I have defined the following function:
$f(z)=\sin^2(π\frac{\Gamma(x)+1}{x})+\sin^2(πx)$
I want to count its zeros along the positive real axis up to a point, call it $x=A$.
By the Cauchy argument principle, the number of zeros should be:
$\frac{1}{2πi}\oint \frac{f'(z)}{f(z)} dz$
..as long as the contour is strictly along the positive real axis, and "some" distance away from $0$, where there is an essential singularity.
I also understand that if the contour is selected such that it does not cross a branch cut of the complex logarithm then the following work holds:
$Z=\frac{1}{2πi}\oint \frac{f'(z)}{f(z)} dz$
$=\frac{1}{2πi}\oint \frac{dL(f(z)}{dz} dz$
$=\frac{1}{2πi}\int_a^b \frac{dL(f(z(t))}{dz}\frac{dz}{dt} dt$
$=\frac{1}{2πi}\int_a^b \frac{dL(f(z(t))}{dt} dt$
$=\frac{1}{2πi}\int_a^b \frac{dL(f(z(t))}{dt} dt$
$=\frac{1}{2πi}[L(f(z(b)))-L(f(z(a)))]$
As long as I carefully choose $z(t)$ to encircle the real axis specifically (thinking along the lines of a Hankel contour, but cut off on both sides), then is this correct working?
I understand that I cannot be near $z=0$ or the negative real axis because of the singularity and the branch cut, respectively. But, other than that, am I correct up to that point?