31. When and are positive integers, it follows from the definition of the gamma function that: 32. Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts. 33. Where the numerator is the upper incomplete gamma function and the denominator is the gamma function. 34. Where the numerator is the upper incomplete gamma function and the denominator is the gamma function . 35. *In general, you have to calculate the Gamma function for the argument plus one. 36. In general, when computing values of the gamma function , we must settle for numerical approximations. 37. C . H . Brown derived rapidly converging infinite series for particular values of the gamma function : 38. Where B ( ) is the Beta function and \ Gamma ( ) is the Gamma function . 39. Which is an entire function, defined for every complex number, just like the reciprocal gamma function . 40. We can replace the factorial by a gamma function to extend any such formula to the complex numbers.
