The Gamma Function

Many important functions in applied sciences are defined via improper integrals. Maybe the most famous among them is the Gamma Function. This is why we thought it would be a good idea to have a page on this function with its basic properties. You may consult any library for more information on this function.

Historically the search for a function generalizing the factorial expression for the natural numbers was on. In dealing with this problem one will come upon the well-known formula

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A very quick approach to this problem suggests to replace n by x in the improper integral to generate the function

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Clearly this definition requires a close look in order to determine the domain of f(x). The only possible bad points are 0 and tex2html_wrap_inline237 . Let us look at the point 0. Since tex2html_wrap_inline239 when tex2html_wrap_inline241 , then we have

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when tex2html_wrap_inline241 . The p-test implies that we have convergence around 0 if and only if -x < 1 (or equivalently x >-1). On the other hand, it is quite easy to show that the improper integral is convergent at tex2html_wrap_inline237 regardless of the value of x. So the domain of f(x) is tex2html_wrap_inline257 . If we like to have tex2html_wrap_inline259 as a domain, we will need to translate the x-axis to get the new function

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which explains somehow the awkward term x-1 in the power of t. Now the domain of this new function (called the Gamma Function) is tex2html_wrap_inline259 . The above formula is also known as Euler's second integral (if you wonder about Euler's first integral, it is coming a little later).

Basic Properties of tex2html_wrap_inline269

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First, from the remarks above we have

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tex2html_wrap_inline271
One of the most important formulas satisfied by the Gamma function is

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for any x > 0. In order to show this formula from the definition of tex2html_wrap_inline281 , we will use the following identity

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(this is just an integration by parts). If we let a goe to 0 and b goe to tex2html_wrap_inline237 , we get the desired identity.

In particular, we get

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for any x > 0 and any integer tex2html_wrap_inline295 . This formula makes it possible for the function tex2html_wrap_inline281 to be extended to tex2html_wrap_inline299 (except for the negative integers). In particular, it is enough to know tex2html_wrap_inline281 on the interval (0,1] to know the function for any x > 0. Note that since

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we get tex2html_wrap_inline309 . Combined with the above identity, we get what we expected before :

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A careful analysis of the Gamma function (especially if we notice that tex2html_wrap_inline315 is a convex function) yields the inequality

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or equivalently

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for every tex2html_wrap_inline295 and x >0. If we let n goe to tex2html_wrap_inline237 , we obtain the identity

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Note that this formula identifies the Gamma function in a unique fashion.

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Weierstrass identity. A simple algebraic manipulation gives

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Knowing that the sequence tex2html_wrap_inline335 converges to the constant -C, where

\begin{displaymath}C = \lim_{n \rightarrow +\infty} 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} - \ln(n)\end{displaymath}

is the Euler's constant. We get

\begin{displaymath}\Gamma(x) = e^{-Cx} \frac{1}{x} \lim_{n \rightarrow +\infty} \prod_{k=1}^{k=n} \frac{e^{x/k}}{1 + x/k}\end{displaymath}

or

\begin{displaymath}\Gamma(x) = e^{-Cx} \frac{1}{x} \prod_{n=1}^{+\infty} \frac{e^{x/n}}{1 + x/n} \cdot\end{displaymath}

tex2html_wrap_inline271
The logarithmic derivative of the Gamma function: Since tex2html_wrap_inline347 for any x >0, we can take the logarithm of the above expression to get

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If we take the derivative we get

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or

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In fact, one can differentiate the Gamma function infinitely often. In "analysis" language we say that tex2html_wrap_inline281 is of tex2html_wrap_inline359 -class. Below you will find the graph of the Gamma function.

The Beta Function

Euler's first integral or the Beta function: In studying the Gamma function, Euler discovered another function, called the Beta function, which is closely related to tex2html_wrap_inline281 . Indeed, consider the function

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It is defined for two variables x and y. This is an improper integral of Type I, where the potential bad points are 0 and 1. First we split the integral and write

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When tex2html_wrap_inline241 , we have

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and when tex2html_wrap_inline377 , we have

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So we have convergence if and only if x > 0 and y >0 (this is done via the p-test). Therefore the domain of B(x,y) is x > 0 and y>0. Note that we have

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Let a and b such that tex2html_wrap_inline397 , we have (via an integration by parts)

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If we let a goe to 0 and b goe to 1, we will get

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Using the properties of the Gamma function, we get

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or

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In particular, if we let x=y = 1/2, we get

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If we set tex2html_wrap_inline415 or equivalently tex2html_wrap_inline417 , then the technique of substitution implies

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Hence we have

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or

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Using this formula, we can now easily calculate the value of tex2html_wrap_inline425 .

Other Important Formulas:


The following formulas are given without detailed proofs. We hope they will be of some interest.

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Asymptotic behavior of the Gamma function when x is large: We have

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where

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If we take, x=n, we get after multiplying by n

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This is a well known result, called Stirling's formula. So for large n, we have

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tex2html_wrap_inline271
The connection with tex2html_wrap_inline447 : For any x > 0, we have

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which implies

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Using the Weierstrass product formula (for tex2html_wrap_inline281 and tex2html_wrap_inline457 ), we get

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If we use the Beta function (B(x,y)), we get the following formulas:

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This page is inspired by Emil Artin's book on the Gamma Function. The exact reference is: Artin, Emil. The Gamma Function. New York, NY: Holt, Rinehart and Winston, 1964.

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Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996

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