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 $\Gamma(i) \approx -.1549498283-.4980156681i$.