Asymptotic formula for complex gamma function at $+\infty+i \times y$

Question

I am currently looking for the behaviour of the complex gamma function at real infinity:

$\lim_{x \to \infty}\Gamma\left(x+i\times y\right)$

and more particularly for asymptotic formulas for the following functions:

$f_1\left(y\right)=\text{Re}\left(\Gamma\left(+\infty+i\times y\right)\right)$

$f_2\left(y\right)=\text{Im}\left(\Gamma\left(+\infty+i\times y\right)\right)$

As a graphical reminder, the behaviour of the complex gamma function for positive real parts is:

And when we cut slices for increasing $x$ values we obtain:

which look likes damped cosine/sine. So, is there asymptotic formulas for these functions at infinity ?

$f_1\left(y\right)=\text{Re}\left(\Gamma\left(+\infty+i\times y\right)\right)$

$f_2\left(y\right)=\text{Im}\left(\Gamma\left(+\infty+i\times y\right)\right)$

Answer 1

This is just Stirling approximation: $$\Gamma(z)=\sqrt{2\pi}\exp\left\{z\ln z-z-\frac{1}{2}\ln z\right\}\Bigl[1+o(1)\Bigr].$$ In particular, as $x\rightarrow+\infty$, one has \begin{align} \mathrm{Re}\,\Gamma(x+iy)&\sim \sqrt{\frac{2\pi}{x}}\,\left(\frac{x}{e}\right)^x\cos\left(y\ln x\right)\Bigl[1+o(1)\Bigr],\\ \mathrm{Im}\,\Gamma(x+iy)&\sim \sqrt{\frac{2\pi}{x}}\,\left(\frac{x}{e}\right)^x\sin\left(y\ln x\right)\Bigl[1+o(1)\Bigr]. \end{align} The observed oscillations are produced by the cosine/sine of $y\ln x$.