If
is analytic and its partial derivatives are continuous throughout some simply connected region
, then
![\begin{displaymath}
\oint_\gamma f(z)\,dz = 0
\end{displaymath}](c1_676.gif) |
(1) |
for any closed Contour
completely contained in
. Writing
as
![\begin{displaymath}
z\equiv x+iy
\end{displaymath}](c1_678.gif) |
(2) |
and
as
![\begin{displaymath}
f(z)\equiv u+iv
\end{displaymath}](c1_679.gif) |
(3) |
then gives
From Green's Theorem,
![\begin{displaymath}
\int_\gamma f(x,y)\,dx-g(x,y)\,dy = -\int\!\!\!\int \left({{...
...over\partial x} + {\partial f\over\partial y}}\right)\,dx\,dy,
\end{displaymath}](c1_683.gif) |
(5) |
![\begin{displaymath}
\int_\gamma f(x,y)\,dx+g(x,y)\,dy = \int\!\!\!\int \left({{\...
...\over\partial x} - {\partial f\over\partial y}}\right)\,dx\,dy
\end{displaymath}](c1_684.gif) |
(6) |
so (4) becomes
![\begin{displaymath}
\oint_\gamma f(z)\,dz = -\int\!\!\!\int \left({{\partial v\o...
...\over\partial x}-{\partial v\over \partial y}}\right)\,dx\,dy.
\end{displaymath}](c1_685.gif) |
(7) |
But the Cauchy-Riemann Equations require that
![\begin{displaymath}
{\partial u\over\partial x} = {\partial v\over\partial y}
\end{displaymath}](c1_686.gif) |
(8) |
![\begin{displaymath}
{\partial u\over\partial y} = - {\partial v\over\partial x},
\end{displaymath}](c1_687.gif) |
(9) |
so
![\begin{displaymath}
\oint_\gamma f(z)\,dz = 0,
\end{displaymath}](c1_688.gif) |
(10) |
Q. E. D.
For a Multiply Connected region,
![\begin{displaymath}
\oint_{C_1} f(z)\,dz = \oint_{C_2} f(z)\,dz.
\end{displaymath}](c1_689.gif) |
(11) |
See also Cauchy Integral Theorem, Morera's Theorem, Residue Theorem (Complex Analysis)
References
Arfken, G. ``Cauchy's Integral Theorem.'' §6.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL:
Academic Press, pp. 365-371, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill,
pp. 363-367, 1953.
© 1996-9 Eric W. Weisstein
1999-05-26