Laplace transforms; Abel's integral equation

[bdforbes]

Using Laplace transforms, find the solution of Abel's integral equation:

$$\int^{x}_{0}\frac{f(u)}{\sqrt{x-u}}du = 1 + x + x^2$$

I recognized that the integral is a Laplace convolution, leading to:

$$(f*g)(x) = 1+x+x^2$$

where $$g(x)=x^{-1/2}$$

So:

$$L(f*g)=L(1)+L(x)+L(x^2)$$

$$L(f)L(g)=\frac{1}{p}+\frac{1}{p^2}+\frac{2}{p^3}$$

I can't figure out the transform of g(x). I tried contour integration in the first quadrant, indenting around the origin and placing the branch cut along the negative real axis, and I got to this:

$$L(g)=\int^{\infty}_{0}\frac{e^{-px}}{\sqrt{x}}dx=i\int^{\infty}_{0}\frac{e^{-ipx}}{\sqrt{ix}}dx$$

Can anyone help me solve this last integral, or suggest another way to find the transform?

 

[bdforbes]

Okay, I realized that I was making it too complicated by trying contour integration. I saw the change of variables x=y^2 for L(g), and got L(g)=sqrt(pi/p).

So now I'm left with:

$$L(f) = \sqrt{\frac{p}{\pi}}(\frac{1}{p}+\frac{1}{p^2}+\frac{2}{p^3})$$

$$= \pi^{-1/2}(p^{-1/2}+p^{-3/2}+2p^{-5/2})$$

Obviously I now know the inverse transform of the first term. The other two I can't get. I tried Bromwich contour integration, but the branch cut presents some difficulties there. Any ideas?

 

[bdforbes]

I found the other transforms using some guesswork. I guess they can't be solved using a Bromwich contour, which to me has some interesting meta-mathematical implications.