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Problem 10.8_2 Use the residue theorem to evaluate the following integral. $. (z+4)³ dz 24+523 +62²

Problem 10.8_2 Use the residue theorem to evaluate the following integral. $. (z+4)³ dz 24+523 +62²

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 10.8_2 Use the residue theorem to evaluate the following integral. (z+4)³ dz 24 +523 +62² |=|=1
Transcribed Image Text:Problem 10.8_2 Use the residue theorem to evaluate the following integral. (z+4)³ dz 24 +523 +62² |=|=1
Expert Solution
Step 1

Here we need to find the integral of given function in unit circle with center at origin by using residue theorem

Now in the given integral of the function

                       f(z)=(z+4)3z4+5z3+6z2 has pole at

        z4+5z3+6z2=0⇒  z2(z2+5z+6)=0⇒ z2(z+2)(z+3)=0⇒                         z=0,0,-2,-3 

But z=-2,-3 are not lies in the unit circle with center at origin,

Therefore, z=0 is the only singularity of f(z) lie in unit circle as a pole of order 2.

Now according to the residue theorem, 

The formula for residue of f(z) at z=a of order m is

  R1=1(m-1)!limz→a dm-1dzm-1 (z-a)m f(z) , as here m=2

Therefore, the residue of f(z) at z=0

 

 

 

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