I am reading a mathematical physics textbook and came across this line that I didn't Understand:
"$$\phi= \frac{1}{2i}\ln\frac{\Gamma(\frac{1}{2}+ia)}{\Gamma(\frac{1}{2}-ia)}$$ with the branch of the logarithm function that is zero when $a=0$ "
I am familiar with the idea that the complex logarithm is multivalued unless we specify a branch, for example $$\ln(z)=\ln(|z|)+i\arg(z)+i2\pi n$$ with $\arg(z)$ between $-\pi$ and $\pi$. If we choose $n=0$ that would be the principle branch. What I don't understand is how the statement that the logarithm is zero when $a=0$ help us determine the branch in the $\phi$ expression. Any help is appreciated.
When $a=0$, $\dfrac{\Gamma\left(\frac12+ia\right)}{\Gamma\left(\frac12-ia\right)}=1$. So, take the branch of the logarithm $\operatorname{Log}\colon\Bbb C\setminus(-\infty,0]\longrightarrow\Bbb C$ defined as follows: if $z\in\Bbb C\setminus(-\infty,0]$, write $z$ as $\rho e^{i\theta}$, with $\theta\in(-\pi,\pi)$, and then $\operatorname{Log}(z)=\log\rho+i\theta$.