Sent Again to Nandakumar on 29.1.2013

Sent Again to Nandakumar on 29.1.2013

4.1.2013

Advanced Calculus Unit X

BETA AND GAMMA FUNCTIONS - Part I

Objectives

From this session a learner is expected to achieve the following

Familiarise with beta and gamma functions
Study the convergence of beta and gamma functions
Learn some properties of beta function
Study the Recurrence Formula For Gamma Function

Contents

1. Introduction

2. The Beta Function

3. Convergence of Beta Function

4. Properties of Beta Function

5. The Gamma Function

6. Convergence of Gamma Function

7. Recurrence Formula For Gamma Function

Introduction

In this session we introudce beta and gamma functions. The convergence of beta and gamma functions will be discussed. We will see that in some cases beta function is a proper integral and in some other cases it is an improper integral. However, in any case beta function is convergent. Some properties of beta function and recurrence formula for gamma function will be discussed.

The Beta Function

If are positive, then the definite integral is called the Beta function, (or Beta Integral) denoted by. That is, . …(1)

The beta integral is some times called Eulerian Integral of the first kind.

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Convergence of Beta Function

For the beta function given by (1) is a proper integral and hence is convergent.
If then is an improper integral of the second kind. The convergence is verified as follows:

We have

Let and .

Then

Convergence of

We take and

Then,as .

i.e.,a non zero finite number.

Hence, by Limit Comparison Test (Quotient test)and converge or diverge together.

Butconverges if and only if i.e., . Hence converges if and only if

Convergence of

We take and .

Then,, a non zero finite number.

Hence, by Limit Comparison Test, both the integrals and converges or diverge together. But converges if and only if i.e., if and only if . Hence converges if and only if

Therefore ifand, both and converges and henceconverges.

Example 1 Express in terms of a Beta function.

Solution

Let

Put . Then and

Also, when , ; when

Hence ,

, using the fact that

Example 2Express in terms of a Beta function.

Solution

Put so that

Hence

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Properties of Beta Function

1.The beta function is symmetric in mand n.

i.e.,

Proof. We have

obtained by taking and in the result

2.When n is a positive integer,

Proof.

Case 1) If n = 1,

Case 2) If n is an integer and n> 1, we have

by integrating by parts

By repeated application, we have

since by Case 1,

... (2)

3.When m is a positive integer, proceeding as in Property 2, we obtain

4.If m and n are positive integers, then… (3)

This can be obtained by multiplying both numerator and denominator of Eq. (2) by .

This can be obtained by putting in Eq.(3).

6. If m and n are positive integers, then

… (4)

Proof. In the given integral, we put so that When x = a, y = 0; and when x = b, y = 1. Hence,

The given integral

This can be obtained by putting in Eq.(4).

8..

Proof. Put so that and . When x = 0,  = 0 and when x = 1,  = /2.



9. .

Proof.Consider the expression. Put so that and. When x = 1, y = 0 and when Hence

10. where

Proof. In the expression for , put so that and . When x = 0, z = 0 and when Hence

since

Example 3 Show that

Solution

By an earlier example, we have

…(5)

In (5), put or and we obtain

The above formula can be used to evaluate integrals. For example,

Example4 Express in terms of Beta function, the integral

(

Solution

Putso that

Also when we have and when

Hence

Example5 Express in terms of beta function.

Solution

We have

Here put so that

Also when , and when Therefore, we have

Example6 Express in terms of Beta Functions, the integral

Solution

Here put so that

Also when and when .Therefore, we get

Example 7 If prove that

(i)

(ii)

Solution

We have

by integrating by parts

… (6)

Also from (6), we obtain

i.e.,

Hence

Therefore, … (7)

From (6) and (7), we get

… (8)

(ii) From (8), we get

… (9)

and … (10)

Adding (9) and (10), we have

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The Gamma Function

If the definite integral is called the Gamma Function and is denoted by Thus

It is also called Eulerian integral of the second kind. Also

Convergence of Gamma Function

The Gamma function is an improper integral of the first kind if and is an improper integral of the third kind . Now we can write

, where

and.

Case1:

When, is a proper integral and is an improper integral of the first kind.

Here

Take.

Then .

Sinceconverges, by Limit Comparison Test, we have

is convergent.

Hence, converges if

Case 2:

When , is an improper integral of the second kind and is an improper integral of the first kind.

Here

Take.

Then, a non zero finite number.

Since converges if i.e., if by Limit Comparison Test, it follows that converges if . As in Case 1 we can show that converges for all

Hence, converges for all

Combining the results obtained in Case 1 and Case 2, we see that converges for all

Example8 (i) is a Gamma function.

(ii) is a Gamma function .

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Recurrence Formula For Gamma Function

Solution

We have

, by integrating by parts

since, by applying L’Hospital Rule repeatedly,

Example 9 When n is a positive integer, show that

Solution We have

When n is a positive integer, by repeated application of the above formula

But

Hence

when n is a positive integer.

Remark

when n is a positive integer.

Example 10 Show that

Solution

Putting in , we have

Hence

Example 11 Show that

Solution

Put ; so that

Also

Also when x = 0, y = 0; and when x = 1, y = 1. Hence

Summary

In this session the concepts of beta and gamma functions have been introduced. The convergence of beta and gamma functions have been discussed. We have seen that in some cases beta function is a proper integral and in some other cases it is an improper integral. Some properties of beta function and recurrence formula for gamma function have been discussed.

Assignments

1. Express in terms of beta functions.

2. Evaluate .

3. Evaluate

4. Show that

5. Prove that

Quiz

1. In terms of beta functions, the integral is ______

(a)

(b)

(c)

(d)

Ans. (b)

2. The value of is ______

(a)

(b)

(c)

(d)

Ans. (a)

3. The value of is _____

(a)

(b)

(c)

(d)

Ans. (c)

4. The value of is ______

(a)

(b)

(c)

(d)

Ans. (a)

5. Fill in the blanks: = ______

(a)

(b)

(c)

(d)

Ans. (d)

FAQ

1. Is beta function a proper integral?

Answer. Not always. For the beta function given by

is a proper integral and hence is convergent. If then is an improper integral of the second kind. In this case also beta function is convergent.

2. Is gamma function a proper integral?

Answer. Gamma function is always an improper integral. The Gamma function is an improper integral of the first kind if and is an improper integral of the third kind .

3. What do you mean by an improper integral?

Answer. The definition or evaluation of the integral

does not follow from the discussion on Riemann integration since the interval is not bounded. Such an integral is called an improper integral of first kind. The theory of this type of integral resembles to a great extent the theory of infinite series. If is continuous on , then

The definition of

does not follow from the discussion on Riemann integration because the function f defined by

is not bounded. Note, however, that f is bounded (and continuous) onfor every This suggests treating

as the

which equals to .

In general, if for all such that but we define as the ordered pair where

We say thatconverges to A if.We say that diverges if does not converge. The integral is called an improper integral of the second kind.

Improper Integrals of the third kind can be expressed in terms of improper integrals of the first and second kinds.

4. What is the statement of limit Comparison Test ( Quotient Test)

Answer: If for , and are unbounded at and if , then

(a) If or i.e., if A is a non zero finite number, then the two integrals and converge or diverge together;

(b) If and converges, then converges;

and

Glossary

Beta function(Beta Integral) :If are positive, then the definite integral is called the Beta function, (or Beta Integral) denoted by. That is,

.…(1)

The beta integral is some times called Eulerian Integral of the first kind.

Gamma Function: If the definite integral is called the Gamma Function and is denoted by Thus

It is also called Eulerian integral of the second kind.

References:

1. T. M. Apostol, Mathematical Analysis, Narosa Publishing House, New Delhi, 1985.

2. R. R. Goldberg, Real Analysis, Oxford & I.B.H. Publishing Co., New Delhi, 1970.

3. D. Soma Sundaram and B. Choudhary, A First Course in Mathematical Analysis, Narosa Publishing House, New Delhi, 1997.