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