“Residue Theorem (Complex Analysis)”的概念、定义、习题解题方法及翻译-数学辞典

单词

Residue Theorem (Complex Analysis)

释义

Residue Theorem (Complex Analysis)

Given a complex function , consider the Laurent Series

(1)

Integrate term by term using a closed contour

encircling


			(2)

The Cauchy Integral Theorem requires that the first and last terms vanish, so we have

(3)

But we can evaluate this function (which has a Pole at

) using the Cauchy Integral Formula,

(4)

This equation must also hold for the constant function

, in which case it is also true that

, so

(5)

(6)

and (3) becomes

(7)

The quantity

is known as the Residue of

. Generalizing to acurve passing through multiple poles, (7) becomes

(8)

where

is the Winding Number and the

superscript denotes the quantitycorresponding to Pole

If the path does not completely encircle the Residue, take the Cauchy Principal Value to obtain

(9)

has only Isolated Singularities, then

(10)

The residues may be found without explicitly expanding into a Laurent Series as follows:

(11)

has a Pole of order

, then

for

and

. Therefore,

(12)

(13)



	(14)


	(15)

Iterating,


	(16)

(17)

and the Residue is

(18)

This amazing theorem says that the value of a Contour Integral in the Complex Plane depends only onthe properties of a few special points inside the contour.

See also Cauchy Integral Formula, Cauchy Integral Theorem, Contour Integral, Laurent Series,Pole, Residue (Complex Analysis)

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