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karan4496
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Homework Statement
a) Find the radius of convergence of the following complex series and the complex point, where the center of the disk of convergence is located:
[tex] \sum_{n=1}^{inf} 4^n (z-i-5)^{2n} [/tex]
b) Find the Laurent series of the following function, f(z), about the singularity, z = 2, and find the residue of f(z)
[tex] f(z) = \frac{1}{z(z-2)^3} [/tex]c) Evaluate the following integral:
[tex] \int_{0}^{inf} \frac{dx}{(x^2 + a^2)^4} [/tex]
Homework Equations
Given
The Attempt at a Solution
a) I gather that 5+i is the center of the disk of convergence? Doing the ratio test I get,
|4(z-(5+i))^2| < 1
I'm a bit lost how to solve this from here.
b) I don't know how to go about expanding this as a Laurent series. If it were a Taylor series, I would factor out a 1/-2^3 from 1/(z-2)^3 and then expand the remaining 1/(1-z/2) and cube it. But this gives me the expansion about z = 0.c) You can extend this integral to the complex plane and write
∫(closed) 1/(z^2+a^2)^4 dz
where singularities would be z = +or- i a
And choosing the upper half of the semi circle contour, I only have to deal with the +'ve i a
Then using the Residue equation for poles of higher order,
I find that the integral is 2∏(0) = 0.
But I'm not sure its correct.
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