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dear sir,
functions of complex variable
Q: ∫1-cos2(Z-3)/(z-3)³ (z-3 is having power of 3)  

it need to be solved by residue theorem

first calculate residue then use theorem 2pi i (sum of residue)

can you provide solution to this I am having exam very soon (i am getting 4pi i) but
the ans given is 8pi i(i is iota)

thanks

Answer
First of all, around what contour are you integrating? That's a pretty important part of this question! I will assume counterclockwise around some closed loop containing z=3.

I suggest you use the residue theorem, if that is what the problem says, but could also use the Cauchy integral formula, so I'm not sure why you think it "need" to be solved by the residue theorem. Here is the correct/best solution, which uses the Cauchy formula:

∫ f(z)/(z-a)^(n+1) = 2 π i f^(n)(a) / n!

where n! is factorial, f^(n) is the nth derivative of f, and i is the imaginary unit, i = &sqrt;(-1).

So f(z) = 1 - cos²(z), a=3, and n=2.

f''(z) = 2 ( cos²(z-3) - sin²(z-3) )

f''(3) = 2 ( cos²(0) - sin²(0) ) = 2

2 π i f''(3) / 2! = 2 π 2 i / 2! = 2 π i

That is the correct answer.

Likewise, if you compute the residue -- which can be done by expanding the series:

( 1 - cos²(z) ) / (z-3)³ = 1/(z-3) - (z-3)/3 + (2/45) (z-3)³ - ...

the residue being the coefficient of the (z-3)^(-1) term, so residue = 1.

Then 2 π i is still the answer.


Since your answer is 4 π i, and the answer given is 8 π i, there are two possibilities:

1. You and the book are wrong.

2. I am wrong, and one of the other answers is right. This would be true if the curve is not a simple loop. If the winding number of the curve is 2, you are right. If it is 4, the book is right.

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