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In about two weeks, I'm going to be giving a presentation on the complex-valued Gamma function $\Gamma(z)$. By definition, I know that $$\Gamma(z)= \int_0^\infty e^{-t}t^{z-1}dt.$$

Now if I let $z=x+iy$, how does the following hold? $$|\Gamma(x+iy)| \leq \Gamma(x).$$

It might actually be something quite simple, but here's what I attempted: $$|\Gamma(x+iy)|= |\int_0^\infty e^{-t}t^{z-1}dt|$$ $$=|\int_0^\infty e^{-t}t^{x+iy-1}dt|$$ $$=|\int_0^\infty e^{-t}t^{x-1}t^{iy}dt|$$ $$\leq|\int_0^\infty e^{-t}t^{x-1}dt|$$ $$=\int_0^\infty e^{-t}t^{x-1 }dt$$ $$=\Gamma(x),$$
where $t \gg 1$. I hope this was the correct procedure.

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Put the absolute value inside the integral (using the triangle inequality $\left|\int f\right| \le \int |f|$), then use $|t^{x+iy}| = t^x$. – Joel Cohen Feb 28 '12 at 22:34
@JoelCohen: thanks! Now I got it. – Libertron Feb 29 '12 at 0:09
You're welcome :) – Joel Cohen Feb 29 '12 at 0:18
up vote 6 down vote accepted

As you've noted, for $x>0$, $$ \begin{align} |\Gamma(x+iy)| &=\left|\int_0^\infty e^{-t}\,t^{x+iy-1}\;\mathrm{d}t\right|\\ &=\left|\int_0^\infty e^{-t}\,t^{x-1}\,e^{iy\log(t)}\;\mathrm{d}t\right|\\ &\le\int_0^\infty\left|e^{-t}\,t^{x-1}\,e^{iy\log(t)}\right|\;\mathrm{d}t\\ &=\int_0^\infty e^{-t}\,t^{x-1}\;\mathrm{d}t\\ &=\Gamma(x)\\ &=|\Gamma(x)|\tag{1} \end{align} $$ For $-1< x<0$, we can use $$ \begin{align} |\Gamma(x+iy)| &=\frac{1}{|x+iy|}|\Gamma(x+1+iy)|\\ &\le\frac{1}{|x|}|\Gamma(x+1)|\\ &=|\Gamma(x)|\tag{2} \end{align} $$ then we can repeat $(2)$ for $-n-1<x<-n$ for $n=1,2,3,\dots$

To finish off, note that $\Gamma(z)$ is not defined (or infinite) for $z$ a non-positive integer.

This yields $|\Gamma(x+iy)|\le|\Gamma(x)|$ for all $x$ where $\Gamma(x)$ is defined (or finite).

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Although the first integral converges for $x>0$ only, the inequality $(2)$ holds because it follows from the functional equation $\Gamma(z+1)=z \, \Gamma(z)$. Is my understanding right? – Américo Tavares Mar 1 '12 at 0:05
@Américo: yes, indeed; you need an application of $(1)$ and two applications of the functional equation for $(2)$. – robjohn Mar 1 '12 at 1:04
Thanks for the explanation. – Américo Tavares Mar 1 '12 at 10:45

Added. For $\text{Re}(z)=x>0$ the Gamma integral converges

$$\Gamma(z) = \int_0^\infty e^{-t} t^{z-1} \,{\rm d}t.$$

Apply the following property of moduli of integrals $$\begin{equation*} \left\vert \int_{0}^{\infty }w(t)dt\right\vert \leq \int_{0}^{\infty }\left\vert w(t)\right\vert dt \end{equation*}$$ to the function $w(t)=e^{-t}t^{x+iy-1}=e^{-t}t^{x-1}t^{iy}$ and observe that

  • $\left\vert t^{iy}\right\vert =\left\vert e^{iy\log t}\right\vert =1,$
  • $\left\vert e^{-t}t^{x-1}\right\vert =e^{-t}t^{x-1}$ for $t>0$.
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