Gamma Function Complex Argument: Problems in Stat Phys & How to Calculate

[Petar Mali]

\Gamma(z)=\int^{\infty}_0x^{z-1}e^{-x}dz
z\in\mathhad{C}

In which problems in statistical physics we need gamma functions of complex argument?

I don't know how to calculate \Gamma(i) for exaple?

 

[mathman]

\Gamma(z)=\int^{\infty}_0x^{z-1}e^{-z}dz
z\in\mathhad{C}

In which problems in statistical physics we need gamma functions of complex argument?

I don't know how to calculate \Gamma(i) for exaple?

Your formula is wrong (typo) Should be

\Gamma(z)=\int^{\infty}_0x^{z-1}e^{-x}dx

 

[Petar Mali]

Your formula is wrong (typo) Should be

\Gamma(z)=\int^{\infty}_0x^{z-1}e^{-x}dx

Yes mistake. I make corrcection!

 

[mathman]

Yes mistake. I make corrcection!

Not quite: you still have dz when it should be dx.

 

[g_edgar]

\Gamma(i)

Well, the integral definition converges only if \Re z > 0, so in particular it does not converge at z=i. So you need to use analytic continuation. But fortunately that is very easy for the \Gamma function, unlike most other functions. Use the functional equation \Gamma(z+1) = z\Gamma(z). So to compute \Gamma(i) we can compute \Gamma(1+i) then apply the formula.

You cannot expect a closed-form answer. \Gamma(1+i) \approx .4980156681-.1549498283i so divide by i to get \Gamma(i) \approx -.1549498283-.4980156681i.