How Can You Use Contour Integration to Evaluate the Integral of 1/(X^4 + a^4)?

[blueyellow]

Use suitable contours in the complex plane and the residue theorem to show that (where a
is a real number):


integral from +infinity to -infinity (1/(X^4+a^4))=pi/((a^3)sqrt(2))


i hav tried to do something similar to what one would do for the integral of 1/(x^2+a^2) but it didnt work.please just help me get started on this question. ta in advance

 

[dextercioby]

I don't understand why it wouldn't work. You normally have 4 poles.

 

[lurflurf]

integrate around a square with corners at
0,R,iR,R+iR
and let R->infinity

 

[uart]

integrate around a square with corners at
0,R,iR,R+iR
and let R->infinity

Isn't it easier to just use the upper infinite semi-circle, -infinity to +infinity along the real axis and then close the contour with a semicircle enclosing two of the functions four poles.

 

[lurflurf]

^Some people like to do it that way, but then you have to deal with two poles instead of one. The savings would be even greater if we wanted to do 1/(x^1024+a^1024).