Chapter 5 Residue Theory —Residue ＆ Application §5.1 Isolated Singularities §5.2 Residue §5.3 Application of Residue Theory to Integrals

§5.1 Isolated Singularities 1.Definition ＆ Classification Def. not analytic at,but analytic on for some,then is called the Isolated Singular point of. Ex. isolated singular point not isolated singular points singular points isolated singular points Where: not isolated singular point

Def: Classification (according to )

In conclusion: We can use the above different situations to judge the types of isolated singular points.

2.Zeros V.S. Poles TH.5.1.1

We can use the above theorem to judge the followings.

TH Corollary 1. Corollary 2.

Ex.5.1.1

Note: Ex:

§5.2 Residue 1.Definition ＆ Evaluation

Def: Note:

Rule: When n=m=1,we can get the rule (1).

Ex.5.2.1

D z1z1 z2z2 z3z3 znzn C1C1 C2C2 C3C3 CnCn C TH Residue Theorem Satisfy the conditions of the residue theorem, then use it. Note: Ex:

And if we know the type of the singular points, then we can get the residue conveniently.

Ex.5.2.2

Homework: P : A1(1)(3)(5)(7)(9)(11),A2.(1)(3)(5), A3-A6, A7(1)(3)(5),A8(1)(3)(5)

§5.3 Application of Residue Theorem to Integrals Residue Theorem is the theorem of complex function and is related to closed loop integral. So if we want to use this theorem in definite integral of real variable function, the real variable function must be transformed to complex function and definite integral must be part of closed loop integral. For example,

1.Trigonometric Integrals over

Ex.5.4.1

Ex:

There are three poles,, inside the circle, is pole of order 2 and pole of order 1.

Ex:

2. z1z1 z2z2 z3z3 y CRCR RR R Ox

Integration path is as the figure. C R is the upper-half circle in the upper-half plane, with O as the center and R as the radius. Let R be big properly to make sure that all the singular points z k of R(z) in the upper-half are in the integration path. z1z1 z2z2 z3z3 y CRCR RR R Ox

Ex.5.4.2

Ex: Solution:

Ex.5.4.3

3. Or:

Ex.5.4.4

Ex.5.4.5

4.Integrals where Integral has poles in C Ex. Solutions ：

Homework: P118: A9-A11, A12(1)(3)(5)(7)

Summary: 1.Classification of isolated singularities. 2.Residue definition and evaluation. 3.Residue Theorem. 4.Application of Residue Theorem to integrals.