# Using the integral equation, find the eigenvalues and eigenfucntions

The integral equation: $$\int_{-\frac{T}{2}}^{\frac{T}{2}}dt' \phi (t')e^{\Gamma\left | t-t' \right |} =\lambda \phi(t)$$

for $(-\frac{1}{2}T< t < \frac{1}{2}T)$

is useful in photon counting statistics theory.

Show that the eigenvalues $\lambda_k$ of the above equation are given by $$\Gamma \lambda_k = \frac{2}{1+u^{2}_k}$$

where the $u_k$ are the roots of the transcendental equation

$$tan(\Gamma T u_k) = \frac{2 u_k}{u^{2}_k - 1}$$

Using this solve for the eigenfunctions associated

The $\Gamma$ here is a constant. I can see that the kernel is an exponential. I've worked this numerous times using Fredholms technique but repeatidly hit a brick wall. Am I missing something?

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