What is the value of $\Gamma(\mathrm{i})$ ?

Question

What is the value of $\Gamma(\mathrm{i})$ ? $\Gamma(z)$ is Gamma function. Here $\mathrm{i}^2=-1$.Can you help me with this problem ?

Answer 1

According to Wikipedia the value is: $\Gamma(i) = (-1+i)! \approx -0.1549 - 0.4980i$.

Now from J.M.'s comment we know that $|\Gamma(i)|^2 = \frac{\pi}{\sinh \pi}$ but I do not think $\Gamma(i)$ can be expressed by elementary functions.

Wikipedia: http://en.wikipedia.org/wiki/Particular_values_of_the_Gamma_function#Imaginary_unit

edit: more identities (including the one above) can be found at http://mathworld.wolfram.com/GammaFunction.html

Answer 2

So: a method for computing it. The integral formula converges to compute $\Gamma(1+i)$, then the functional equation will give us $\Gamma(i)$ from that. $$ \Gamma(i) = \int_{0}^{\infty} \operatorname{e} ^{-x} \operatorname{sin} \bigl(\operatorname{log} (x)\bigr) d x - i \int_{0}^{\infty} \operatorname{e} ^{-x} \operatorname{cos} \bigl(\operatorname{log} (x)\bigr) d x $$