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

单词

Cauchy Integral Formula

释义

Cauchy Integral Formula

Given a Contour Integral of the form

(1)

define a path

as an infinitesimal Circle around the point

(the dot in the above illustration).Define the path

as an arbitrary loop with a cut line (on which the forward and reverse contributions canceleach other out) so as to go around

The total path is then

(2)

(3)

From the Cauchy Integral Theorem, the Contour Integral along any path not enclosing a Pole is 0.Therefore, the first term in the above equation is 0 since

does not enclose the Pole, and we are leftwith

(4)

Now, let

, so

. Then


			(5)

But we are free to allow the radius

to shrink to 0, so


			(6)

and

(7)

If multiple loops are made around the Pole, then equation (7) becomes

(8)

where

is the Winding Number.

A similar formula holds for the derivatives of ,





			(9)

Iterating again,

(10)

Continuing the process and adding the Winding Number

(11)

See also Morera's Theorem

References

Arfken, G. ``Cauchy's Integral Formula.'' §6.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 371-376, 1985.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 367-372, 1953.

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