Gamma Function Problems

• Jan 24th 2010, 01:09 PM
mathisfunforme
Gamma Function Problems
Hi,
I am having trouble with problems from the following book: Mathematical Methods in the Physical Sciences, Third Edition, Mary L. Boas. Specifically, I am looking at problem 11.3.8 and 11.3.9.

The directions state, "Express each of the following integrals as a $\displaystyle \Gamma$ function.

For #8, the problem is stated

$\displaystyle \int_a^\infty x^{2/3} e^{-x} dx$

I get $\displaystyle \Gamma(\frac{5}{3})$ based on the definition of the gamma function as defined by the text.

The gamma function is defined as follows in the text, $\displaystyle \Gamma(p) = \int_0^\infty x^{p-1}e^{-x}dx.$

However, I am not sure if this is correct and is someone willing to verify the answer?

For #9, I am not sure where to begin:

The problem is stated $\displaystyle \int_0^\infty e^{-x^4}dx$ Hint: Put $\displaystyle x^4 = u$

What am I supposed to do with this? The hint doesn't help me much. I still have to integrate with respect to u and I don't know how to put dx in terms of du if this is the case. Does anyone have any suggestions of where to begin or how to set-up this problem?

Thank you.
• Jan 24th 2010, 01:19 PM
tonio
Quote:

Originally Posted by mathisfunforme
Hi,
I am having trouble with problems from the following book: Mathematical Methods in the Physical Sciences, Third Edition, Mary L. Boas. Specifically, I am looking at problem 11.3.8 and 11.3.9.

The directions state, "Express each of the following integrals as a $\displaystyle \Gamma$ function.

For #8, the problem is stated

$\displaystyle \int_a^\infty x^{2/3} e^{-x} dx$

I get $\displaystyle \Gamma(\frac{5}{3})$ based on the definition of the gamma function as defined by the text.

The gamma function is defined as follows in the text, $\displaystyle \Gamma(p) = \int_0^\infty x^{p-1}e^{-x}dx.$

However, I am not sure if this is correct and is someone willing to verify the answer?

For #9, I am not sure where to begin:

The problem is stated $\displaystyle \int_0^\infty e^{-x^4}dx$ Hint: Put $\displaystyle x^4 = u$

Well, do what they say: $\displaystyle u=x^4 \Longrightarrow du = 4x^3dx\Longrightarrow dx= \frac{du}{4u^{3\slash 4}}$ , so:

$\displaystyle \int\limits_0^\infty e^{-x^4}\,dx=\frac{1}{4}\,\int\limits_0^\infty e^{-u}u^{-3\slash 4}du=\frac{1}{4}\Gamma\!\!\left(\frac{1}{4}\right)$.

Tonio

What am I supposed to do with this? The hint doesn't help me much. I still have to integrate with respect to u and I don't know how to put dx in terms of du if this is the case. Does anyone have any suggestions of where to begin or how to set-up this problem?

Thank you.

.