the gamma function.docx

7/27/2019 The Gamma Function.docx

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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 wethought 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

A very quick approach to this problem suggests to replace n byx in the improperintegral to generate the function

Clearly this definition requires a close look in order to determine the domain

off(x). The only possible bad points are 0 and . Let us look at the point 0.

Since when , then we have

when . The p-test implies that we have convergence around 0 if and only if -x < 1 (or equivalentlyx >-1). On the other hand, it is quite easy to show that the

improper integral is convergent at regardless of the value ofx. So the domain

off(x) is . If we like to have as a domain, we will need totranslate the x-axis to get the new function

which explains somehow the awkward termx-1 in the power oft. Now the domain

of this new function (called the Gamma Function) is . The aboveformula is also known as Euler's second integral (if you wonder about Euler's

first integral, it is coming a little later).


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Basic Properties of

First, from the remarks above we have

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

for anyx > 0. In order to show this formula from the definition of , we

will use the following identity

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

In particular, we get

for anyx > 0 and any integer . This formula makes it possible for the

function to be extended to (except for the negative integers).

In particular, it is enough to know on the interval (0,1] to know thefunction for anyx > 0. Note that since

we get . Combined with the above identity, we get what we

expected before :

A careful analysis of the Gamma function (especially if we notice

that is a convex function) yields the inequality


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or equivalently

for every andx >0. If we let n goe to , we obtain the identity

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

Weierstrass identity. A simple algebraic manipulation gives

Knowing that the sequence converges to

the constant -C, where

is the Euler's constant. We get

or


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The logarithmic derivative of the Gamma function: Since foranyx >0, we can take the logarithm of the above expression to get

If we take the derivative we get

or

In fact, one can differentiate the Gamma function infinitely often. In

"analysis" language we say that is of -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 . Indeed, consider the function


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

write

When , we have

and when , we have

So we have convergence if and only ifx > 0 andy >0 (this is done via the p-

test). Therefore the domain ofB(x,y) isx > 0 andy>0. Note that we have

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

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

Using the properties of the Gamma function, we get

or


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In particular, if we letx=y = 1/2, we get

If we set or equivalently , then the technique of

substitution implies

Hence we have

or

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

Other Important Formulas:

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

some interest.

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

where

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, wehave

The connection with : For anyx > 0, we have

which implies

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

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

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, Rinehartand Winston, 1964.

the gamma function.docx

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