I am trying to determine whether $\Gamma(x+iy)\rightarrow 0$ as $y\rightarrow\infty$. How should I go about doing it?
I was trying to see if I could get anything from $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin\pi z}$ but although $|\sin z|\rightarrow\infty$ as $y\rightarrow\infty$, I think it does not follow that $\Gamma(z)\rightarrow 0$. Am I right?
Another approach I was trying was a change of variables by letting $u=\ln t$ so that (for $x>0$,) $\Gamma(x+iy)=\int_{-\infty}^{\infty}e^{xu}e^{-e^{u}}e^{iyu}du$. I have a couple of questions about this. First, is $e^{xu}e^{-e^{u}}$ integrable over the real line? Next, is there something about Fourier transforms that I can use here (perhaps the Riemann-Lebesgue lemma)?
First a simple estimate: $$ \begin{align} |\Gamma(z)| &=\left|\;\int_0^\infty t^{z-1}e^{-t}\,\mathrm{d}t\;\right|\\ &\le\int_0^\infty t^{\mathrm{Re}(z)-1}e^{-t}\,\mathrm{d}t\\ &=\Gamma(\mathrm{Re}(z))\tag{1} \end{align} $$ As CYC mentions, this estimate can be used to show that $$ |\Gamma(z)|=\frac{|\Gamma(z+1)|}{|z|}\le\frac{|\Gamma(\mathrm{Re}(z+1))|}{|z|}\tag{2} $$ Which gives the desired decay.
Polynomial Decay
In fact, for $z=x+iy$, we get $$ \begin{align} |\Gamma(x+iy)| &=\frac{|\Gamma(z+n)|}{|z(z+1)(z+2)\dots(z+n-1)|}\\[6pt] &\le\frac{|\Gamma(x+n)|}{|y|^n}\tag{3} \end{align} $$ which gives decay faster than any power of $1/|y|$.
Exponential Decay
Using estimate $(3)$ and $\frac{\Gamma(x+n)}{n!}\le(n+x)^x$, we get $$ \begin{align} e^{\alpha|y|}|\Gamma(x+iy)| &\le\sum_{n=0}^\infty\frac{\alpha^n|y|^n}{n!}\frac{|\Gamma(x+n)|}{|y|^n}\\ &\le\sum_{n=0}^\infty\alpha^n(n+x)^x\\ &=C(\alpha,x)\tag{4} \end{align} $$ which converges for $\alpha<1$. Thus, for $\alpha<1$ $$ |\Gamma(x+iy)|\le C(\alpha,x)e^{-\alpha|y|}\tag{5} $$
Particular Value
Since $\Gamma$ is real on the real axis, $$ \Gamma\left(x-iy\right)=\overline{\Gamma\left(x+iy\right)}\tag{6} $$ Therefore, applying the reflection formula for $\Gamma$ yields $$ \left|\Gamma\left(\tfrac12+iy\right)\right|^2=\frac{\pi}{\cosh(\pi y)}\tag{7} $$