Solve Complex Integral: Residual Calculus?

[aaaa202]

Homework Statement

I have an integral of the form:

∫₀^∞exp(ax+ibx)/x dx
What is the general method for solving an integral of this kind.

Homework Equations

Maybe residual calculus?

The Attempt at a Solution

 

[brmath]

Residues sound like the only way. However, make sure you are using $e^{-z}/z$; with a positive exponent your integral will not converge.

 

[Dick]

I don't see how it's going to converge no matter what a is. It's also divergent near x=0.

 

[brmath]

I don't see how it's going to converge no matter what a is. It's also divergent near x=0.

So true. Wish I had read the lower limit as 0 instead of 1. Can I blame it on bad eyesight?

 

[aaaa202]

I need it to converge badly. But I know what mistake I made. I wanted the integral to be the imaginary part of the above. At least I think so. I have attached the whole exercise now as pdf. Is it correct what I have done so far and how do I evaluate the integral?

 

[Ray Vickson]

I need it to converge badly. But I know what mistake I made. I wanted the integral to be the imaginary part of the above. At least I think so. I have attached the whole exercise now as pdf. Is it correct what I have done so far and how do I evaluate the integral?

It all looks incorrect. You wanted to integrate something like $\exp(-cr)/r \, \exp(ikr\ cos(\theta))$ over $R^3$ in spherical corrdinates. The volume element in spherical coordinates is not $dr \, d\theta \, d\phi$; it is $r^2 \sin(\theta)\, dr \, d \theta \, d \phi$.

 

[aaaa202]

Right, I wrote that in a rush I can see. So basically I forgot the sin(theta) in the first line but it should be there or I couldn't make the substitution dcos(theta). Also the r^2 should be there and 1/r I forgot too so I would end up with having to integrate the imaginary part of r times the expression on the last line. But still with that, I don't see how I can solve that integral.

 

[Ray Vickson]

I need it to converge badly. But I know what mistake I made. I wanted the integral to be the imaginary part of the above. At least I think so. I have attached the whole exercise now as pdf. Is it correct what I have done so far and how do I evaluate the integral?

Your final integral is "elementary" and is the type of thing you learned to do in Calculus 101. Look at it again.