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.