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## Homework Statement

whose Fourier transform is f

^{~}(p) = 1/(a

^{2}+ p

^{2})

## Homework Equations

f(x) = 1/√2π ∫

^{∞}

_{-∞}e

^{ipx}f

^{~}(p)

## The Attempt at a Solution

First of all I let f(z) = e

^{ixz}/(z

^{2}+ a

^{2})

and γ = γ

_{1}+ γ

_{2}

with the ϒ's parametrised by:

γ

_{1}: {z=t, -R<t<R}

γ

_{2}: {z=Re

^{it}, 0<t<π}

(So a semicircle of radius R)

In this contour the only pole that lies inside is the z= +ia

so using Cauchy's residue theorem:

∫

_{ϒ}f(z)dz = 2πi (Res(f, z=ia))

I found the residue of z=ia to be Rez(f,z=ia)= -i e

^{-ax}/2a

So ∫

_{ϒ}f(z)dz= π e

^{-ax}/a

And I have something in my notes about the fact you have to check that the integral of f(z) on γ

_{2}goes to zero as R goes to infinity, which I vaguely understand because we actually want the integral from -∞ to +∞.

So the checks I have in my notes are:

Suppose γ has length L and on γ |f(z)|<M

Then |∫

_{γ}f(z)dz|≤M⋅L

With γ ≡ {z=γ(t), a≤t≤b}

L = ∫

_{a}

^{b}dt|γ'(t)|

On γ

_{2}: |z| =R

Using ||a|-|b|| ≤ |a+b| ≤ |a|+|b|

R

^{2}-a

^{2}≤ |z

^{2}+a

^{2}| ≤ R

^{2}+a

^{2}

So 1/R

^{2}-a

^{2}≥ 1|z

^{2}+a

^{2}| ≥ 1R

^{2}+a

^{2}

Therefore |∫

_{γ2}f(z)dz|≤πR/(R

^{2}-a

^{2})

So ∫

_{γ2}f(z)dz → 0 as R → ∞

as you have |∫

_{γ2}f(z)dz| ≤ e

^{-Rxsin(t)}/(R

^{2}- a

^{2})

and if x ≥0 the exponent is, at most, equal to 1, but if x<0, it blows up and we need a different contour for x<0.

Let γ = γ

_{1}+ γ

_{3}

with ϒ

_{3}: {z=Re

^{it}, π<t<2π}

so a semicircle in the negative part, and this time it's a clockwise contour so Cauchy's residue must carry a minus sign on the right hand side.

So doing everything as before I found ∫

_{ϒ}= πe

^{ax}/a

But I'm having trouble with the checks, I found that the inequality for the modulus of the integral over ϒ

_{3}comes out the same? So effectively it still blows up?

But I know this is the right answer I just can't follow through with the checks.

I appreciate this is quite long, but a lot of it is just background to set the scene, and I think I just have a lack of understanding of the checks and just know the procedure.

Thanks in advance