Two identities involving contour integrals in the presence of a branch point where the integrand explodes, and the Kummer function

Question

I need to understand how to establish two identities. The first is

$$ \int_{C} z^{-1-q}(1-z)^{-1-\lambda } dz=\frac{2 \pi \Gamma (q+\lambda +1)}{\Gamma (\lambda +1) \Gamma (q+1)}, q\geq 0, \lambda >0$$ where C is a contour $z=1/2+ i t, t \in (-\infty,\infty) $

Mathematica believes this

With[{z = l + I t}, 
 Integrate [  
  z^(-q - 1) (1 - z)^(-λ - 1), {t, -Infinity, Infinity}, 
  Assumptions -> {q > 0, λ > 0, 0 < l < 1}]]

When q=0 (RHS is just 2 \pi), this comes from the pole at 0, by using a half circle contour which surrounds it, and blowing it to $\infty$. With q>0, my first idea was to use a rectangle excluding the branch point at 0, but this cannot work since the integrand explodes there

The second identity is a contour integral representation of the HypergeometricU function with powers integrand (unlike the ones given at http://functions.wolfram.com/HypergeometricFunctions/HypergeometricU/07/02/ )

$$ \int_{C} e^{-x z} z^{-1-q}(1-z)^{-\lambda } dz= \frac{2 \pi\ \ e^{-x} x^{q+\lambda} \ U(q+1 , q+\lambda+1,x)}{\Gamma(\lambda ) }, q\geq 0, \lambda >0, x>0$$

Mathematica's NIntegrate confirms this

cnS = {q -> 5/Pi, \[Lambda] -> 4 Pi, l -> 1/3};
int[x_] := Exp[ -x  z] z^(-q - 1) (1 - z)^(-\[Lambda]) /. cnS;
RBr[x_] := 
  Chop[NIntegrate [
    int[x] /. z -> I t + l /. cnS, {t, -Infinity, Infinity}], 10^(-9)];
R[z_] := 2 Pi  Exp[- z] z^
    n HypergeometricU[q + 1, q + \[Lambda] + 1, z]/Gamma[\[Lambda]];
Print["R Bromwich=", RBr[1], " R exact= ", Chop[R[1] //. cnS // N]]

but the hard question for me is how to prove this by providing the right integration contour.

I also had some intreaguing problems when $\lambda$ is not an integer, but they disappeared on my minimal example above:) I am still wondering though whether NIntegrate could give a wrong answer in cases when a parameter switches from integer to noninteger.

Answer 1

Version 12.1 solves the first integral without difficulty.

With[{z = 1/2 + I t}, Integrate[z^(-q - 1) (1 - z)^(-λ - 1), {t, -Infinity, Infinity}, 
    Assumptions -> {q >= 0, λ > 0}]]

(* (2 π Gamma[1 + q + λ])/(Gamma[1 + q] Gamma[1 + λ]) *)

I shall give the second integral some thought tomorrow.

Answer 2

Actually, I do believe that this question is of pure mathematics character and does not belong here. Nonetheless, I provide a short answer. A lot of information on special functions can be found in the so called Bateman Manuscript Project. There are download links at the bottom of the page. Specifically, you need the volume I, Sec. 6.5

As I mentioned in the comment above, another comprehensive source on special functions is the DLMF project. The advantage is that they follow modern notations. The disadvantage is that they do not present all derivations.