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 .
5.
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
7.
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
(c) If and diverges, then diverges.
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.
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