Chapter 5 Residue Theory

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Chapter 5 Residue Theory
Residue Application
5.1 Isolated Singularities 5.2 Residue 5.3
Application of Residue Theory to Integrals
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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
Where
isolated singular points
not isolated singular point
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Def Classification (according to
)
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In conclusion
We can use the above different situations to
judge the types of isolated singular points.
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2.Zeros V.S. Poles
TH.5.1.1
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We can use the above theorem to judge the
followings.
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TH.5.1.2
Corollary 1.
Corollary 2.
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Ex.5.1.1
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Ex
Note
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5.2 Residue
1.Definition Evaluation
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Def
Note
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Rule
When nm1,we can get the rule (1).
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Ex.5.2.1
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TH.5.2.1 Residue Theorem
Note
Satisfy the conditions of the residue theorem,
then use it.
Ex
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And if we know the type of the singular points,
then we can get the residue conveniently.
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Ex.5.2.2
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Homework
P117-118 A1(1)(3)(5)(7)(9)(11),A2.(1)(3)(5),A3-A6
, A7(1)(3)(5),A8(1)(3)(5)
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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,
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1.Trigonometric Integrals over
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Ex.5.4.1
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Ex
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There are three poles, , inside
the circle , is pole of order 2 and
pole of order 1.
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Ex
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2.
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Integration path is as the figure. CR 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 zk of R(z) in the upper-half are in the
integration path.
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Ex.5.4.2
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Solution
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Ex.5.4.3
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3.
Or
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Ex.5.4.4
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Ex.5.4.5
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4.Integrals where Integral has poles in C
Ex.
Solutions
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Homework
P118 A9-A11, A12(1)(3)(5)(7)
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Summary
1.Classification of isolated singularities. 2.Resi
due definition and evaluation. 3.Residue
Theorem. 4.Application of Residue Theorem to
integrals.