The Gamma Function and Its DerivativeDate: 05/26/98 at 12:39:33 From: Eric Geib Subject: Derivative of n! N! = the integral from 0 to infinity of (x ^ n)(e ^ -x) dx Does this hold true for all real numbers and not just non-negative integers? If so, does the derivative of n! with respect to n exist? Date: 05/26/98 at 15:56:10 From: Doctor Rob Subject: Re: Derivative of n! The function given by the above integral is called the Gamma function. Actually, you have written Gamma(n+1). It is true that for natural numbers n, Gamma(n+1) = n!. It is also true that Gamma is defined for all nonnegative real numbers. Furthermore, Gamma(n) is differentiable, so you could define the derivative of n! with respect to n to be the derivative of Gamma(n+1) with respect to n. Gamma can be extended to a function defined for all complex numbers except for 0 and negative integral values, at which points it has a pole. This function is also called the Gamma function, and it is also infinitely differentiable wherever it is defined. -Doctor Rob, The Math Forum http://mathforum.org/dr.math/ Date: 05/26/98 at 16:39:47 From: Evan M Glazer Subject: Re: Derivative of n! Dr. Math, Thanks for the quick response to my question. I have one more follow up. What is (and how would you determine) the derivative of the Gamma function? Thanks! -Eric Geib Date: 05/27/98 at 17:02:42 From: Doctor Rob Subject: Re: Derivative of n! When the integral converges absolutely, as this one does, you can find the derivative by interchanging the integration and differentiation: Gamma(n) = Integral x^(n-1)*e^(-x) dx from 0 to infinity dGamma(n)/dn = Integral d(x^[n-1]*e^[-x])/dn dx from 0 to infinity = Integral d(e^[{n-1}*ln(x)-x])/dn dx from 0 to infinity = Integral x^(n-1)*e^(-x)*ln(x) dx from 0 to infinity In general, for any integer k > 0, d^k(Gamma[n])/dn^k = Integral x^(n-1)*e^(-x)*ln(x)^k dx from 0 to infinity I am quite sure that this cannot be integrated in closed form in terms of the familiar functions of calculus, so it must be evaluated either by means of an infinite series or by numerical methods. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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