Searching for roots of complex function

Question

I'm searching for the roots of a complex function of $x$ with parameters $k$ and $q$ $$ \begin{align*} f(x;q,k)&= 2 i q\log(-2 ik)+i\pi-2 i\;\text{Im}\big(\log(\Gamma(1+2 i q))\big)\\ &+\log\left(\frac{\Gamma(1+i q+i qx/k)}{\Gamma(1+iq+iqx/k)}\right)+\log\left(\frac{\sqrt{1-x}-\sqrt{-1-x}}{\sqrt{1-x}+\sqrt{-x-1}}\right) \end{align*} $$ where $k=\sqrt{(-x+1)(-x-1)}$, $x$ is complex with $\text{Re}(x)<-1$ and $\text{Im}(x)>0$ and $q$ is real and positive parameter. If I change the seed in FindRoot slightly, the result changes dramatically. How may I efficiently find these roots?

Code:

2 I*q*Log[-2*I*Sqrt[(-x + 1) (-x - 1)]] + I*π -  2*I*Arg[
Gamma[1 + 2*I*q]] + Log[Gamma[1 + I*q - I*q*x/Sqrt[(-x + 1) (-x - 1)]]] - 
Log[Gamma[1 - I*q - I*q*x/Sqrt[(-x + 1) (-x - 1)]]] + Log[Sqrt[-x + 1] - 
Sqrt[-x - 1]] - Log[Sqrt[-x + 1] + Sqrt[-x - 1]]

Answer 1

one of the best way to finding the roots of complex functions is Davidenko's Method of Complex Root Search in this way you can find the exact root of complex functions by changing your equation into a differential equation. Here is a sample of this method you can use this way.

(*Davidenko's Method of Complex Root Search*)
Clear[γ]
Pm = D[P, γ];
Intv = .0045 + .0041 I;
γ = a[t] + I *b[t];
g = Re[P];
h = Im[P];
gb = Re[Pm];
ga = -Im[Pm];
Pg = (gb)^2 + (ga)^2;
tf = 20
sol = NDSolve[{a'[t] == ((-gb*g + ga*h)/Pg), 
   b'[t] == ((-ga*g + gb*h)/Pg), a[0] == Re[Intv], 
   b[0] == Im[Intv]}, {b, a}, {t, 0, tf}]
Print[Plot[Evaluate[{b[t], a[t]} /. sol], {t, 0, tf}]]
a[tf] /. sol
b[tf] /. sol

In this method P is your equation that should be equal to zero and Gamma is the variable of your equation and Intv is the initial value that you want to find the root near.