Integral Representations
If Re b > Re a > 0, M(a,b,z) can be represented as an integral
thus is the characteristic function of the beta distribution. For a with positive real part U can be obtained by the Laplace integral
The integral defines a solution in the right half-plane re z > 0.
They can also be represented as Barnes integrals
where the contour passes to one side of the poles of Γ(−s) and to the other side of the poles of Γ(a+s).
Read more about this topic: Confluent Hypergeometric Function
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