:
3907:
3877:
3875:
3874:
3869:
3855:
3854:
3826:
3815:
3813:
3812:
3807:
3805:
3801:
3799:
3798:
3797:
3792:
3791:
3783:
3775:
3767:
3759:
3757:
3756:
3755:
3750:
3749:
3741:
3733:
3725:
3713:
3709:
3688:
3686:
3685:
3684:
3679:
3678:
3670:
3659:
3653:
3651:
3650:
3649:
3636:
3628:
3620:
3618:
3617:
3616:
3603:
3595:
3583:
3579:
3558:
3556:
3555:
3554:
3538:
3522:
3520:
3519:
3514:
3480:
3478:
3477:
3472:
3454:
3452:
3451:
3446:
3435:
3434:
3406:
3375:
3373:
3372:
3367:
3362:
3361:
3356:
3336:
3334:
3333:
3328:
3326:
3325:
3317:
3294:
3292:
3291:
3286:
3263:
3261:
3260:
3255:
3237:
3235:
3234:
3229:
3224:
3223:
3205:
3204:
3192:
3191:
3173:
3172:
3125:
3123:
3122:
3117:
3115:
3114:
3109:
3087:Basic properties
3082:
3080:
3079:
3074:
3066:
3064:
3063:
3062:
3054:
3047:
3039:
3027:
3025:
3024:
3019:
3017:
3015:
3007:
2999:
2991:
2989:
2981:
2973:
2968:
2966:
2958:
2950:
2945:
2943:
2935:
2927:
2915:
2913:
2912:
2907:
2905:
2901:
2899:
2891:
2883:
2878:
2876:
2868:
2860:
2852:
2850:
2842:
2834:
2826:
2822:
2818:
2817:
2815:
2807:
2799:
2794:
2792:
2784:
2776:
2768:
2766:
2758:
2750:
2742:
2740:
2732:
2724:
2717:
2709:
2701:
2697:
2693:
2692:
2690:
2682:
2674:
2666:
2664:
2656:
2648:
2641:
2633:
2624:
2622:
2614:
2606:
2590:
2588:
2587:
2582:
2524:
2522:
2521:
2516:
2508:
2490:
2488:
2487:
2482:
2474:
2452:
2450:
2449:
2444:
2433:When a function
2408:
2406:
2405:
2400:
2395:
2394:
2386:
2362:
2360:
2359:
2354:
2352:
2351:
2328:
2326:
2325:
2320:
2315:
2314:
2308:
2304:
2303:
2301:
2300:
2299:
2283:
2275:
2273:
2272:
2271:
2255:
2248:
2240:
2235:
2233:
2232:
2231:
2226:
2225:
2217:
2206:
2192:
2188:
2187:
2185:
2184:
2183:
2167:
2159:
2157:
2156:
2155:
2139:
2132:
2124:
2119:
2117:
2116:
2115:
2110:
2109:
2101:
2090:
2076:
2075:
2069:
2065:
2064:
2062:
2061:
2060:
2044:
2036:
2034:
2033:
2032:
2016:
2009:
2001:
1996:
1994:
1993:
1992:
1976:
1962:
1958:
1957:
1955:
1954:
1953:
1937:
1929:
1927:
1926:
1925:
1909:
1902:
1894:
1889:
1887:
1886:
1885:
1869:
1853:of first order:
1845:
1843:
1842:
1837:
1832:
1828:
1827:
1826:
1821:
1812:
1804:
1796:
1792:
1791:
1790:
1772:
1771:
1759:
1758:
1740:
1739:
1722:
1718:
1717:
1709:
1691:
1690:
1682:
1673:
1672:
1667:
1647:
1616:
1614:
1613:
1608:
1603:
1602:
1597:
1573:
1571:
1570:
1565:
1563:
1562:
1538:
1536:
1535:
1530:
1528:
1524:
1520:
1519:
1517:
1506:
1498:
1496:
1485:
1478:
1470:
1461:
1459:
1458:
1457:
1449:
1439:
1433:
1429:
1428:
1426:
1415:
1407:
1405:
1394:
1387:
1379:
1370:
1368:
1357:
1342:of first order:
1334:
1332:
1331:
1326:
1314:
1312:
1311:
1306:
1294:
1292:
1291:
1286:
1259:
1257:
1256:
1251:
1246:
1205:
1204:
1199:
1190:
1175:
1087:sense of Sobolev
1072:
1070:
1069:
1064:
1059:
1058:
1017:
1015:
1014:
1009:
981:
979:
978:
973:
953:
951:
950:
945:
934:
933:
903:
902:
877:
875:
874:
869:
861:
844:
843:
833:
832:
812:
810:
809:
808:
786:
783:
768:
766:
752:
750:
746:
745:
733:
731:
730:
729:
721:
714:
706:
686:
684:
683:
678:
673:
665:
664:
641:
639:
638:
633:
612:complex variable
595:: in the paper (
593:Dimitrie Pompeiu
545:complex variable
515:all fundamental
481:
479:
478:
473:
465:
463:
462:
461:
448:
447:
434:
430:
429:
419:
407:
405:
404:
399:
387:
385:
384:
379:
361:
359:
358:
353:
351:
350:
338:
337:
307:
305:
304:
299:
284:
282:
281:
276:
255:
254:
248:
247:
235:
234:
219:
218:
205:
204:
192:
191:
176:
175:
144:
142:
141:
136:
134:
133:
128:
114:complex variable
94:complex analysis
83:Historical notes
7762:
7761:
7757:
7756:
7755:
7753:
7752:
7751:
7727:
7726:
7725:
7716:Mario Benedicty
7696:
7663:
7653:Springer Verlag
7641:
7621:
7619:
7603:
7581:
7553:Kreyszig, Erwin
7550:
7528:
7509:
7487:
7469:Hörmander, Lars
7467:
7445:
7423:
7401:
7381:
7359:
7333:
7301:
7294:Beniamino Segre
7281:
7279:
7262:Andreotti, Aldo
7260:
7257:
7238:, available at
7195:
7164:
7130:
7088:
7052:
7045:, available at
6986:
6946:
6906:
6852:
6786:
6744:
6722:
6712:Springer Verlag
6703:
6645:
6642:
6636:
6631:
6585:
6580:
6579:
6564:
6560:
6533:
6528:
6527:
6525:
6521:
6515:sheaf theoretic
6508:
6504:
6479:
6475:
6471:, p. 1,23.
6462:
6458:
6423:
6418:
6417:
6387:
6386:
6354:
6353:
6322:
6317:
6316:
6282:
6277:
6276:
6268:
6264:
6255:
6251:
6242:
6238:
6201:
6197:
6182:
6178:
6161:
6157:
6127:
6126:
6120:
6116:
6089:
6085:
6077:See reference (
6076:
6072:
6051:
6047:
6034:
6030:
6018:See references
6017:
6013:
6009:
5982:
5963:
5962:
5949:
5945:
5928:
5919:
5888:
5884:
5876:
5870:
5865:
5864:
5842:
5838:
5821:
5812:
5790:
5786:
5778:
5772:
5761:
5760:
5720:
5719:
5686:
5675:
5674:
5670:
5668:
5649:
5648:
5626:
5622:
5602:
5598:
5563:
5559:
5551:
5548:
5544:
5498:
5494:
5483:
5479:
5453:
5449:
5441:
5438:
5434:
5406:
5391:
5387:
5365:
5361:
5353:
5352:
5339:
5335:
5315:
5311:
5276:
5272:
5264:
5261:
5257:
5220:
5216:
5205:
5201:
5175:
5171:
5163:
5160:
5156:
5128:
5113:
5109:
5096:
5092:
5078:
5077:
5033:
5032:
5007:
4988:
4977:
4976:
4948:
4935:
4924:
4923:
4919:
4917:
4894:
4893:
4874:
4857:
4825:
4817:
4814:
4810:
4788:
4780:
4757:
4749:
4746:
4742:
4735:
4701:
4693:
4692:
4682:
4665:
4633:
4625:
4622:
4618:
4605:
4597:
4574:
4566:
4563:
4559:
4552:
4527:
4513:
4512:
4468:
4467:
4434:
4417:
4416:
4412:
4410:
4400:having natural
4382:
4365:
4364:
4339:
4328:
4327:
4301:
4290:
4285:
4284:
4261:
4250:
4245:
4244:
4226:
4192:
4191:
4169:
4165:
4157:
4118:
4114:
4106:
4097:
4060:
4056:
4048:
4047:
4034:
4030:
4022:
3992:
3988:
3980:
3971:
3943:
3939:
3925:
3924:
3880:
3879:
3846:
3829:
3828:
3824:
3822:
3803:
3802:
3780:
3776:
3768:
3738:
3734:
3726:
3714:
3693:
3689:
3667:
3663:
3655:
3654:
3641:
3637:
3629:
3608:
3604:
3596:
3584:
3563:
3559:
3546:
3542:
3528:
3527:
3487:
3486:
3483:complex numbers
3457:
3456:
3426:
3409:
3408:
3404:
3402:
3351:
3346:
3345:
3312:
3307:
3306:
3304:euclidean space
3277:
3276:
3240:
3239:
3215:
3196:
3183:
3164:
3132:
3131:
3104:
3093:
3092:
3089:
3048:
3040:
3033:
3032:
3008:
3000:
2982:
2974:
2959:
2951:
2936:
2928:
2921:
2920:
2903:
2902:
2892:
2884:
2869:
2861:
2843:
2835:
2824:
2823:
2808:
2800:
2785:
2777:
2759:
2751:
2733:
2725:
2722:
2718:
2699:
2698:
2683:
2675:
2657:
2649:
2646:
2642:
2625:
2615:
2607:
2596:
2595:
2531:
2530:
2493:
2492:
2459:
2458:
2435:
2434:
2431:
2381:
2370:
2369:
2343:
2338:
2337:
2310:
2309:
2291:
2287:
2263:
2259:
2253:
2249:
2214:
2210:
2202:
2201:
2194:
2193:
2175:
2171:
2147:
2143:
2137:
2133:
2098:
2094:
2082:
2071:
2070:
2052:
2048:
2024:
2020:
2014:
2010:
1984:
1980:
1972:
1971:
1964:
1963:
1945:
1941:
1917:
1913:
1907:
1903:
1877:
1873:
1861:
1855:
1854:
1816:
1782:
1763:
1750:
1731:
1730:
1726:
1704:
1700:
1699:
1695:
1677:
1662:
1657:
1656:
1650:Euclidean space
1645:
1643:
1592:
1581:
1580:
1554:
1549:
1548:
1526:
1525:
1510:
1489:
1483:
1479:
1462:
1443:
1435:
1434:
1419:
1398:
1392:
1388:
1371:
1361:
1347:
1346:
1317:
1316:
1297:
1296:
1262:
1261:
1194:
1181:
1180:
1173:
1171:
1151:Andreotti (1976
1143:
1111:
1083:weak derivative
1050:
1039:
1038:
988:
987:
964:
963:
925:
894:
883:
882:
824:
813:
800:
790:
737:
715:
707:
699:
698:
687:he defines the
656:
651:
650:
642:defined in the
615:
614:
585:
529:imaginary parts
523:respect to the
453:
439:
435:
421:
420:
413:
412:
390:
389:
364:
363:
342:
329:
324:
323:
290:
289:
250:
249:
239:
226:
210:
207:
206:
196:
183:
167:
160:
154:
153:
123:
118:
117:
90:
85:
69:complex domains
35:), named after
17:
12:
11:
5:
7760:
7758:
7750:
7749:
7744:
7739:
7729:
7728:
7724:
7723:
7694:
7661:
7639:
7601:
7579:
7548:
7526:
7507:
7485:
7465:
7443:
7425:Henrici, Peter
7421:
7399:
7379:
7357:
7331:
7299:
7256:
7253:
7252:
7251:
7193:
7174:Pergamon Press
7162:
7144:(1): 277–281,
7128:
7102:(1): 108–113,
7086:
7050:
6984:
6944:
6937:Cauchy problem
6922:(2): 492–499,
6904:
6850:
6784:
6754:(in Italian),
6742:
6720:
6701:
6659:(in Italian),
6647:Amoroso, Luigi
6641:
6638:
6637:
6635:
6632:
6630:
6629:
6613:Andreotti 1976
6592:
6588:
6558:
6540:
6536:
6519:
6502:
6496:under certain
6482:Aldo Andreotti
6473:
6469:Hörmander 1990
6456:
6442:
6435:
6432:
6426:
6394:
6370:
6367:
6364:
6361:
6340:
6337:
6334:
6329:
6325:
6304:
6301:
6294:
6291:
6285:
6262:
6249:
6236:
6216:Pompeiu (1912)
6208:Pompeiu's work
6195:
6176:
6162:The corrected
6155:
6134:
6114:
6083:
6070:
6058:Henri Poincaré
6045:
6028:
6010:
6008:
6005:
6004:
6003:
5998:
5993:
5988:
5981:
5978:
5977:
5976:
5956:
5952:
5948:
5940:
5937:
5931:
5925:
5922:
5920:
5916:
5912:
5904:
5897:
5894:
5887:
5882:
5879:
5873:
5867:
5866:
5858:
5851:
5848:
5841:
5833:
5830:
5824:
5818:
5815:
5813:
5809:
5805:
5797:
5793:
5789:
5784:
5781:
5775:
5769:
5768:
5745:
5742:
5739:
5736:
5733:
5730:
5727:
5707:
5704:
5701:
5698:
5693:
5689:
5685:
5682:
5667:
5664:
5663:
5662:
5642:
5635:
5632:
5625:
5618:
5611:
5608:
5601:
5594:
5590:
5587:
5579:
5572:
5569:
5562:
5557:
5554:
5547:
5541:
5536:
5533:
5530:
5526:
5522:
5514:
5507:
5504:
5497:
5490:
5486:
5482:
5475:
5471:
5468:
5460:
5456:
5452:
5447:
5444:
5437:
5431:
5426:
5423:
5420:
5416:
5412:
5409:
5407:
5404:
5400:
5397:
5394:
5390:
5381:
5374:
5371:
5364:
5360:
5355:
5354:
5346:
5342:
5338:
5331:
5324:
5321:
5314:
5307:
5303:
5300:
5292:
5285:
5282:
5275:
5270:
5267:
5260:
5254:
5249:
5246:
5243:
5239:
5235:
5227:
5223:
5219:
5212:
5208:
5204:
5197:
5193:
5190:
5182:
5178:
5174:
5169:
5166:
5159:
5153:
5148:
5145:
5142:
5138:
5134:
5131:
5129:
5126:
5122:
5119:
5116:
5112:
5103:
5099:
5095:
5091:
5086:
5085:
5058:
5055:
5052:
5049:
5046:
5043:
5040:
5020:
5017:
5013:
5010:
5006:
5003:
5000:
4995:
4991:
4987:
4984:
4964:
4961:
4958:
4954:
4951:
4947:
4942:
4938:
4934:
4931:
4916:
4909:
4908:
4907:
4886:
4883:
4877:
4869:
4866:
4860:
4853:
4849:
4846:
4837:
4834:
4828:
4823:
4820:
4813:
4809:
4800:
4797:
4791:
4786:
4783:
4776:
4772:
4769:
4763:
4760:
4755:
4752:
4745:
4741:
4738:
4736:
4734:
4731:
4728:
4725:
4722:
4713:
4710:
4704:
4700:
4695:
4694:
4688:
4685:
4677:
4674:
4668:
4661:
4657:
4654:
4645:
4642:
4636:
4631:
4628:
4621:
4617:
4611:
4608:
4603:
4600:
4593:
4589:
4586:
4580:
4577:
4572:
4569:
4562:
4558:
4555:
4553:
4551:
4548:
4545:
4542:
4539:
4533:
4530:
4526:
4521:
4520:
4493:
4490:
4487:
4484:
4481:
4478:
4475:
4455:
4452:
4449:
4446:
4441:
4437:
4433:
4430:
4427:
4424:
4409:
4406:
4404:requirements.
4388:
4385:
4381:
4378:
4375:
4372:
4352:
4349:
4345:
4342:
4338:
4335:
4310:
4305:
4300:
4296:
4293:
4270:
4265:
4260:
4256:
4253:
4225:
4222:
4206:
4205:
4185:
4178:
4175:
4168:
4163:
4160:
4154:
4151:
4148:
4145:
4142:
4134:
4127:
4124:
4117:
4112:
4109:
4103:
4100:
4098:
4096:
4093:
4090:
4087:
4084:
4076:
4069:
4066:
4059:
4055:
4050:
4049:
4041:
4037:
4033:
4028:
4025:
4019:
4016:
4013:
4010:
4007:
3999:
3995:
3991:
3986:
3983:
3977:
3974:
3972:
3970:
3967:
3964:
3961:
3958:
3950:
3946:
3942:
3938:
3933:
3932:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3867:
3864:
3861:
3858:
3853:
3849:
3845:
3842:
3839:
3836:
3821:
3818:
3817:
3816:
3796:
3789:
3786:
3779:
3774:
3771:
3765:
3762:
3754:
3747:
3744:
3737:
3732:
3729:
3723:
3720:
3717:
3715:
3712:
3708:
3705:
3702:
3699:
3696:
3692:
3683:
3676:
3673:
3666:
3662:
3657:
3656:
3648:
3644:
3640:
3635:
3632:
3626:
3623:
3615:
3611:
3607:
3602:
3599:
3593:
3590:
3587:
3585:
3582:
3578:
3575:
3572:
3569:
3566:
3562:
3553:
3549:
3545:
3541:
3536:
3535:
3512:
3509:
3506:
3503:
3500:
3497:
3494:
3470:
3467:
3464:
3444:
3441:
3438:
3433:
3429:
3425:
3422:
3419:
3416:
3401:
3398:
3365:
3360:
3355:
3324:
3321:
3316:
3284:
3253:
3250:
3247:
3227:
3222:
3218:
3214:
3211:
3208:
3203:
3199:
3195:
3190:
3186:
3182:
3179:
3176:
3171:
3167:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3128:complex vector
3113:
3108:
3103:
3100:
3088:
3085:
3072:
3069:
3060:
3057:
3051:
3046:
3043:
3014:
3011:
3006:
3003:
2997:
2994:
2988:
2985:
2980:
2977:
2971:
2965:
2962:
2957:
2954:
2948:
2942:
2939:
2934:
2931:
2917:
2916:
2898:
2895:
2890:
2887:
2881:
2875:
2872:
2867:
2864:
2858:
2855:
2849:
2846:
2841:
2838:
2832:
2829:
2827:
2825:
2821:
2814:
2811:
2806:
2803:
2797:
2791:
2788:
2783:
2780:
2774:
2771:
2765:
2762:
2757:
2754:
2748:
2745:
2739:
2736:
2731:
2728:
2721:
2715:
2712:
2707:
2704:
2702:
2700:
2696:
2689:
2686:
2681:
2678:
2672:
2669:
2663:
2660:
2655:
2652:
2645:
2639:
2636:
2631:
2628:
2626:
2621:
2618:
2613:
2610:
2604:
2603:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2514:
2511:
2507:
2503:
2500:
2480:
2477:
2473:
2469:
2466:
2442:
2430:
2427:
2398:
2393:
2390:
2385:
2380:
2377:
2350:
2346:
2318:
2313:
2307:
2298:
2294:
2290:
2286:
2281:
2278:
2270:
2266:
2262:
2258:
2252:
2246:
2243:
2238:
2230:
2223:
2220:
2213:
2209:
2204:
2203:
2200:
2196:
2195:
2191:
2182:
2178:
2174:
2170:
2165:
2162:
2154:
2150:
2146:
2142:
2136:
2130:
2127:
2122:
2114:
2107:
2104:
2097:
2093:
2088:
2087:
2085:
2079:
2074:
2068:
2059:
2055:
2051:
2047:
2042:
2039:
2031:
2027:
2023:
2019:
2013:
2007:
2004:
1999:
1991:
1987:
1983:
1979:
1974:
1973:
1970:
1966:
1965:
1961:
1952:
1948:
1944:
1940:
1935:
1932:
1924:
1920:
1916:
1912:
1906:
1900:
1897:
1892:
1884:
1880:
1876:
1872:
1867:
1866:
1864:
1835:
1831:
1825:
1820:
1815:
1811:
1807:
1803:
1799:
1795:
1789:
1785:
1781:
1778:
1775:
1770:
1766:
1762:
1757:
1753:
1749:
1746:
1743:
1738:
1734:
1729:
1725:
1721:
1716:
1712:
1708:
1703:
1698:
1694:
1689:
1686:
1681:
1676:
1671:
1666:
1642:
1635:
1606:
1601:
1596:
1591:
1588:
1561:
1557:
1540:
1539:
1523:
1516:
1513:
1509:
1504:
1501:
1495:
1492:
1488:
1482:
1476:
1473:
1468:
1465:
1463:
1455:
1452:
1446:
1442:
1437:
1436:
1432:
1425:
1422:
1418:
1413:
1410:
1404:
1401:
1397:
1391:
1385:
1382:
1377:
1374:
1372:
1367:
1364:
1360:
1355:
1354:
1324:
1304:
1284:
1281:
1278:
1275:
1272:
1269:
1249:
1245:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1203:
1198:
1193:
1189:
1170:
1167:
1142:
1139:
1119:Wirtinger 1927
1110:
1107:
1095:Pompeiu (1913)
1062:
1057:
1053:
1049:
1046:
1035:differentiable
1007:
1004:
1001:
998:
995:
971:
943:
940:
937:
932:
928:
924:
921:
918:
915:
912:
909:
906:
901:
897:
893:
890:
879:
878:
867:
864:
860:
856:
853:
850:
847:
842:
839:
836:
831:
827:
823:
820:
816:
807:
803:
799:
796:
793:
789:
782:
779:
776:
772:
765:
762:
759:
755:
749:
744:
740:
736:
727:
724:
718:
713:
710:
676:
672:
668:
663:
659:
631:
628:
625:
622:
601:complex valued
584:
581:
559:expresses the
513:Amoroso (1912)
483:
482:
471:
468:
460:
456:
452:
446:
442:
438:
433:
428:
424:
397:
377:
374:
371:
349:
345:
341:
336:
332:
297:
286:
285:
274:
271:
268:
265:
262:
259:
253:
246:
242:
238:
233:
229:
225:
222:
217:
213:
209:
208:
203:
199:
195:
190:
186:
182:
179:
174:
170:
166:
165:
163:
132:
127:
110:Henri Poincaré
89:
86:
84:
81:
15:
13:
10:
9:
6:
4:
3:
2:
7759:
7748:
7745:
7743:
7740:
7738:
7735:
7734:
7732:
7721:
7717:
7713:
7708:
7704:
7700:
7695:
7692:
7688:
7684:
7680:
7676:
7672:
7668:
7664:
7662:0-387-97195-5
7658:
7654:
7650:
7649:
7644:
7640:
7637:
7633:
7629:
7618:on 2011-09-27
7617:
7613:
7609:
7608:
7602:
7598:
7594:
7590:
7586:
7582:
7580:0-471-83091-7
7576:
7572:
7568:
7564:
7560:
7559:
7554:
7549:
7545:
7541:
7537:
7533:
7529:
7523:
7519:
7515:
7514:
7508:
7504:
7500:
7496:
7492:
7488:
7486:0-444-88446-7
7482:
7478:
7477:North-Holland
7474:
7470:
7466:
7462:
7458:
7454:
7450:
7446:
7444:0-471-58986-1
7440:
7436:
7432:
7431:
7426:
7422:
7418:
7414:
7410:
7406:
7402:
7400:0-534-13308-8
7396:
7392:
7388:
7384:
7380:
7376:
7372:
7368:
7364:
7360:
7358:9780821869536
7354:
7350:
7349:Prentice-Hall
7346:
7342:
7341:
7336:
7332:
7328:
7324:
7320:
7316:
7312:
7308:
7304:
7300:
7297:
7295:
7289:
7278:on 2012-03-07
7277:
7273:
7269:
7268:
7263:
7259:
7258:
7254:
7249:
7245:
7241:
7236:
7232:
7228:
7224:
7220:
7216:
7212:
7209:(in German),
7208:
7207:
7202:
7198:
7194:
7191:
7187:
7183:
7179:
7175:
7171:
7167:
7163:
7159:
7155:
7151:
7147:
7143:
7140:(in French),
7139:
7138:
7133:
7129:
7125:
7121:
7117:
7113:
7109:
7105:
7101:
7098:(in French),
7097:
7096:
7091:
7087:
7083:
7079:
7074:
7069:
7066:(1): 89–178,
7065:
7062:(in French),
7061:
7060:
7055:
7051:
7048:
7043:
7039:
7035:
7031:
7027:
7023:
7019:
7015:
7011:
7007:
7003:
7000:(in German),
6999:
6998:
6993:
6989:
6988:Peschl, Ernst
6985:
6981:
6977:
6973:
6969:
6965:
6961:
6957:
6953:
6949:
6945:
6942:
6938:
6934:
6929:
6925:
6921:
6917:
6913:
6909:
6905:
6902:
6898:
6894:
6889:
6885:
6881:
6877:
6873:
6869:
6865:
6861:
6860:
6855:
6851:
6848:
6844:
6840:
6836:
6832:
6827:
6823:
6819:
6815:
6811:
6807:
6803:
6799:
6798:
6793:
6789:
6785:
6782:
6778:
6773:
6769:
6765:
6761:
6757:
6753:
6752:
6747:
6743:
6739:
6735:
6731:
6727:
6723:
6717:
6713:
6709:
6708:
6702:
6699:
6695:
6691:
6686:
6682:
6678:
6674:
6670:
6666:
6662:
6658:
6657:
6652:
6648:
6644:
6643:
6639:
6633:
6626:
6622:
6618:
6614:
6610:
6607:but only for
6606:
6590:
6586:
6576:
6573:, p. 5:
6572:
6568:
6562:
6559:
6555:
6538:
6534:
6523:
6520:
6516:
6512:
6506:
6503:
6499:
6495:
6491:
6487:
6483:
6477:
6474:
6470:
6466:
6460:
6457:
6453:
6440:
6430:
6416:
6412:
6408:
6392:
6384:
6365:
6359:
6351:
6327:
6323:
6302:
6299:
6289:
6272:
6266:
6263:
6259:
6253:
6250:
6246:
6240:
6237:
6233:
6229:
6225:
6221:
6217:
6213:
6212:Fichera (1969
6209:
6205:
6204:Henrici (1993
6199:
6196:
6192:
6191:
6186:
6180:
6177:
6173:
6169:
6165:
6164:Dover edition
6159:
6156:
6152:
6148:
6132:
6124:
6123:Poincaré 1899
6118:
6115:
6111:
6108:
6106:
6101:
6100:Poincaré 1899
6097:
6093:
6087:
6084:
6080:
6079:Poincaré 1899
6074:
6071:
6067:
6063:
6059:
6055:
6054:Poincaré 1899
6049:
6046:
6042:
6038:
6032:
6029:
6026:, p. 10.
6025:
6021:
6015:
6012:
6006:
6002:
5999:
5997:
5994:
5992:
5989:
5987:
5984:
5983:
5979:
5954:
5950:
5935:
5923:
5921:
5910:
5902:
5892:
5880:
5871:
5856:
5846:
5828:
5816:
5814:
5803:
5795:
5791:
5782:
5773:
5759:
5758:
5757:
5743:
5740:
5737:
5734:
5731:
5728:
5725:
5705:
5691:
5687:
5683:
5680:
5665:
5640:
5630:
5616:
5606:
5592:
5588:
5585:
5577:
5567:
5555:
5545:
5539:
5534:
5531:
5528:
5524:
5520:
5512:
5502:
5488:
5484:
5473:
5469:
5466:
5458:
5454:
5445:
5435:
5429:
5424:
5421:
5418:
5414:
5410:
5408:
5402:
5398:
5395:
5392:
5388:
5379:
5369:
5344:
5340:
5329:
5319:
5305:
5301:
5298:
5290:
5280:
5268:
5258:
5252:
5247:
5244:
5241:
5237:
5233:
5225:
5221:
5210:
5206:
5195:
5191:
5188:
5180:
5176:
5167:
5157:
5151:
5146:
5143:
5140:
5136:
5132:
5130:
5124:
5120:
5117:
5114:
5110:
5101:
5097:
5076:
5075:
5074:
5072:
5056:
5053:
5050:
5047:
5044:
5041:
5038:
5018:
5011:
5004:
4993:
4989:
4985:
4982:
4956:
4952:
4940:
4936:
4932:
4929:
4914:
4911:Functions of
4910:
4881:
4864:
4851:
4847:
4844:
4832:
4821:
4811:
4807:
4795:
4784:
4774:
4770:
4767:
4761:
4753:
4743:
4739:
4737:
4729:
4726:
4723:
4708:
4686:
4672:
4659:
4655:
4652:
4640:
4629:
4619:
4615:
4609:
4601:
4591:
4587:
4584:
4578:
4570:
4560:
4556:
4554:
4546:
4543:
4540:
4531:
4511:
4510:
4509:
4507:
4491:
4485:
4473:
4453:
4439:
4435:
4431:
4428:
4425:
4422:
4407:
4405:
4403:
4386:
4373:
4370:
4343:
4336:
4333:
4326:
4308:
4298:
4294:
4268:
4258:
4254:
4243:
4239:
4235:
4231:
4223:
4221:
4219:
4215:
4211:
4183:
4173:
4161:
4152:
4149:
4146:
4143:
4140:
4132:
4122:
4110:
4101:
4099:
4091:
4088:
4085:
4074:
4064:
4039:
4035:
4026:
4017:
4014:
4011:
4008:
4005:
3997:
3993:
3984:
3975:
3973:
3965:
3962:
3959:
3948:
3944:
3923:
3922:
3921:
3919:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3865:
3851:
3847:
3843:
3840:
3837:
3834:
3819:
3794:
3784:
3772:
3763:
3760:
3752:
3742:
3730:
3721:
3718:
3716:
3710:
3706:
3703:
3700:
3697:
3694:
3690:
3681:
3671:
3646:
3642:
3633:
3624:
3621:
3613:
3609:
3600:
3591:
3588:
3586:
3580:
3576:
3573:
3570:
3567:
3564:
3560:
3551:
3547:
3526:
3525:
3524:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3484:
3468:
3465:
3462:
3431:
3427:
3423:
3420:
3417:
3414:
3399:
3397:
3395:
3392:(ordinary or
3391:
3387:
3386:
3381:
3380:
3363:
3358:
3343:
3340:
3322:
3319:
3305:
3302:
3298:
3275:
3271:
3267:
3251:
3248:
3245:
3220:
3216:
3212:
3209:
3206:
3201:
3197:
3193:
3188:
3184:
3180:
3177:
3174:
3169:
3165:
3158:
3152:
3149:
3146:
3140:
3137:
3129:
3111:
3101:
3098:
3086:
3084:
3070:
3067:
3055:
3044:
3029:
3012:
3004:
2995:
2992:
2986:
2978:
2969:
2963:
2955:
2946:
2940:
2932:
2896:
2893:
2888:
2885:
2879:
2873:
2865:
2856:
2853:
2847:
2839:
2830:
2828:
2819:
2812:
2804:
2795:
2789:
2781:
2772:
2769:
2763:
2755:
2746:
2743:
2737:
2729:
2719:
2713:
2710:
2705:
2703:
2694:
2687:
2679:
2670:
2667:
2661:
2653:
2643:
2637:
2634:
2629:
2627:
2619:
2611:
2594:
2593:
2592:
2575:
2569:
2566:
2563:
2557:
2551:
2548:
2542:
2536:
2528:
2512:
2509:
2505:
2501:
2498:
2478:
2471:
2467:
2456:
2440:
2428:
2426:
2424:
2420:
2416:
2412:
2396:
2391:
2388:
2378:
2368:
2364:
2348:
2344:
2334:
2329:
2316:
2305:
2296:
2292:
2279:
2276:
2268:
2264:
2250:
2244:
2241:
2236:
2228:
2218:
2198:
2189:
2180:
2176:
2163:
2160:
2152:
2148:
2134:
2128:
2125:
2120:
2112:
2102:
2083:
2077:
2066:
2057:
2053:
2040:
2037:
2029:
2025:
2011:
2005:
2002:
1997:
1989:
1985:
1968:
1959:
1950:
1946:
1933:
1930:
1922:
1918:
1904:
1898:
1895:
1890:
1882:
1878:
1862:
1852:
1849:
1833:
1829:
1823:
1813:
1805:
1797:
1793:
1787:
1783:
1779:
1776:
1773:
1768:
1764:
1760:
1755:
1751:
1747:
1744:
1741:
1736:
1732:
1727:
1723:
1719:
1710:
1701:
1696:
1692:
1687:
1684:
1674:
1669:
1655:
1654:complex field
1651:
1648:Consider the
1646:Definition 2.
1640:
1637:Functions of
1636:
1634:
1632:
1628:
1624:
1620:
1604:
1599:
1589:
1579:
1575:
1559:
1555:
1545:
1521:
1514:
1502:
1499:
1493:
1480:
1474:
1471:
1466:
1464:
1450:
1430:
1423:
1411:
1408:
1402:
1389:
1383:
1380:
1375:
1373:
1365:
1345:
1344:
1343:
1341:
1338:
1322:
1302:
1282:
1279:
1276:
1273:
1270:
1267:
1239:
1236:
1233:
1230:
1227:
1221:
1218:
1215:
1206:
1201:
1191:
1179:
1178:complex plane
1176:Consider the
1174:Definition 1.
1168:
1166:
1164:
1160:
1156:
1152:
1148:
1140:
1138:
1136:
1132:
1131:Levi operator
1128:
1124:
1120:
1117:in the paper
1116:
1108:
1106:
1104:
1100:
1096:
1092:
1088:
1084:
1080:
1076:
1075:Fichera (1969
1073:According to
1060:
1055:
1051:
1047:
1044:
1036:
1032:
1031:Henrici (1993
1028:
1025:
1021:
1005:
999:
993:
985:
969:
961:
957:
938:
935:
930:
926:
919:
913:
907:
904:
899:
895:
865:
862:
851:
845:
837:
834:
829:
825:
814:
805:
801:
797:
794:
791:
787:
780:
774:
753:
742:
738:
722:
711:
697:
696:
695:
694:
690:
674:
666:
661:
657:
649:
645:
644:neighbourhood
626:
620:
613:
609:
608:real analysis
605:
602:
598:
594:
590:
589:Henrici (1993
587:According to
582:
580:
578:
574:
570:
566:
565:Levi operator
562:
558:
554:
550:
546:
542:
538:
537:Osgood (1966)
534:
530:
526:
522:
518:
514:
510:
506:
502:
498:
494:
493:Poincaré 1899
490:
489:
469:
466:
458:
454:
450:
444:
440:
436:
431:
426:
422:
411:
410:
409:
395:
375:
372:
369:
347:
343:
339:
334:
330:
322:
319:
315:
311:
295:
272:
269:
266:
263:
260:
257:
244:
240:
236:
231:
227:
223:
220:
215:
211:
201:
197:
193:
188:
184:
180:
177:
172:
168:
161:
152:
151:
150:
148:
130:
115:
111:
107:
106:Remmert (1991
103:
99:
98:Poincaré 1899
95:
87:
82:
80:
78:
74:
70:
66:
62:
58:
54:
53:real variable
50:
46:
42:
38:
34:
30:
26:
22:
7719:
7698:
7647:
7635:
7627:
7620:, retrieved
7616:the original
7606:
7557:
7512:
7472:
7429:
7386:
7339:
7313:(3): 61–83,
7310:
7306:
7293:
7287:
7280:, retrieved
7276:the original
7266:
7210:
7204:
7169:
7166:Vekua, I. N.
7141:
7135:
7099:
7093:
7063:
7057:
7054:Poincaré, H.
7001:
6995:
6951:
6932:
6919:
6915:
6892:
6866:(1): 69–79,
6863:
6857:
6839:hypersurface
6830:
6804:(1): 61–87,
6801:
6795:
6776:
6758:(1): 27–37,
6755:
6749:
6706:
6689:
6663:(1): 75–85,
6660:
6654:
6624:
6608:
6571:Gunning 1990
6561:
6522:
6505:
6476:
6459:
6274:
6265:
6258:Henrici 1993
6252:
6239:
6198:
6188:
6185:Osgood (1966
6179:
6158:
6117:
6107:-dimensional
6104:
6086:
6073:
6048:
6031:
6020:Fichera 1986
6014:
5669:
4918:
4912:
4411:
4233:
4227:
4207:
3918:product rule
3823:
3820:Product rule
3403:
3385:definition 2
3383:
3379:definition 1
3377:
3344:counterpart
3269:
3266:real vectors
3090:
3030:
2918:
2432:
2330:
1644:
1638:
1541:
1172:
1144:
1112:
880:
597:Pompeiu 1912
586:
488:definition 2
486:
484:
310:biharmonique
309:
287:
91:
32:
28:
18:
7569:, pp.
7213:: 357–375,
7132:Pompeiu, D.
7090:Pompeiu, D.
7004:: 574–594,
6841:can be the
6271:Vekua (1962
6224:integration
6168:Osgood 1966
6037:derivatives
5986:CR–function
5666:Conjugation
4218:derivatives
4210:derivations
3485:, then for
3390:derivatives
646:of a given
599:), given a
577:Levi-Civita
505:Levi (1910)
149:as follows
49:derivatives
7731:Categories
7707:0094.28002
7679:0780.30001
7622:2010-08-24
7597:0644.35005
7544:0528.32001
7503:0685.32001
7461:1107.30300
7417:0699.32001
7375:0141.08601
7327:0705.32006
7282:2010-08-28
7227:52.0342.03
7190:0100.07603
7116:43.0481.01
7082:29.0370.02
7042:0004.30001
7018:58.1096.05
6980:0138.30901
6964:45.0661.02
6928:36.0482.01
6880:42.0449.02
6818:41.0487.01
6772:0201.10002
6738:0981.30001
6677:43.0453.03
6634:References
6498:operations
6094:, and the
5071:chain rule
4506:chain rule
4402:smoothness
4238:chain rule
4232:: for the
4224:Chain rule
3339:isomorphic
3337:or in its
1091:Ilia Vekua
962:of radius
63:or simply
7471:(1990) ,
7427:(1993) ,
7235:121149132
7158:121616964
7124:120717465
7034:127138808
6950:(1966) ,
6943:is given.
6901:Levi 1910
6888:120133326
6826:122678686
6700:is given.
6685:122956910
6605:functions
6554:functions
6434:¯
6425:∂
6336:Ω
6303:∈
6293:¯
6284:∂
5947:∂
5939:¯
5930:∂
5915:¯
5896:¯
5886:∂
5878:∂
5850:¯
5840:∂
5832:¯
5823:∂
5808:¯
5788:∂
5780:∂
5738:…
5718:then for
5700:Ω
5684:∈
5634:¯
5624:∂
5610:¯
5600:∂
5586:∘
5571:¯
5561:∂
5553:∂
5525:∑
5506:¯
5496:∂
5481:∂
5467:∘
5451:∂
5443:∂
5415:∑
5396:∘
5373:¯
5363:∂
5359:∂
5337:∂
5323:¯
5313:∂
5299:∘
5284:¯
5274:∂
5266:∂
5238:∑
5218:∂
5203:∂
5189:∘
5173:∂
5165:∂
5137:∑
5118:∘
5094:∂
5090:∂
5051:…
5031:then for
5009:Ω
5002:Ω
4986:∈
4960:Ω
4950:Ω
4933:∈
4920:Lemma 3.2
4885:¯
4876:∂
4868:¯
4859:∂
4845:∘
4836:¯
4827:∂
4819:∂
4799:¯
4790:∂
4782:∂
4768:∘
4759:∂
4751:∂
4727:∘
4712:¯
4703:∂
4699:∂
4684:∂
4676:¯
4667:∂
4653:∘
4644:¯
4635:∂
4627:∂
4607:∂
4599:∂
4585:∘
4576:∂
4568:∂
4544:∘
4529:∂
4525:∂
4504:then the
4489:Ω
4486:⊆
4480:Ω
4448:Ω
4432:∈
4413:Lemma 3.1
4384:Ω
4380:→
4377:Ω
4351:Ω
4348:→
4341:Ω
4299:⊆
4292:Ω
4259:⊆
4252:Ω
4212:from the
4177:¯
4167:∂
4159:∂
4153:⋅
4141:⋅
4126:¯
4116:∂
4108:∂
4089:⋅
4068:¯
4058:∂
4054:∂
4032:∂
4024:∂
4018:⋅
4006:⋅
3990:∂
3982:∂
3963:⋅
3941:∂
3937:∂
3898:…
3878:then for
3860:Ω
3844:∈
3788:¯
3778:∂
3770:∂
3764:β
3746:¯
3736:∂
3728:∂
3722:α
3704:β
3695:α
3675:¯
3665:∂
3661:∂
3639:∂
3631:∂
3625:β
3606:∂
3598:∂
3592:α
3574:β
3565:α
3544:∂
3540:∂
3505:…
3469:β
3463:α
3440:Ω
3424:∈
3400:Linearity
3283:Ω
3210:…
3178:…
3141:≡
3130:and that
3102:∈
3059:¯
3050:∂
3042:∂
3010:∂
3002:∂
2996:−
2984:∂
2976:∂
2961:∂
2953:∂
2938:∂
2930:∂
2871:∂
2863:∂
2845:∂
2837:∂
2810:∂
2802:∂
2787:∂
2779:∂
2770:−
2761:∂
2753:∂
2735:∂
2727:∂
2685:∂
2677:∂
2668:−
2659:∂
2651:∂
2617:∂
2609:∂
2476:∂
2465:∂
2413:and have
2379:⊂
2376:Ω
2363:functions
2289:∂
2285:∂
2261:∂
2257:∂
2222:¯
2212:∂
2208:∂
2199:⋮
2173:∂
2169:∂
2145:∂
2141:∂
2106:¯
2096:∂
2092:∂
2050:∂
2046:∂
2038:−
2022:∂
2018:∂
1982:∂
1978:∂
1969:⋮
1943:∂
1939:∂
1931:−
1915:∂
1911:∂
1875:∂
1871:∂
1814:∈
1798:∣
1777:…
1745:…
1621:and have
1590:⊆
1587:Ω
1574:functions
1512:∂
1508:∂
1491:∂
1487:∂
1454:¯
1445:∂
1441:∂
1421:∂
1417:∂
1409:−
1400:∂
1396:∂
1363:∂
1359:∂
1240:∈
1228:∣
1192:≡
1155:monograph
917:∂
889:Γ
819:Γ
815:∮
795:π
778:→
726:¯
717:∂
709:∂
667:∈
610:) of one
511:) and of
509:Levi 1911
321:variables
308:he calls
267:⩽
261:⩽
221:−
7645:(1991),
7555:(1988),
7385:(1990),
7347:, N.J.:
7264:(1976),
7199:(1927),
7168:(1962),
6990:(1932),
6910:(1905),
6843:boundary
6790:(1910),
6649:(1912),
6226:to be a
6151:variable
6064:, while
5980:See also
5671:Lemma 4.
5012:″
4953:′
4387:″
4344:′
4323:and two
4295:″
4255:′
3825:Lemma 2.
3405:Lemma 1.
1133:and the
1027:variable
956:boundary
563:and the
145:and its
7671:1084167
7589:0941372
7571:xiv+394
7536:0716497
7495:1045639
7453:0822470
7409:1052649
7367:0180696
7319:0917525
7182:0150320
7026:1512774
6972:0201668
6764:0265616
6730:1831783
6575:Gunning
6490:algebra
6488:of the
6486:closure
6415:Sobolev
6407:Pompeiu
4242:domains
3394:partial
3342:complex
3299:in the
3268:, with
1652:on the
1085:in the
954:is the
569:Amoroso
531:of the
7705:
7685:
7677:
7669:
7659:
7595:
7587:
7577:
7542:
7534:
7524:
7501:
7493:
7483:
7459:
7451:
7441:
7415:
7407:
7397:
7373:
7365:
7355:
7325:
7317:
7233:
7225:
7188:
7180:
7156:
7122:
7114:
7080:
7040:
7032:
7024:
7016:
6978:
6970:
6962:
6926:
6886:
6878:
6824:
6816:
6770:
6762:
6736:
6728:
6718:
6683:
6675:
6228:circle
6220:domain
5073:holds
4508:holds
3920:holds
3297:domain
3274:subset
3238:where
2411:linear
2367:domain
2333:domain
1848:linear
1619:linear
1578:domain
1544:domain
1337:linear
1020:circle
881:where
557:Osgood
43:, are
7309:, 8,
7231:S2CID
7154:S2CID
7120:S2CID
7030:S2CID
6956:Dover
6884:S2CID
6864:XVIII
6845:of a
6822:S2CID
6681:S2CID
6623:with
6210:: as
6121:See (
6007:Notes
4220:are.
3126:is a
2419:space
2365:on a
1627:space
1576:on a
1081:as a
958:of a
693:limit
648:point
547:of a
507:(and
362:with
7683:ISBN
7657:ISBN
7575:ISBN
7522:ISBN
7481:ISBN
7439:ISBN
7395:ISBN
7353:ISBN
6939:for
6802:XVII
6716:ISBN
6696:for
6565:See
6381:has
6183:See
4975:and
4466:and
4363:and
4325:maps
4283:and
3916:the
3481:are
3455:and
3382:and
3301:real
3264:are
1315:and
1089:was
960:disk
575:and
573:Levi
527:and
525:real
318:real
23:and
7703:Zbl
7675:Zbl
7626:. "
7593:Zbl
7540:Zbl
7499:Zbl
7457:Zbl
7413:Zbl
7371:Zbl
7323:Zbl
7223:JFM
7215:doi
7186:Zbl
7146:doi
7112:JFM
7104:doi
7078:JFM
7068:doi
7038:Zbl
7014:JFM
7006:doi
7002:106
6976:Zbl
6960:JFM
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