If is analytic in some simply connected region , then

(1)

for any closed contour completely contained in . Writing as

(2)

and as

(3)

then gives

			(4)
			(5)

From Green's theorem,

			(6)
			(7)
			(8)
			(9)

so (◇) becomes

(10)

But the Cauchy-Riemann equations require that

			(11)
			(12)

so

(13)

Q.E.D.

For a multiply connected region,

(14)

Arfken, G. "Cauchy's Integral Theorem." §6.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 365-371, 1985.

Kaplan, W. "Integrals of Analytic Functions. Cauchy Integral Theorem." §9.8 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 594-598, 1991.

Knopp, K. "Cauchy's Integral Theorem." Ch. 4 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 47-60, 1996.

Krantz, S. G. "The Cauchy Integral Theorem and Formula." §2.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 26-29, 1999.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 363-367, 1953.

Woods, F. S. "Integral of a Complex Function." §145 in Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathematics. Boston, MA: Ginn, pp. 351-352, 1926.

CITE THIS AS:

Weisstein, Eric W. "Cauchy Integral Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyIntegralTheorem.html