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## Homework Statement

1. Evaluate the following integrals using residues:

a)

[tex]\int _0 ^{\infty} \frac{x^{1/4}}{1 + x^3}dx[/tex]

b)

[tex]\int _{-\infty} ^{\infty} \frac{\cos (x)}{1 + x^4}dx[/tex]

c)

[tex]\int _0 ^{\infty} \frac{dx}{p(x)}[/tex]

where p(x) is a poly. with no zeros on {x

__>__0}

d)

[tex]\int _{-\infty} ^{\infty}\frac{\sin ^2(x)}{x^2}dx[/tex]

2. Let A be a complex constant lying outside the real interval [-1,1]. Using residues, prove that:

[tex]\int _{-1} ^1 \frac{dx}{(x-A)\sqrt{1-x^2}} = \frac{\pi }{\sqrt{A^2 - 1}}[/tex],

with the appropriate determination of [itex]\sqrt{A^2 - 1}[/itex].

## Homework Equations

*Let f(z) be analytic except for isolated singularities a*

[tex]\frac{1}{2\pi i}\int _{\gamma }f(z)dz = \sum _j n(\gamma , a_j)\mbox{Res} _{z=a_j}f(z)[/tex]

for any cycle [itex]\gamma[/itex] which is homologous to zero in [itex]\Omega[/itex] and does not pass through any of the points a

_{j}in a region [itex]\Omega[/itex]. Then[tex]\frac{1}{2\pi i}\int _{\gamma }f(z)dz = \sum _j n(\gamma , a_j)\mbox{Res} _{z=a_j}f(z)[/tex]

for any cycle [itex]\gamma[/itex] which is homologous to zero in [itex]\Omega[/itex] and does not pass through any of the points a

_{j}.## The Attempt at a Solution

1.a) I made the substitution z = x

^{1/4}, giving:

[tex]\int _0 _{\infty} \frac{x^{1/4}}{1 + x^3}dx[/tex]

[tex]= 4\int _0 ^{\infty} \frac{z^4}{1 + z^{12}}dz[/tex]

[tex] = 2\int _{-\infty} ^{\infty} \frac{z^4}{1 + z^{12}}dz[/tex]

[tex] = 4\pi i\sum _{\mbox{Im} (z) > 0}\mbox{Res}f(z)[/tex]

I know how to give expressions for these residues, but I don't know a good way to compute this thing. I've used rotationaly symmetry to express this as (a sum of 6 things) times (one of the residues) but it's still ugly.

b)

[tex]\int _{-\infty} ^{\infty} \frac{\cos x}{1 + x^4}dx[/tex]

[tex] = \mbox{Re}\left (\int _{-\infty} ^{\infty} \frac{e^{ix}}{1 + x^4}dx \right )[/tex]

[tex] = \mbox{Re}\left (2\pi i \sum _{\mbox{Im} (z) > 0} \mbox{Res} \frac{e^{iz}}{1 + z^4} \right )[/tex]

I know the relevant poles are [itex]e^{3i\pi /4}[/itex] and [itex]e^{i\pi /4}[/itex], so I know how to find expressions for the residues at these poles, but again I don't have a neat way to compute this.

c) If p is constant or linear, the integral doesn't exist. Otherwise, the integral does exist, but I have no clue really how to compute it for arbitrary p.

d) Again, not much clue.

2. Well I can compute that the residue at A is (1 - A

^{2})

^{-1/2}. It's a matter of making a clever choice of arc over which to integrate, or possibly a parametrized family of arcs and then taking the limits as the parameters of the family tend to desired limits, but I can't see what this clever choice would be. Any hints?

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