An integral and a deravative of a simple factorial

• Jan 2, 2004

• #1

MathematicalPhysicist

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how can you find the integral and the derevative of a simple factorial f(x)=x! (to find what f'(x) equals and what Sf(x)dx equals)? as i see it you have progressive multiplications, f(x)=x(x-1)(x-2)...*(x-k), which is the product of x-k where k=0 till infinity, should i take logarithms on both sides? if i have asked this before link me to the thread.
thanks in advance.

 

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• Jan 2, 2004

• #2

master_coda

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Well, you can't integrate or differentiate the traditional factorial function. You would have to take advantage of the fact that [itex]n!=\Gamma(n+1)[/itex] and operate on the gamma function instead.

I can't help you with that though. I don't know a lot about the gamma function.

 

• Jan 2, 2004

• #3

HallsofIvy

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Integration and differentiation require, at least, that the function be defined on some interval of real numbers. The factorial function is only defined for non-negative integers. You can, as master_coda said, use the gamma function instead.

 

• Sep 20, 2009

• #4

elsapt

1

0

you cannot differentiate a factorial...
For a function to be differentiable, it has to be continuous. For discrete functions like x! the derivative does not exist.

As for the integration goes, theoretically, it is possible to integrate x!.
I am not sure though, that the gamma function approach will work. The result of the gamma function integration gamma(x+1) leads to x!. Hence replacing x! by its gamma forms leads to a double integral which will be more difficult to solve.

 

• Sep 21, 2009

• #5

The_Fool

2

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The factorial function is not continuous, so you just use [itex](x-1)!=\Gamma(x)[/itex]. There is no known indefinite integral of the Gamma function. However, it does have a derivative in terms of itself and another function.
[itex]\Gamma '(x)=\Gamma(x) \psi(x)[/itex] where [itex]\psi[/itex] is known as the digamma function.

 

• Sep 21, 2009

• #6

Count Iblis

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You can express the integral of Log[Gamma(z)] in terms of the Barnes G-function:

http://mathworld.wolfram.com/BarnesG-Function.html

 

• Sep 21, 2009

• #7

The_Fool

2

0

Count Iblis said:

You can express the integral of Log[Gamma(z)] in terms of the Barnes G-function:

http://mathworld.wolfram.com/BarnesG-Function.html

That can also be expressed with the polygamma function:

[itex]\int log[\Gamma(z)]dx=\psi^{(-2)}(z)+C[/itex]

 

• Sep 21, 2009

• #8

Count Iblis

1,863

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The_Fool said:

That can also be expressed with the polygamma function:

[itex]\int log[\Gamma(z)]dx=\psi^{(-2)}(z)+C[/itex]

Yes, but then the poygamma function of order minus two or smaller is nothing more than the (repeated) integral of Log(Gamma). The properties of these functions are not trivial. Barnes and others investigated the related Barnes G-function, otherwise Barnes would not have been bothered to do that.

 

1. What is an integral of a simple factorial?

An integral of a simple factorial is a mathematical operation that allows us to find the area under the curve of a factorial function. It is represented by the symbol ∫ and is the inverse operation of a derivative.

2. What is a derivative of a simple factorial?

A derivative of a simple factorial is a mathematical operation that allows us to find the rate of change of a factorial function at a specific point. It is represented by the symbol d/dx and is the inverse operation of an integral.

3. How do you calculate the integral of a simple factorial?

To calculate the integral of a simple factorial, you can use the formula ∫x^n = (x^(n+1))/(n+1) + C, where C is a constant. You can also use integration rules and techniques such as substitution, integration by parts, and partial fractions.

4. How do you find the derivative of a simple factorial?

To find the derivative of a simple factorial, you can use the formula d/dx(x^n) = nx^(n-1), where n is the power of the factorial function. You can also use differentiation rules and techniques such as the power rule, product rule, and quotient rule.

5. What is the relationship between the integral and derivative of a simple factorial?

The integral and derivative of a simple factorial are inverse operations of each other. This means that if you take the integral of a factorial function and then the derivative of the resulting function, you will get back the original function. Similarly, if you take the derivative of a factorial function and then the integral of the resulting function, you will get back the original function. This relationship is known as the Fundamental Theorem of Calculus.