argument of functions

Question

can anyone explain what the $\arg\Gamma (ix)$ is?

I am largely unclear on what the gamma function is also and how it is defined for complex numbers.

I know how the argument of a function is normally defined, but I am unclear on this gamma function so can't really decide what to do. I have tried a google search and looked at the wiki page but am getting know where really.

score 0 · Accepted Answer · answered Mar 26 '13 at 10:33

0

$$\Gamma(z) = \int_0^{\infty} dt \: t^{z-1} e^{-t}$$

$$\Gamma(i x) = \int_0^{\infty} dt \: t^{i x-1} e^{-t} = \int_0^{\infty} \frac{dt}{t} \: e^{i x \log{t}-t}$$

Expand into sines and cosines

$$\Gamma(i x) = \int_0^{\infty} \frac{dt}{t} \: e^{-t} [\cos{(x \log{t})} + i \sin{(x \log{t})}]$$

From this, in principle, $\arg{\Gamma(i x)}$ may be computed. I would worry about the potential divergence of the integrals near $t=0$.

answered Mar 26 '13 at 10:33

Ron Gordon

138,521

thankyou, did you find this in a text book, if so can you give me the name? – Mar 26 '13 at 11:20
No, that's my own work. – Ron Gordon Mar 26 '13 at 12:16
what would be the best textbook, to find stuff like this in do you think? – Mar 26 '13 at 15:30
Off the top of my head, Arfken for starters: http://www.amazon.com/Mathematical-Methods-Physicists-Seventh-Comprehensive/dp/0123846544 – Ron Gordon Mar 26 '13 at 15:32

argument of functions

1 Answers1