“Contour Integration”的概念、定义、习题解题方法及翻译-数学辞典

单词

Contour Integration

释义

Contour Integration

Let and be Polynomials of Degrees and withCoefficients , ..., and , ..., . Take the contour in the upper half-plane, replace by ,and write . Then

(1)

Define a path

which is straight along the Real axis from

and makes a circular arc to connect thetwo ends in the upper half of the Complex Plane. The Residue Theorem thengives

where Res denotes the Residues. Solving,

Define

and set

(4)

then equation (3) becomes

(5)

Now,

(6)

for

. That means that for

, or

, so

(7)

for

. Apply Jordan's Lemma with

. We must have

(8)

so we require

. Then

(9)

for

Since this must hold separately for Real and Imaginary Parts, this result can beextended to

(10)

(11)

It is also true that

(12)

See also Cauchy Integral Formula, Cauchy Integral Theorem,Inside-Outside Theorem, Jordan's Lemma, Residue (Complex Analysis), Sine Integral

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 353-356, 1953.

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