Why is the gamma function equal to (n-1) ?

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Homework Help Overview

The discussion centers around the equality of the gamma function and factorials, specifically the relationship \(\Gamma(n) = (n-1)!\). The subject area involves properties of the gamma function and its connection to calculus and factorials.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the conditions under which the equality holds, particularly focusing on integer values of \(n\). Questions arise regarding the necessary calculus properties for understanding the gamma function, and some mention integration techniques such as integration by parts.

Discussion Status

There is an ongoing exploration of the gamma function's properties, with some participants providing insights into its relationship with factorials and integration methods. However, there is no explicit consensus on the understanding of the topic, as some participants express confusion and seek further clarification.

Contextual Notes

Some participants note that the equality holds specifically for integer values of \(n\), and there is mention of the need for integration skills to grasp the concepts discussed. Additionally, a reference to a talk on the gamma function introduces an external perspective that may influence the discussion.

michonamona
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Homework Statement



Why is the equality below true?

\Gamma(n) = (n-1)!

Where \Gamma(n) = \int^{\infty}_{0} x^{n-1} e^{-x}dx

Homework Equations


The Attempt at a Solution



I've read the article on wikipedia but I cannot understand it. Is there any special properties in calculus that I must know in order to comprehend this?

Thank you
M
 
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It is true only if n is integer

Solve:

\int_0^\infty t^{4-1}e^{-t} dt

Compare if it's equal to
(4-1)!

Gamma function is really beautiful beacuse it extends the concept of factorial out of integer numbers.
I've read the article on wikipedia but I cannot understand it. Is there any special properties in calculus that I must know in order to comprehend this?

Yep, being able to integrate. :()
 
You can use integration by parts to show that

\Gamma(n) = (n-1)\Gamma(n-1)

If n is an integer, you can use this to prove by induction that

\Gamma(n) = (n-1)!
 
I went to a talk by John Chapman on this and he said said that the Gamma function is related to "Runge phenomenon".
 

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