Cauchy Integral Theorem

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

$$\oint_\gamma f(z)\,dz = 0$$ (1)

for any closed Contour $\gamma$ completely contained in $R$. Writing $z$ as

$$z\equiv x+iy$$ (2)

and $f(z)$ as

$$f(z)\equiv u+iv$$ (3)

then gives

$$\oint_\gamma f(z)\,dz = \int_\gamma (u+iv)(dx+i\,dy)$$

$$= \int_\gamma u\,dx-v\,dy+ i\int_\gamma v\,dx+u\,dy.$$ (4)

From Green's Theorem,

$$\int_\gamma f(x,y)\,dx-g(x,y)\,dy = -\int\!\!\!\int \left({{... ...over\partial x} + {\partial f\over\partial y}}\right)\,dx\,dy,$$ (5)

$$\int_\gamma f(x,y)\,dx+g(x,y)\,dy = \int\!\!\!\int \left({{\... ...\over\partial x} - {\partial f\over\partial y}}\right)\,dx\,dy$$ (6)

so (4) becomes

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

But the Cauchy-Riemann Equations require that

$${\partial u\over\partial x} = {\partial v\over\partial y}$$ (8)

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

so

$$\oint_\gamma f(z)\,dz = 0,$$ (10)

Q. E. D.

For a Multiply Connected region,

$$\oint_{C_1} f(z)\,dz = \oint_{C_2} f(z)\,dz.$$ (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.