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Contents
 Introduction to Laurent and Taylor series
 convergence of series
 residue at a pole, residue at infinity
 methods of finding residues
 residue theorem
 solving of definite Integrals by contour integration

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 residue
In Laurent expansion of any function f(z), the co-efficient of (z-z0)-1 about
isolated singularity z=z0 is defined as the residue of f(z) at z=z0.

R A
f(z) = + Z0

General term
(contains positive Principal term
power of z) (contains negative
power of z)

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 Types of Isolated singularity
Isolated Singulariy

Removable Singulariy Poles Essential Singulariy

• If principal part of f(z) does not exist • If there are finite no. of terms in • If there are infinite no. of terms in
• b1=b2=…=bn = 0 principal part of f(z) principal part of f(z)
• Lt f(Z) exist & finite • bm≠ 0 and bm+1= bm+2 = bm+3 =…= 0, • Lt f(Z) does not exist
Z → Z0
Z → Z0
• In the expansion of f(z) all the then z = z0 is called pole of order m
• Limit point of zeros (means at which
powers are non-negative • If m = 1, then it is simple pole
value the function becomes zero) is
• We can remove the singularity by • Lt f(Z) exist & infinite
Z → Z0 essential singularity
redefining the function

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 residue
 If a function has removable singularity at z=a, the residue of f(z) at z=a is 0
 If a function is analytic at z=a, the residue of f(z) at z=a is 0
R
Z0

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 Solve

Find the residue of z-4 exp(iaz) at z = 0

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 solution
exp(iaz) 1 (iaz ) 2
(iaz ) 3
f(z) = z-4 exp(iaz) =
z4
= 4 (1 + iaz +
z + + ….)
2 6

(ia)3 (a)3
Residue = the co-efficient of (z-0)-1 is = -i
6 6

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 Solve

Find the residue of sin z/z at z=0

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 solution

sin 𝑧 1
𝑓 𝑧 = = ( )
𝑧 𝑧

Residue = the co-efficient of (z-0)-1 is 0

 The function has removable singularity at z=0, the residue of f(z) at z=0 is 0

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 1) Residue – at simple pole

R
Z0

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 1) Residue –pole of order ‘m’

R
Z0

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 1) Residue –pole of order ‘m’

R
Z0

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 Cauchy's residue theorem
If f(z) be analytic within and on a simple closed curve C except for a finite
number of isolated singularities inside C then

R
Z0

Sum of residue of f(z) at all the

poles inside C

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 Points to remember

Residue for simple pole

Residue for pole of R

Z0
order n

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 Solve

Find the residue of z -4 exp(iaz) at z = 0

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 solution

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 Solve

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 solution

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 Solve

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 solution

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 Solve

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 solution

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 Solve

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 solution

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 Solve

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 solution

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 Solve

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 solution

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 Solve

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 solution

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 Solve

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 solution

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 solution

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