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.

 

[CellCoree]

for your question! I am always happy to clear up any misconceptions or confusion about mathematical concepts. So, let's dive into the relationship between the gamma function and the factorial function.

First of all, it is important to understand that the gamma function and the factorial function are two different mathematical functions. The factorial function, denoted by the symbol "!", is defined as the product of all positive integers less than or equal to a given number. For example, 5! = 5*4*3*2*1 = 120.

On the other hand, the gamma function, denoted by the symbol "Γ", is a continuous function that extends the concept of factorial to all real and complex numbers (except for negative integers). It is defined as Γ(z) = ∫0∞ t^(z-1) * e^(-t) dt, where z is a complex number.

Now, to answer your question, the gamma function is not the derivative of the factorial function. However, there is a relationship between the two functions. The gamma function can be thought of as an extension of the factorial function, as it contains the same values as the factorial function for positive integers.

In fact, one can show that Γ(n+1) = n!, where n is a positive integer. This relationship is often used in mathematical equations and calculations, but it does not mean that the gamma function is the derivative of the factorial function.

In conclusion, the gamma function and the factorial function are two distinct mathematical functions with a relationship that extends to positive integers. The gamma function is not the derivative of the factorial function. I hope this explanation helps to clarify any confusion. Happy exploring!