Factorial Function: Does it Hold for Non-Integer N?

In summary, the factorial function is a mathematical function denoted by the symbol ! and is used to calculate the product of all positive integers from 1 up to a given number n. It is often used in combinatorics and probability to calculate the number of ways to arrange a set of objects or events. The factorial function does not hold for non-integer values of n and cannot be used for negative values of n. Alternative functions, such as the Gamma function and the Pochhammer symbol, can be used for non-integer values. However, the factorial function has limitations, including its applicability only to small values of n and potential computational errors for large values of n.
  • #1
Pacopag
197
4

Homework Statement



I was just wondering if
[tex]n!=n(n-1)![/tex]
is completely general. Does it hold even for non-integer n?


Homework Equations





The Attempt at a Solution

 
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  • #2
In general, n! is only for integers.

(But it is possible to compute nCr for n being negative or a non-integer.)
 
  • #3
How would you define n! for non-integer in order to have it hold?

The gamma functions, [itex]\Gamma(z)= \int_0^\infty t^{z-1} e^{-t}dt[/itex] has the property that [itex]\Gamma(z)= (z-1)![/itex] for z a positive integer, and is defined for all z except negative integers.
 

1. What is the factorial function and how is it typically used?

The factorial function is a mathematical function denoted by the symbol ! and is used to calculate the product of all positive integers from 1 up to a given number n. It is often used in combinatorics and probability to calculate the number of ways to arrange a set of objects or events.

2. Does the factorial function hold for non-integer values of n?

No, the factorial function is only defined for positive integers. It does not hold for non-integer values of n, as there is no concept of taking the product of non-integer numbers.

3. Are there any alternative functions that can be used for non-integer values?

Yes, there are alternative functions such as the Gamma function and the Pochhammer symbol that can be used to extend the concept of factorial to non-integer values. These functions are commonly used in statistics and mathematical analysis.

4. Can the factorial function be used for negative values of n?

No, the factorial function is only defined for positive integers. It cannot be used for negative values of n, as there is no concept of taking the product of negative numbers.

5. Are there any limitations to using the factorial function?

Yes, the factorial function can only be used for relatively small values of n due to the rapid growth of the function. It also cannot be used for non-integer or negative values of n. Additionally, the factorial function can lead to computational errors for large values of n due to overflow of the computer's memory.

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