Integral (related to Laplace transform)

[Pzi]

Hi.

Homework Statement

$$\int\limits_0^{ + \infty } {\frac{{dt}}{{{e^{st}} \cdot (1 + {e^t})}}}$$

Homework Equations

The same as...
$$Laplace\left[ {\frac{1}{{1 + {e^t}}}} \right]$$

The Attempt at a Solution

Found no elegant properties related to Laplace transform here.
So figured my best shot would be to integrate directly...

Any suggestions?

 

[Ray Vickson]

Hi.

Homework Statement

$$\int\limits_0^{ + \infty } {\frac{{dt}}{{{e^{st}} \cdot (1 + {e^t})}}}$$

Homework Equations

The same as...
$$Laplace\left[ {\frac{1}{{1 + {e^t}}}} \right]$$

The Attempt at a Solution

Found no elegant properties related to Laplace transform here.
So figured my best shot would be to integrate directly...

Any suggestions?

I think it is a non-elementary integral. By changing variables to u = exp(t), and recognizing that for s > 0 we have exp(-s*t) = exp(-s*ln(u)) = u^(-s), the integral becomes
$$J = \int_1^{\infty} \frac{ u^{-s-1}}{1+u} \, du,$$ which Maple 11 evaluates as
$$J = -\text{LerchPhi}(-1,1,-s) - \pi \csc(\pi s),$$
where LerchPhi is the function defined as
$$\text{LerchPhi}(z,a,v) = \sum_{n=0}^{\infty} \frac{z^n}{(v+n)^a}$$ if $|z| < 1$ or $|z|=1$ and $\text{Re}(a) > 1.$ It is extended to the whole complex z-plane by analytic continuation.

RGV

 

[Pzi]

I think it is a non-elementary integral. By changing variables to u = exp(t), and recognizing that for s > 0 we have exp(-s*t) = exp(-s*ln(u)) = u^(-s), the integral becomes
$$J = \int_1^{\infty} \frac{ u^{-s-1}}{1+u} \, du,$$ which Maple 11 evaluates as
$$J = -\text{LerchPhi}(-1,1,-s) - \pi \csc(\pi s),$$
where LerchPhi is the function defined as
$$\text{LerchPhi}(z,a,v) = \sum_{n=0}^{\infty} \frac{z^n}{(v+n)^a}$$ if $|z| < 1$ or $|z|=1$ and $\text{Re}(a) > 1.$ It is extended to the whole complex z-plane by analytic continuation.
RGV

Yea, looks like it's indeed a non-elementary case here...
As this is a part of some differential equations I will work around it (using convolution), but it will be ugly.

Thanks for clarification!