Is gamma function derivative of factorial?

iVenky

I was searching for derivative of factorial. Many say that gamma function is the derivative of the factorial. Is that true because I searched about gamma function and it doesn't say anything like that.

Thanks a lot

Mute

No, the Gamma function is a generalization of the factorial to non-integer values, it is not the derivative of it. The factorial, strictly speaking, has no derivative because it is not a continuous function. However, the Gamma function is a continuous function, and so does have a derivative (except where the Gamma function is singular). This is the closest you will get to something which can be described as "the derivative of the factorial".

arildno

The gamma function can be regarded as an EXTENSION of the factorial function, in the sense that whenever we let the gamma function have a natural number "n" as its argument, the function value of the gamma function equals the factorial value (n-1)!

The gamma function extends the factorial function in that the gamma function is well defined for a lot of other numbers as well, not just for the naturals, to which the factorial is restricted.

JJacquelin

Hi !

As already said :
- The factorial function is not derivable.
- The extension of the factorial function, i.e. the Gamma function, is derivable.
About the derivative of the Gamma function, see:
http://mathworld.wolfram.com/DigammaFunction.html
The logarithmic derivative of the Gamma function is the digamma function.