The discussion revolves around the gamma function with complex arguments, particularly its application in statistical physics and the calculation of specific values such as \(\Gamma(i)\). Participants explore the definition, convergence issues, and methods for calculating the gamma function in the complex plane.
Participants express disagreement regarding the initial formula presented, with corrections made. There is also a consensus on the need for analytic continuation for calculating \(\Gamma(i)\), but no consensus on the broader implications or applications in statistical physics.
The discussion highlights limitations related to the convergence of the gamma function's integral definition and the necessity of analytic continuation for complex arguments. The specific conditions under which the integral converges are not fully resolved.
This discussion may be useful for those interested in complex analysis, statistical physics, and the properties of special functions like the gamma function.
Your formula is wrong (typo) Should bePetar Mali said:\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?
mathman said:Your formula is wrong (typo) Should be
\Gamma(z)=\int^{\infty}_0x^{z-1}e^{-x}dx
Petar Mali said:Yes mistake. I make corrcection!