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)
|
Q.E.D.
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.Referenced
on Wolfram|Alpha
Cauchy Integral Theorem
Cite this as:
Weisstein, Eric W. "Cauchy Integral Theorem."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CauchyIntegralTheorem.html
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