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
From Green's theorem,
so (◇) becomes
 |
(8) |
But the Cauchy-Riemann equations require that
so
 |
(11) |
For a multiply connected region,
 |
(12) |
See also
Argument Principle, Cauchy Integral Formula, Contour Integral, Morera's Theorem, Residue Theorem
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References
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.
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Cite this as:
Weisstein, Eric W. "Cauchy Integral Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CauchyIntegralTheorem.html