I am trying to solve a system of $2$ equations in $(x,\gamma)$ but I can't really proceed further to simplify the system in the following:
\begin{equation} c = \int_{k=1}^\infty2k^{-3} \dfrac{\gamma( (1 - x)^k -1 )}{\gamma( (1-x)^k - x^k) - (1-x^k)} \end{equation} \begin{equation} x = \int^\infty_{k=1}k^{-2} \dfrac{\gamma( (1 - x)^k -1 )}{\gamma( (1-x)^k - x^k) - (1-x^k)} \end{equation}
In which $c$ is a positive constant smaller than $1$. I considered using approximations, of which the most meaningful was a log transformation which in simulations showed a closed fit. However, given that the log transformation would be in the form of $\log(1 + \cdot\cdot\cdot)$ it prevents me from simplifying the integral by taking out the denominator of either ratio. Any idea would be greatly appreciated!
For those who are interested in the above, I solved my problem by deriving a simple bound on the main object common to both integrals and studied the problem for the two bonds:
\begin{equation} \gamma(1 - (1-x)^k) \leq \dfrac{\gamma((1-x)^k -1)}{\gamma((1-x)^k - x^k) - (1-x^k)} \leq x^k + \gamma(1 - x^k) \end{equation}