Cauchy-Riemann conditions
d'Alembert–Euler conditions
Conditions that must be satisfied by the real part and the imaginary part
of a complex function
,
, for it to be monogenic and analytic as a function of a complex variable.
A function , defined in some domain
in the complex
-plane, is monogenic at a point
, i.e. has a derivative at
as a function of the complex variable
, if and only if its real and imaginary parts
and
are differentiable at
as functions of the real variables
and
, and if, moreover, the Cauchy–Riemann equations hold at that point:
![]() | (1) |
If the Cauchy–Riemann equations are satisfied, then the derivative can be expressed in any of the following forms:
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A function , defined and single-valued in a domain
, is analytic in
if and only if its real and imaginary parts are differentiable functions satisfying the Cauchy–Riemann equations throughout
. Each of the two functions
and
of class
satisfying the Cauchy–Riemann equations (1) is a harmonic function of
and
; the conditions (1) constitute conjugacy conditions of these two harmonic functions: Knowing one of them, the other may be found by integration.
The conditions (1) are valid for any two orthogonal directions and
, with the same mutual orientations as the
- and
-axes, in the form:
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For example, in polar coordinates , for
:
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Defining the complex differential operators by
![]() |
one can rewrite the Cauchy–Riemann equations (1) as
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Thus, a differentiable function of the variables
and
is an analytic function of
if and only if
.
For analytic functions of several complex variables ,
,
, the Cauchy–Riemann equations constitute a system of partial differential equations (overdetermined when
) for the functions
![]() |
![]() |
![]() | (2) |
or, in terms of the complex differentiation operators:
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Each of the two functions and
of class
satisfying the conditions (2) is a pluriharmonic function of the variables
and
(
). When
the pluriharmonic functions constitute a proper subclass of the class of harmonic functions. The conditions (2) are conjugacy conditions for two pluriharmonic functions
and
: Knowing one of them, one can determine the other by integration.
The conditions (1) apparently occurred for the first time in the works of J. d'Alembert [1]. Their first appearance as a criterion for analyticity was in a paper of L. Euler, delivered at the Petersburg Academy of Sciences in 1777 [2]. A.L. Cauchy utilized the conditions (1) to construct the theory of functions, beginning with a memoir presented to the Paris Academy in 1814 (see [3]). The celebrated dissertation of B. Riemann on the fundamentals of function theory dates from 1851 (see [4]).
References
[1] | J. d'Alembert, "Essai d'une nouvelle théorie de la résistance des fluides" , Paris (1752) |
[2] | L. Euler, Nova Acta Acad. Sci. Petrop. , 10 (1797) pp. 3–19 |
[3] | A.L. Cauchy, "Mémoire sur les intégrales définies" , Oeuvres complètes Ser. 1 , 1 , Paris (1882) pp. 319–506 |
[4] | "Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse" H. Weber (ed.) , Riemann's gesammelte math. Werke , Dover, reprint (1953) pp. 3–48 (Dover, reprint, 1953) |
[5] | A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) pp. Chapt. 1 (Translated from Russian) |
[6] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) pp. 1, Chapt. 1; 2, Chapt. 1 (In Russian) |
Comments
References
[a1] | L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 24–26 |
Cauchy-Riemann conditions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy-Riemann_conditions&oldid=31183