Express Gamma(i) in terms of elementary functions, if possible.

(i is the imaginary unit and Gamma is Euler's Gamma Function)

(Elementary functions are as defined in Wickepedia)

Bob.

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- Sep 24th 2007, 12:05 AMbobbykGamma(i)
Express Gamma(i) in terms of elementary functions, if possible.

(i is the imaginary unit and Gamma is Euler's Gamma Function)

(Elementary functions are as defined in Wickepedia)

Bob. - Sep 24th 2007, 03:43 AMJhevon
- Sep 24th 2007, 10:00 AMThePerfectHacker
(It should be $\displaystyle \Gamma(p)$).

Yes, but the beautiful Gamma function is defined for complex numbers with real part positive as well.

$\displaystyle \Gamma(i) = \int_0^{\infty} e^{-z}z^{1-i} dz$

But I am afraid I cannot do this. I do not even want to try. MathWorld has so many identities and non involving imaginary numbers. - Sep 24th 2007, 01:24 PMtopsquark
- Sep 24th 2007, 01:47 PMbobbykGamma(i)
Thanks, all!

I have the numerical value of Gamma(i). Its modulus is an elementary

function: Sqrt(pi/sinh(pi)).

So its real and imaginary parts COULD be elementary functions too, but

I haven't been able to find them.

Bob. - Sep 24th 2007, 04:57 PMThePerfectHacker
- Sep 24th 2007, 05:05 PMThePerfectHacker
$\displaystyle \Gamma (i) = \int_0^{\infty} e^{-z} z^{i-1} dz$

Now, $\displaystyle z^{i} = e^{\log(z)i} = \cos (\log z)+i\sin (\log z)$

Thus,

$\displaystyle \Gamma (i) = \int_0^{\infty} z^{-1}e^{-z}\cos (\log z) dz + i\int_0^{\infty} z^{-1}e^{-z} \sin (\log z) dz$

So we end up with,

$\displaystyle \int_{-\infty}^{\infty} e^{-e^x} \cos x dx + i \int_{-\infty}^{\infty} e^{-e^x} \sin x dx$

What next? - Sep 25th 2007, 05:47 AMtopsquark