Question

Complex analysisss
Where l is an integer, and |ξ| is a complex number with |ξ| < 1, consider the integral (detail in picture)

Answer 1

\[f(l,\xi)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\frac{e^{-il\Theta }d\Theta }{1-\xi \cos(\Theta )}\]

Prove, without evaluating the integral, that

\[f(l,\xi)=f(-l,\xi)=f(\left | l\right |,\xi)=f(-\left | l \right |,\xi)\]

Answer 2

Hint: use the change of variable:
\[\phi = -\theta\]
Is this hint sufficient?

Answer 3

Note: If l is an integer, the integral is a countour integral.

Answer 4

Use the hint by Bobo-i-bo and it should be very obvious. Also note that l equal |l| or -|l| and -l equals the other one of |l| or -|l|.

Answer 5

yes i found out after i make it into 2 integrand(cos and isin), isin part will=0, and by even function of cos, it can proof this question

Answer 6

You are not allowed to evaluate the integral though...

Answer 7

Just use the substitution \(\theta=-\phi\) or \(\phi=-\theta\). They may make a difference in other integrals but not this one