- #1
shebbbbo
- 17
- 0
The question asks to show using the residue theorem that
[itex]\int[/itex]cos(x) / (x2 +1)2 dx = [itex]\pi[/itex] / e
(the terminals of the integral are -∞ to ∞ but i didnt know the code to write that)
I found the singularities at -i and +i
so i think we change the function inside the integral to cos(z) / (z2 +1)2
i expanded the cos(z) as cosh(1) - isinh(1)(z-i) -0.5cosh(1)(z-i)2 +...
and i expanded (z2+1)2 as -(1/4)(z-i)2 - i/4(z-i) + 3/16 +...
I did the same for the singularity at x=-i and when i added both the residues together i got
(9/16e + e/16) (this is multiplied by 2[itex]\pi[/itex]i to find residues)
this doesn't seem right? i don't know if what I've done is the right method. please help, I've spent soooo many hours on this one stupid question :(
[itex]\int[/itex]cos(x) / (x2 +1)2 dx = [itex]\pi[/itex] / e
(the terminals of the integral are -∞ to ∞ but i didnt know the code to write that)
I found the singularities at -i and +i
so i think we change the function inside the integral to cos(z) / (z2 +1)2
i expanded the cos(z) as cosh(1) - isinh(1)(z-i) -0.5cosh(1)(z-i)2 +...
and i expanded (z2+1)2 as -(1/4)(z-i)2 - i/4(z-i) + 3/16 +...
I did the same for the singularity at x=-i and when i added both the residues together i got
(9/16e + e/16) (this is multiplied by 2[itex]\pi[/itex]i to find residues)
this doesn't seem right? i don't know if what I've done is the right method. please help, I've spent soooo many hours on this one stupid question :(