Proving a dependence between variables

[MathsNewb]

Homework Statement

I want to find the relationship between $\psi$ and $\gamma$ from this equation
$$\Lambda=\sqrt{\gamma ^2+4 \pi ^2 \psi ^2} \sum _{j=0}^{\infty } \frac{\left(\frac{(2 j-1)\text{!} (2 \pi \psi )^j}{(2 j)\text{!} \left(\gamma ^2+4 \pi ^2 \psi ^2\right)^{j/2}}\right)^2}{1-2 j}$$ thereby proving a dependence between $\psi$ and $\gamma$ (where $\psi$,$\gamma$ and $\Lambda$ are variables)

Homework Equations

The above derives from $$\Lambda=\frac{2 \sqrt{\gamma ^2+4 \pi ^2 \psi ^2} E\left(\frac{4 \psi ^2 \pi ^2}{\gamma ^2+4 \pi ^2 \psi ^2}\right)}{\pi }$$

The Attempt at a Solution

Trying to find a closed form approximation.

 

[lippyka]

I tried to find the Taylor series of $E(x)$ and substitute in the equation, however I was unable to obtain a closed form solution. Can anyone help? Thanks in advance.