Mellin Transforms -What could I be doing wrong?

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  • #1
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Main Question or Discussion Point

I need to compute integral x^(1/4)/(x^2+9)dx from 0 to infinity

so I use a formula

-pie^(-5pi/4i)/sin(5pi4)SUM residues z^(5/4-1)/(z^2+9) so thats sqrt2pie^(-5pi/4i)3^(1/4)/(6i)(e^(pi/8i)-e^(3pi/8i)) (with the 0<arg<2pi branch of sqrt) but e^(-7pi/4)(e^(pi/8i))-e^(3pi/8i)) which is some horrible number...and thats not the answer according to my book

=(
 

Answers and Replies

  • #2
1,838
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Looks like you made a mistake with the phase factors. It is not wise to put in the sin(5 pi/4) factor in there in the way you did.

You choose a convention for the polar angles and that chaoice then determines both the phase factors of the residues and the two phase factors of the two integrals along the real axis. The sum of the latter two phase factors will be proportional to a sinus term up to some phase factor.
 
  • #3
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Looks like you made a mistake with the phase factors. It is not wise to put in the sin(5 pi/4) factor in there in the way you did.

You choose a convention for the polar angles and that chaoice then determines both the phase factors of the residues and the two phase factors of the two integrals along the real axis. The sum of the latter two phase factors will be proportional to a sinus term up to some phase factor.
i tried what you said

but i sitill cant seem to find where i went wrong

the formula in the text says -pie^(-pia)/sin(pia) SUM residues of fz^(a-1) excluding 0

= Integral -inf to inf z^(a-1)f(z)
 
  • #4
1,838
7
Just write down the contour, and precisely write down the the residues and the contribution from the two integrals along the real axis, without talking any shortcuts. If you do that, then I can see where you go wrong if you don't get the right answer.
 

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