improper integral #5

Printable View

June 30th 2011, 02:37 PM
Random Variable

improper integral #5

Evaluate $\int^{\infty}_{0} \frac{e^{-x^{p}} - e^{-x^{q}}}{x} \ dx \ \ p,q>0$ .

Then argue that $\int^{\infty}_{0} \frac{e^{-x^{p}} - e^{-x^{q}}}{x} \ dx = \int^{\infty}_{0} \frac{\cos (x^{p}) - \cos(x^{q})}{x} \ dx$ .
June 30th 2011, 06:49 PM
Drexel28

Re: improper integral #5

Quote:

Originally Posted by Random Variable

Evaluate $\int^{\infty}_{0} \frac{e^{-x^{p}} - e^{-x^{q}}}{x} \ dx \ \ p,q>0$ .

Then argue that $\int^{\infty}_{0} \frac{e^{-x^{p}} - e^{-x^{q}}}{x} \ dx = \int^{\infty}_{0} \frac{\cos (x^{p}) - \cos(x^{q})}{x} \ dx$ .

Spoiler:

Let $I_{p,q}$ denote our integral. Then, evidently

$\displaystyle I(p,q)=-\int_0^{\infty}\int_q^p e^{-x^y}x^{y-1}\log(x)\;dy\,dx$

Switch the order of integration (Fubini's theorem is satisfied) to get

$\displaystyle I(p,q)=-\int_0^{\infty}\int_q^p x e^{-x^y}x^y\log(x)\;dx\,dy$

Rewrite this as

$\displaystyle I(p,q)=-\int_q^p\frac{1}{y^2}\int_0^{\infty}ye^{-x^y}x^{y-1}\log\left(x^y\right)\; dx\,dy$

Make the obvious substitution ( $z=x^y$ ) on the inside to turn it into

$\displaystyle I(p,q)=\left(-\int_q^p\frac{dy}{y^2}\right)\left(\int_0^{\infty} e^{-x}\log(x)\text{ }dx\right)$

Thus, the first integral being easy, and the second being the definition of the Euler-Mascheroni constant we find that

$\displaystyle I(p,q)=\frac{q-p}{pq}\gamma$

To prove the second assertion merely write the exponentials in complex notation and separate the real and imaginary parts. The theorem then becomes immediate from the common theorem that

$\displaystyle \int_0^{\infty}\frac{\sin(ax)-\sin(bx)}{x}\;dx=0\quad a\ne b$
June 30th 2011, 07:40 PM
Random Variable

Re: improper integral #5

I didn't even think there was a way to write it as a double integral that would make evaluating it easier.

I used another integral representation of $\gamma$ , namely that $\int^{\infty}_{0} \Big(\frac{1}{1+x} - e^{-x} \Big) \ \frac{dx}{x} = -\psi_{0}(1) = \gamma$ .

Can you also show that $\int^{\infty}_{0} \frac{ e^{-x^{p}} - e^{-x^{q}}}{x} \ dx = \int_{0}^{\infty} \frac{\cos(x^{p}) - e^{-x^{q}}}{x} \ dx$ ?
July 1st 2011, 01:29 PM
Random Variable

Re: improper integral #5

Drexel28

Could you explain your reasoning for the second part of the question in more detail? I feel really stupid because I don't understand where the integral of the form you posted even comes into play.