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Math Help - Gamma Function Problems

  1. #1
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    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  \Gamma function.

    For #8, the problem is stated

     <br /> <br />
\int_a^\infty x^{2/3} e^{-x} dx <br /> <br />

    I get  \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,  \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  \int_0^\infty e^{-x^4}dx Hint: Put 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.
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  2. #2
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    Oct 2009
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    Quote Originally Posted by mathisfunforme View Post
    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  \Gamma function.

    For #8, the problem is stated

     <br /> <br />
\int_a^\infty x^{2/3} e^{-x} dx <br /> <br />

    I get  \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,  \Gamma(p) = \int_0^\infty x^{p-1}e^{-x}dx.


    Both your answer and the Gamma Function's definition are correct


    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  \int_0^\infty e^{-x^4}dx Hint: Put x^4 = u


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


     \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.
    .
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