# Is it even related to the Gamma Function?

Calculus Level 4

Let $$f(z, s)$$ be an analytic function of $$z$$ defined for $$\Re (z) >0$$, and by the summation formula defined for $$0<x<1$$ and $$s > 0$$ as - $\large f(x,s) = \sum_{n=0}^\infty \frac{ (-1)^nx^{n+s}}{n+s}$ Using analytic continuation and extending the domain of $$x$$ to all positive real numbers in $$\dfrac{\partial}{\partial x} f(x,s)$$, if $$y = \displaystyle \int_0^\infty \frac{\partial}{\partial x}f\left(x,\frac16\right) dx$$, find $$\cos\left(\dfrac{3y}{2}\right)$$. Give your answer to two decimal places 

Relevant wiki: Gamma function

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