How would I go about proving that, for a curve in the complex plane ##\alpha## and a real number ##\beta##,(adsbygoogle = window.adsbygoogle || []).push({});

$$\exists\alpha,\beta: \frac{x}{2\pi i}\int\limits_\alpha \frac{\Gamma(z+\frac{1}{2})\Gamma(-z)x^{\beta z}}{\Gamma(\frac{3}{2}-z)}\, dz = \arctan{x}?$$

The poles of the integrand are ##z_0\in\left\{-(n+\frac{1}{2}), n, n+\frac{3}{2}\right\}## for a non-negative integer n. This is as far as I've gotten in terms of work. I can't think where to go from here. I don't know if it's true, but either way I intend to prove or disprove it.

And then, if I can prove the existence of a curve and a real number that satisfy the relation, how might I find those?

Thanks for any help in advance.

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# Proving a conjecture in complex analysis

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