Solving Integrals from Hell for n = 1, 1/2, 2, 3/2

[l'Hôpital]

Hi, I was just wondering if these integrals could be solved analytically, or if I would just have to resort to approximations.

$$\int_{0}^{\infty} \sqrt{1 + \omega E^2} E^n ln(1 + \omega E^2) \frac{e^{\phi E}}{(\lambda e^{\phi E} + 1)^2} dE$$

For

$$n = 1, 1/2, 2, 3/2$$

 

[jasonRF]

any restrictions on any parameters? For example, is it true that $$\omega$$ is real and positive? Is $$\phi$$ real? Any restrictions on $$\lambda$$?

 

[l'Hôpital]

No restrictions. But if you'd like to set some, go for it.

I'd just rather avoid approximation unless they wouldn't perturb the value by much.

 

[jasonRF]

Surely there must be some restrictions. For example, if $$\lambda = 0$$ then the real part of $$\phi$$ must be negative, otherwise the integral doesn't converge. Are there any special cases that are particularly interesting?

I am playing with it, but don't have my hopes up.

By the way, where did this integral come from? It is unusual to have so many arbitrary parameters (meaning all can be arbitrary complex numbers) in the practical problems I usually run into. jason

 

[l'Hôpital]

Truth be told, I don't know if there are any restrictions.

A friend asked me for my opinion on these integrals, if there was a way to solve them analytically. Of course, I was stumped, so I figured I'd do him the favor of posting these up in PF.

I'll try to see if I can find some restrictions on them.