Can Gamma(i) be Expressed in Terms of Elementary Functions? Bob

[bobbyk]

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

 

[EnumaElish]

Look under "Alternative definitions" in: http://en.wikipedia.org/wiki/Gamma_function

 

[mathman]

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

Obviously it is useful only for integers.

 

[bobbyk]

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

 

[mathman]

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

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.

 

[Math Jeans]

Do you consider an integral to be an elementary function?

Da Jeans

 

[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.
I don't understand your question about an integral.
Thanks for responding, though!
Bob

 

[bobbyk]

Gamma(i)

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.

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

 

[EnumaElish]

You have an answer, in the Wikipedia link I posted.

 

[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.

 

[Gib Z]

http://en.wikipedia.org/wiki/Particular_values_of_the_Gamma_function

It's not in there, and nothing I can find on Google either. Lose all hope now.

 

[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

 

[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

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

You should look here for more info (functions.wolfram.com).

Also http://dlmf.nist.gov/Contents/GA/ .