Hello,
i like to find an analytical solution for the following integral:
integral from [0 to pi]
{cos(2nθ)/ (x^2+ cos^2(θ))}dθ
x is constant and n is integer
Appreciate your help!
mopen
I took a look @wolphram alpha:
int_0^(pi) cos(2nt)/(x^2+cos^2 0;t))dt - Wolfram|Alpha
There are two ways to [try] to solve the integral...
a) use the variable transformation...
(1)
b) use the identity...
(2)
... and integrate along the unit circle in the complex plane using the Cauchy integral formulas...
Neither a) nor b) is particulary comfortable but... what do You prefer?...
Kind regards
For the second option for the moment You can keep the term n and the first step is to set so that the integral becomes...
(1)
... that has limits and . Now You use the identities...
(2)
(3)
... and take into Account that is , Your integral, if no mistakes of me, becomes...
(4)
... where is the unit circle. Now all that You have to do is to apply the residue theorem... difficult task but not superior to human possibilities ...
Kind regards
In the previous post we are arrived to the identity...
(1)
... where is the unit circle. Now if we apply the residue theorem the integral (1) is given by...
(2)
... where the are the residues of the poles of the f(z) in (1) internal to the unit circle. In the f(z) has inside the unit circle a pole of order n in and a simple pole in...
(3)
May be is useful to proceed 'step by step', so that we start with n=0. In that case we have only the pole (3) and its residue is...
(4)
... so that is...
(5)
In next post[s] we will consider the cases n>0...
Kind regards
Dear chisigma,
Thank you very much, i just take time to check the validity of my derivation and that the integral is correct. Your solution using the residue theorem is interesting !, but what about n=1?
Best Regards
Mopen
First we rewritre the general formula...
(1)
... where is the unit circle. If and f(z) has two poles of order 1 inside the unit circle: and and their residues are...
(2)
(3)
Now is...
(4)
The result looks like the 'Orient Express' and , because in pure calculus I'm very poor, I suggest mopen and all the people of MHF to control my work. In any case what I can say is that for n=1 the computation efforts are remarkable and for n>1 I prefer to 'pass the ball' ...
Kind regards