Engineering Maths

1. (1) Prepared by Mr.Zalak Patel
Lecturer, Mathematics
BETA AND GAMMA FUNCTIONS
Introduction to topic :
Weightage for university exam: 06Marks
No. of lectures required to teach: 02hrs
Definition of Gamma function :
The gamma function is denoted and defined by the integral
)0(1
0
mdxxem mx
Properties of Gamma function :
(1) mmm )1(
(2) !)1( mm when m is a positive integer.
(3) aaamamam ).........2)(1()( ,when n is a positive integer.
(4) m = 2 )0(12
0
2
mdxxe mx
(5) )0(1
0
mdxxe
t
m mtx
m
(6)
2
1
(7) .
20
2
dxe x
Definition of Beta function :
The beta function is denoted and defined by the integral
)0,0(,)1(),(
1
0
11
nmdxxxnmB nm
),( nmB is also denoted by ),( nm .
Properties of Beta function :
(1) ),( nmB = ),( mnB
(2) ),( nmB = 2 dnm
2
0
1212
cossin
(3) ),( nmB = dx
x
x
nm
m
0
1
)1(
(4) ),( nmB = dx
x
xx
nm
nm1
0
11
)1(

2. (2) Prepared by Mr.Zalak Patel
Lecturer, Mathematics
Relation between Beta and Gamma functions:
Relation between Beta and Gamma functions is
nm
nm
nmB
.
),(
Using above relation we can derive following results
dqp
2
0
cossin =
2
1
2
2
2
2
1
.
2
1
2
1
,
2
1
qp
qp
qp
B
2
1
Euler s formula:
n
nn
sin
1.
Legendre s formula (or Duplication formula):
12
2
)2(
2
1
. n
n
nn

3. This document was created with Win2PDF available at http://www.daneprairie.com.
The unregistered version of Win2PDF is for evaluation or non-commercial use only.