“Cauchy Integral Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Cauchy Integral Theorem

释义

Cauchy Integral Theorem

If is analytic and its partial derivatives are continuous throughout some simply connected region , then

(1)

for any closed Contour

completely contained in

. Writing

(2)

and

(3)

then gives


			(4)

From Green's Theorem,

(5)

(6)

so (4) becomes

(7)

But the Cauchy-Riemann Equations require that

(8)

(9)

(10)

Q. E. D.

For a Multiply Connected region,

(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.

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