# Gamma function

1. Sep 3, 2007

### bobbyk

Can Gamma(i) be expressed in terms of elementary functions? I know Mod(Gamma(i)) can.
Bob.

2. Sep 17, 2007

### EnumaElish

3. Sep 17, 2007

### mathman

Gamma(n+1)=n! Is this elementary enough?

Obviously it is useful only for integers.

4. Sep 17, 2007

### bobbyk

mathman, I was asking about Gamma(i), where i is the imaginary unit, not an integer.
Bob

5. Sep 18, 2007

### mathman

Minor quibble - you should have said so in the first place. i (unfortunately) has multiple uses as a symbol. sqrt(-1) or simply an index are two examples.

6. Sep 19, 2007

### Math Jeans

Do you consider an integral to be an elementary function?

Da Jeans

7. Sep 19, 2007

### bobbyk

I'm using the term "Elementary Function" in the technical sense. It's not what I
consider it to mean - the term is well-defined (see wikipedia) and is agreed on by
(almost) all mathematians.
Thanks for responding, though!
Bob

8. Dec 10, 2007

### bobbyk

Gamma(i)

OK, now that you know what I meant, do you have an answer? Or don't you know
what I mean by "elementary function" ?

Bob

9. Dec 11, 2007

### EnumaElish

10. Dec 13, 2007

### Xevarion

I think the OP wants to know if there's a known closed form for $$\Gamma(i)$$, meaning (presumably) a finite combination of elementary functions and algebraic numbers. (So no infinite products, no integrals, and no "So-and-so's constant".)

The answer: I have no idea. My guess is that if there even is one, it would be pretty hard to find.

11. Dec 13, 2007

### Gib Z

12. Dec 14, 2007

### Avodyne

Seems unlikely, since the evaluation of $|\Gamma(i)|^2=\Gamma(i)\Gamma(-i)$ relies on the standard identity $\Gamma(x)\Gamma(-x)=-\pi/x\sin(\pi x)$.

And, $\Gamma(i)$ is not listed here: http://mathworld.wolfram.com/GammaFunction.html

13. Dec 14, 2007

### benorin

I know it's not what you wanted, but the absolute value of the constant is here:

For $$y\in\mathbb{R},$$ we have

$$\left| \Gamma (iy)\right| = \sqrt{\frac{\pi}{y\sinh \pi y}}$$​

this follows from mirror symmetry, i.e. $$\Gamma (\overline{z}) = \overline{\Gamma (z)}$$, and from the formula $$\Gamma (z)\Gamma (-z)=-\frac{\pi }{z\sin \pi z}$$.

Hence we have that

$$\boxed{\left| \Gamma (i)\right| = \sqrt{\frac{2\pi}{e^{\pi} -e^{-\pi}}}}$$​