How do I find the zeros of:
1/Gamma[1 - I x]
The zeros should be $ -i\, n$ with $ n>0 $
I tried with Solve[1/Gamma[1 - I x]==0,x], but I get the message:
Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
So I tried with Reduce[1/Gamma[1 - I x]==0,x], but I get False as a result
f[x_] = 1/Gamma[1 - I x];
Plot[f[y*I], {y, -5, 5}, PlotRange -> All]
FindRoot[f[y*I] == 0, {y, #}] & /@ Range[-5.25, -1.25]
(* {{y -> -5.}, {y -> -4.}, {y -> -3.}, {y -> -2.}, {y -> -1.}} *)
Assuming[{Element[n, Integers], n > 0},
Limit[f[-m*I], m -> n]]
(* 0 *)
x has to be pure imaginary to give a real argument to Gamma. Plot of the function suggests that the roots are near the negative integers. FindRoot verified that starting the search near a negative integer would converge to the integer. However, Gamma is undetermined (ComplexInfinity) at negative integers, so the roots only exist in the Limit. The last Mathematica statement verifies root for all negative integers.
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Apr 15, 2018 at 14:45
Here is my proposal. The product of two gamma functions $\Gamma(z)$ and $\Gamma(w)$, with arguments satisfying the condition that $z+w$ is an integer, as in our case with $z=ix$ and $w=1-ix$, then:
$$\Gamma(z) \Gamma(1 - z) =\frac{\pi}{\sin(\pi z)}$$
So:
Reduce[Gamma[I x] Sin[I \[Pi] x]/\[Pi] == 0, x]
$$c_1\in \mathbb{Z}\land \left(\left(c_1\leq -1\land \left(x=2 i c_1\lor x=2 i c_1+i\right)\right)\lor \left(i x\notin \mathbb{Z}\land \left(x=2 i c_1\lor x=\frac{2 i \pi c_1+i \pi }{\pi }\right)\right)\right)$$
which includes your expected result.
Gamma[z] Gamma[1 - z] == Pi/Sin[Pi z]withz=I x... $\endgroup$