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Cauchy Integral Theorem

If $f(z)$ is analytic and its partial derivatives are continuous throughout some simply connected region $R$, then

\oint_\gamma f(z)\,dz = 0
\end{displaymath} (1)

for any closed Contour $\gamma$ completely contained in $R$. Writing $z$ as
z\equiv x+iy
\end{displaymath} (2)

and $f(z)$ as
f(z)\equiv u+iv
\end{displaymath} (3)

then gives
$\displaystyle \oint_\gamma f(z)\,dz$ $\textstyle =$ $\displaystyle \int_\gamma (u+iv)(dx+i\,dy)$  
  $\textstyle =$ $\displaystyle \int_\gamma u\,dx-v\,dy+ i\int_\gamma v\,dx+u\,dy.$ (4)

From Green's Theorem,
\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} (5)

\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} (6)

so (4) becomes

\oint_\gamma f(z)\,dz = -\int\!\!\!\int \left({{\partial v\o...
...\over\partial x}-{\partial v\over \partial y}}\right)\,dx\,dy.
\end{displaymath} (7)

But the Cauchy-Riemann Equations require that
{\partial u\over\partial x} = {\partial v\over\partial y}
\end{displaymath} (8)

{\partial u\over\partial y} = - {\partial v\over\partial x},
\end{displaymath} (9)

\oint_\gamma f(z)\,dz = 0,
\end{displaymath} (10)

Q. E. D.

For a Multiply Connected region,

\oint_{C_1} f(z)\,dz = \oint_{C_2} f(z)\,dz.
\end{displaymath} (11)

See also Cauchy Integral Theorem, Morera's Theorem, Residue Theorem (Complex Analysis)


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.

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© 1996-9 Eric W. Weisstein