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Date: 11-12-2018
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Date: 28-11-2018
395
Date: 25-11-2018
385
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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) |
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(5) |
From Green's theorem,
(6) |
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(7) |
so (◇) becomes
(8) |
But the Cauchy-Riemann equations require that
(9) |
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(10) |
so
(11) |
Q.E.D.
For a multiply connected region,
(12) |
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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