Created by T. Madas
GAMMA
FUNCTION
Created by T. Madas
Created by T. Madas
SUMMARY OF THE GAMMA FUNCTION
The Gamma function Γ ( x ) , is defined as
Γ( x) ≡

∞
t x −1 e −t dt ,
0
where x ≠ ... − 4, − 3, − 2, − 1, 0 .
Gamma function common rules and facts
•
Γ ( x + 1) ≡ x Γ ( x )
•
Γ ( n + 1) ≡ n ! , n ∈ »
•
Γ (1) = 1 , Γ ( 2 ) = 1 , Γ 1 = π
2
•
Γ ( 0 ) = ±∞ , Γ ( −1) = ±∞ , Γ ( −2 ) = ±∞ , Γ ( −3) = ±∞ , etc
•
Γ ( z ) Γ (1 − z ) ≡
•

1
Γ′ ( x ) = Γ ( x )  −γ + +

x

x∈»
Γ ( n ) ≡ ( n − 1)! , n ∈ »
or
( )
π
, z ∈»
sin π z
∞

k =1

1 
1
−


 k x + k 

Γ( x)
Created by T. Madas
Created by T. Madas
Question 1
Evaluate each of the following expressions, leaving the final answer in exact
simplified form.
a)
3Γ ( 6 )
.
Γ ( 4)
( )
b) Γ 3 .
2
c)

∞
x7 e − x dx .
0
V , 60 , 1 π , 7! = 5040
2
Created by T. Madas
Created by T. Madas
Question 2
Evaluate each of the following expressions, leaving the final answer in exact
simplified form.
a)
60Γ ( 5 )
.
Γ (7)
( )
b) Γ 5 .
2
c)

∞
2 x 4 e1− x dx .
0
V , 2 , 3 π , 48e
4
Created by T. Madas
Created by T. Madas
Question 3
Evaluate each of the following expressions, leaving the final answer in exact
simplified form.
( )
( )
Γ −1
2 .
a)
1
Γ
2
( )
b) Γ − 3 .
2
c)

∞
x3 e−4 x dx .
0
V , −2 , 4 π , 3
3
128
Created by T. Madas
Created by T. Madas
Question 4
By using techniques involving the Gamma function, find the value of

∞
x3 e
− 12 x 2
dx .
0
V , MM3-B , 2
Question 5
By using techniques involving the Gamma function, find the value of

∞
x e− x dx .
0
V, 4
Created by T. Madas
Created by T. Madas
Question 6
By using techniques involving the Gamma function, find the value of

∞
1
( ln x )3
x2
dx .
V, 6
Question 7
By using techniques involving the Gamma function, find the exact value of

∞
2
2 x e − x dx .
0
Give the answer in the form Γ ( k ) , where k is a rational constant.
( )
V, Γ 3
4
Created by T. Madas
Created by T. Madas
Question 8
By using techniques involving the Gamma function, find the exact value of

∞
2
x 6 e −4 x dx .
0
Give the answer in the form k π , where k is a rational constant.
V,
Created by T. Madas
15 π
2048
Created by T. Madas
Question 9
By using techniques involving the Gamma function, find the exact value of

1
0
  1 
ln  x  
  
a −1
dx ,
where a ≠ 1, 0, − 1, − 2, − 3, ...
Give the answer in terms of a Gamma function.
V , MM3-A , Γ ( a )
Question 10
By using techniques involving the Gamma function, show that

1
[ln x ]n
n
dx = ( −1) n ! , n ∈ » .
0
proof
Created by T. Madas
Created by T. Madas
Question 11
By using techniques involving the Gamma function, show that

∞
− k2
e σ
σ
0
6
dσ =
3 π
5
, k ≠ 0.
8k 2
proof
Question 12
a) Show clearly that
Γ ( x + 1) ≡ x Γ ( x ) , x ≠ 0 , x ∉ − » .
b) Hence show further that
Γ ( n + 1) ≡ n ! , n ∈ » .
proof
Created by T. Madas
Created by T. Madas
Question 13
By using techniques involving the Gamma function, show that

∞
m − ax n
x e
0
dx =
(
Γ mn+1
m+1
na n
).
proof
Question 14
By using techniques involving the Gamma function, show that

k
−∞
e ax
(k − x)
n
dx = eak a n−1 Γ (1 − n ) , 0 < n < 1 .
proof
Created by T. Madas
Created by T. Madas
Question 15
The gamma function is defined as
Γ( x) =

∞
e−t t x −1 dt ,
x ≠ 0, − 1, − 2, − 3, ...
0
i.
Show that for x > 0
Γ ( x + 1) = x Γ ( x ) .
ii.
Express the integral
I (s) =

∞
e− p
s
dp ,
s >0,
0
in terms of Gamma functions.
iii. Deduce that
lim  I ( s )  = 1 .
s →∞
1 1
I (s) = Γ 
s s
Created by T. Madas
Created by T. Madas
Question 16
a) Write down the definition of the Gamma function Γ ( x ) , for Re ( x ) > 0 .
b) Use the standard recurrence relation of the Gamma function to extent Γ ( x ) ,
for Re ( x ) ≤ 0 , and hence find the residues at the simple poles at x = −n ,
n = 0, 1, 2, 3 ...
( −1)n
n!
Created by T. Madas
Created by T. Madas
Question 17
I =2

∞
2
e − x dx
and
I =2
0

∞
e− y
2
dy .
0
By finding an expression for I , show that
( )
Γ 1 = π.
2
proof
Created by T. Madas
Created by T. Madas
Question 18
It is given that the following integral converges

1
ln x
− 12
dx .
0
Evaluate the above integral by a suitable substitution introducing Gamma Functions.
You may assume that
( )
Γ 1 = π.
2
π
Created by T. Madas
Created by T. Madas
Question 19
It is given that the following integral converges

1
( x ln x )n
dx , n ∈ » .
0
Evaluate the above integral by a suitable substitution introducing Gamma Functions.
MM3-E ,
Created by T. Madas
( −1)n n!
( n + 1)n+1
Created by T. Madas
Question 20
It is given that the following integral converges

∞
e
− 12 t
ln t dt .
0
Evaluate the above integral by introducing a new parametric term in the integrand and
carrying out a suitable differentiation under the integral sign.
You may assume that

1
Γ′ ( x ) = Γ ( x )  −γ + +

x

∞

k =1

1 
1
−

 .
 k x + k 

2 ( −γ + ln 2 )
Created by T. Madas
Created by T. Madas
Question 21
Prove the validity of Legendre’s duplication formula for the Gamma function, which
states that
Γ ( 2n ) π
, n∈».
Γ n + 1 ≡ 2 n−1
2
2
Γ ( n)
(
)
MM3-C , proof
Question 22
Legendre’s duplication formula for the Gamma function, states that
Γ ( 2m ) π
, m∈» .
Γ m + 1 ≡ 2 m−1
2
2
Γ (m)
(
)
Show that
( 2 m )! π , m ∈ »
Γ m + 1 ≡ 2m
2
2 m!
(
)
proof
Created by T. Madas
Created by T. Madas
Question 23
I (λ, x) =

∞
e −λt t x−1 ln t dt .
0
a) By carrying a suitable differentiation over the integral sign, show that
I ( λ , x ) = λ − x Γ′ ( x ) − Γ ( x ) ln λ  .
b) Find simplified expressions for I ( λ ,1) , I ( λ , 2 ) and I ( λ ,3) .

1
′
You may assume that Γ ( x ) = Γ ( x )  −γ + +

x

MM3-D , I ( λ ,1) = −
1
λ
∞

k =1

1 
1
 −
 .
 k x + k 

[γ + ln λ ] , I ( λ , 2 ) =
I ( λ ,3) =
Created by T. Madas
1
λ2
1
λ3
[1 − γ − ln λ ] ,
[3 − 2γ − 2ln λ ]
Created by T. Madas
Question 24
By considering standard power series expansions and using Gamma functions, show

∞
1
0
x
x
dx =

r =1
 ( −1)r −1 

.
r
 r

You may assume that integration and summation commute.
MM3F , proof
Created by T. Madas
Created by T. Madas
Question 25
By considering standard power series expansions and using Gamma functions, show

1
0
1
x
x
dx = 1 +
1
2
2
+
1
3
3
+
1
4
4
+
1
55
+ ...
You may assume that integration and summation commute.
MM3J , proof
Created by T. Madas
Created by T. Madas
Question 26
I=

∞
1
( ln x )3
x 2 ( x − 1)
dx .
Show by the means of partial fractions and a suitable substitution that
I = −6 +

∞
(
u 3 e −u 1 − e −u
0
and hence show that
I=
π4
15
)
−1
du ,
−6.
You may assume without proof that
∞
ζ ( 4) =

r =1
π4
1
=
.
 r 4 
90
proof
Created by T. Madas
Created by T. Madas
Question 27
By using the substitution u = x + u x in the integral definition of Γ ( x + 1) , show that
for large values of n
n! ≈
2π e− n n
n + 12
, n∈».
proof
Created by T. Madas
Created by T. Madas
BETA FUNCTION
Created by T. Madas
Created by T. Madas
SUMMARY OF THE BETA FUNCTION
The Beta function B ( m, n ) , is defined as
B ( m, n ) ≡

1
x m−1 (1 − x )
n −1
dx .
0
Alternative definitions of B ( m, n ) are
B ( m, n ) ≡
π

2
2 sin 2m−1 θ cos 2 n−1 θ dθ .
0
B ( m, n ) ≡

∞
0
x m−1
dx
( x + 1)m+n
Beta function common rules and facts
•
B ( m, n ) ≡ B ( n, m )
•
B ( m, n ) ≡
Γ ( m) Γ ( n)
,
Γ ( m + n)
or
B ( m, n ) ≡
or B ( m, n ) ≡
( m − 1)!( n − 1)! ,
( m + n + 1)!
m∈» , n∈»
( m − 1)( n − 1) B m − 1, n − 1
(
)
( m + n − 1)( m + n − 2 )
Created by T. Madas
Created by T. Madas
Question 1
Show clearly that
B ( m, n ) = 2

π
2
( sin θ )2m−1 ( sin θ )2n−1 dθ .
0
proof
Question 2
By using techniques involving the Beta function, find the exact value of

1
4
7 x5 (1 − x ) dx .
0
1
180
Created by T. Madas
Created by T. Madas
Question 3
By using techniques involving the Beta function, find the exact value of

π
2
sin 5 θ cos 4 θ dθ .
0
8
315
Question 4
By using techniques involving the Beta function, find the exact value of

1
x 4 1 − x 2 dx .
0
π
32
Created by T. Madas
Created by T. Madas
Question 5

π
2
sin x dx .
0
Express the value of the above integral in terms of Gamma functions, in their simplest
form
( )
( )
Γ 3 π
4
2Γ 5
4
Question 6
By using techniques involving the Beta function, find the exact value of

π
2
sin 7 θ dθ .
0
8
35
Created by T. Madas
Created by T. Madas
Question 7
By using techniques involving the Beta function, find the exact value of

4
0
x3
dx .
4− x
4096
35
Question 8
By using techniques involving the Beta function, find the exact value of

π
4
sin 2 2 x cos 4 2 x dx .
0
1
35
Created by T. Madas
Created by T. Madas
Question 9
By using techniques involving the Beta function, find the exact value of

1
4
4
0
1 − x4
dx .
π 2
Question 10
By using techniques involving the Beta function, find the exact value of

π
0
( )
( )
sin 5 1 x cos 2 1 x dx .
2
2
16
105
Created by T. Madas
Created by T. Madas
Question 11
a) Show clearly that

1
0
(
x m−1 1 − x n
)
p −1
dx =
1 m 
B , p  , n ≠ 0 .
n n 
b) Hence, find the exact value of

1
(
x 5 1 − x3
0
)
2
dx .
1
24
Question 12
By using techniques involving the Beta function, find the exact value of

1
11
1 − 3 x dx .
0
1331
1564
Created by T. Madas
Created by T. Madas
Question 13
In =

1
0
(1 − x )
n
dx .
Use Beta functions to show that
In =
2
( n + 2 )( n + 1)
.
proof
Created by T. Madas
Created by T. Madas
Question 14

a
1
x 2 a − x dx .
0
Find an exact simplified value for the integral above, by using a suitable substitution
to transform into a Beta function.
You may assume that a is a positive constant.
1 π a2
8
Created by T. Madas
Created by T. Madas
Question 15
I m ,n =

π
2
sin m θ cos n θ dθ , m ∈ » , n ∈ » .
0
a) Show clearly that
I m, n =
m −1
I m−2, n
m+n
b) Hence, show further that
B ( m, n ) =
( m − 1)( n − 1) B m − 1, n − 1 .
(
)
( m + n − 1)( m + n − 2 )
proof
Created by T. Madas
Created by T. Madas
Question 16
Show clearly that
B ( m, n ) =
Γ ( m) Γ ( n)
.
Γ ( m + n)
proof
Question 17
By using techniques involving the Beta function, find the exact value of

π
2
tan θ dθ .
0
π
2
Created by T. Madas
Created by T. Madas
Question 18
By using techniques involving the Beta function, find the exact value of

4
x 2 16 − x 2 dx .
0
16π
Question 19
By using techniques involving the Beta function, find the exact value of

2
16x 2 − x 6 dx .
0
2π
Created by T. Madas
Created by T. Madas
Question 20
By using techniques involving the Beta function, find the exact value of

1
0
1
3
x 2 − x3
dx .
2π
3
Question 21
By using techniques involving the Beta function, find the exact value of

π
2
1
7
( sin θ ) 2 ( cosθ ) 2
dx .
0
5π 2
64
Created by T. Madas
Created by T. Madas
Question 22
By using techniques involving the Beta function, find the exact value of

2
(
2x − x 2
)
5
2
dx .
0
5π
16
Question 23
By using techniques involving the Beta function, find the exact value of
2
(
4− x
)
7
2 2
dx .
0
35π
Created by T. Madas
Created by T. Madas
Question 24
By using techniques involving the Beta function, find the exact value of

1
−1
1+ x
dx .
1− x
π
Question 25
By using techniques involving the Beta function, find the exact value of

∞
0
1
1 + x2
dx .
π
2
Created by T. Madas
Created by T. Madas
Question 26
Use the substitution x 2 = sin θ to show that

1
( )
1
1 − x4
0
2
Γ 1 
4 
.
dx = 
32π
proof
Question 27
By using techniques involving the Beta function, find the exact value of

π
0
3
( )
sin 3 x
2
8sin 3 x dx .
2
π
3
Created by T. Madas
Created by T. Madas
Question 28
By using techniques involving the Beta function, find the exact value of

2
−2
2− x
dx .
2+ x
2π
Created by T. Madas
Created by T. Madas
Question 29
By using techniques involving the Beta function, find the exact value of
∞
(
0
x3
1 + 8x
3 2
)
dx .
MM3-A ,
Created by T. Madas
π
72 3
Created by T. Madas
Question 30
By using techniques involving the Beta function, find the exact value of
4

1
( x − 2 )( 4 − x )
2
dx .
π
Question 31
By using techniques involving the Beta function, find the exact value of

2π
cos 4 θ + sin 6 θ dθ .
0
11π
4
Created by T. Madas
Created by T. Madas
Question 32
By using techniques involving the Beta function, show that

5
1
( )
2
2 Γ 1 
 4  .
4 ( 5 − x )( x − 1) dx =
3 π
proof
Created by T. Madas
Created by T. Madas
Question 33

B ( m, n ) ≡
1
x m−1 (1 − x )
n −1
dx .
0
a) Use the substitution x =
y
to show that
y +1
B ( m, n ) ≡

∞
0
y m−1
( y + 1)
m+ n
dy .
b) Hence find the exact value of
∞
(
0
x3
4 + x)
6
dx .
1
320
Created by T. Madas
Created by T. Madas
Question 34
a) Show clearly that

b
( b − x )m−1 ( x − a )n−1
dx = ( b − a )
m + n −1
B ( m, n ) .
a
b) Hence find the exact value of

5
3
5− x
dx .
x −3
π
Created by T. Madas
Created by T. Madas
Question 35
By using techniques involving the Beta function, find the exact value of

∞
1
3 3
(1 + x )
0
dx .
10π
27 3
Question 36
By using techniques involving the Beta function, find the exact value of

∞
−∞
x2
1 + x4
dx .
MM3-B ,
Created by T. Madas
π
2
Created by T. Madas
Question 37
By using techniques involving the Beta function, find the exact value of

∞
1
dx .
1 + x6
0
π
3
Question 38
By using techniques involving the Beta function, find the exact value of

∞
0
1
(
2 + x2
)
4
dx .
5π 2
512
Created by T. Madas
Created by T. Madas
Question 39
y
x
O
The figure above shows the curve with parametric equations
x = 8cos3 t ,
y = sin 3 t , 0 ≤ t ≤ 1 π .
2
The finite region bounded by the curve and the coordinate axes is revolved fully about
the x axis, forming a solid of revolution S .
Determine the x coordinate of the centre of mass of S .
MM3-C , x =
Created by T. Madas
21
16
Created by T. Madas
Question 40
By using techniques involving the Beta function, show that

1
π
1
n
0
1− x
n
dx =
n
sin πn
( )
.
proof
Question 41
By using techniques involving the Beta function, show that

1
0
1
n
x +1
dx =
π 
cosec   .
n
n
π
proof
Created by T. Madas
Created by T. Madas
Question 42

B ( m, n ) ≡
1
x m−1 (1 − x )
n −1
dx .
0
a) Use the substitution x =
y
to show that
y +1
B ( m, n ) ≡

∞
0
y m−1
( y + 1)
m+ n
dy .
b) Hence find the exact value of

∞
0
x
16 + x 2
dx .
π 2
4
Created by T. Madas
Created by T. Madas
Question 43
By using techniques involving the Beta function, find the exact value of

π
0
2
tan x
( cos x + sin x )
2
dx .
MM3-D ,
Created by T. Madas
π
2
Created by T. Madas
Question 44
By using techniques involving the Beta function, find the exact value of

∞
1
1
x
2
x −1
dx .
π
2
Created by T. Madas
Created by T. Madas
Question 45
1
I=
1 − x2 )
(
−1
n
dx , n ∈ » .
Use techniques involving the Beta function to show that
2
I
22 n+1 ( n!)
=
.
( 2n + 1)!
MM3-E , proof
Created by T. Madas
Created by T. Madas
Question 46
By using techniques involving the Beta function, find the exact value of

∞
0
1
dx .
( x + 1) x
MM3-C , π
Created by T. Madas
Created by T. Madas
Question 47
By using techniques involving the Beta function, find the exact value of

π
0
2
tan x
2
cos x + 4sin 2 x
dx .
π
4
Created by T. Madas
Created by T. Madas
Question 48

∞
2
e−u u 2 x −1 du .
0
a) Determine the value of the above integral in terms of Gamma functions.
b) Show that

1π
2
( cos θ )2 x−1 ( sin θ )2 y −1 dθ
≡ 1 B ( x, y ) .
2
0
c) Use the results of part (a) and (b) to show further that
B ( x, y ) =
Γ( x)Γ( y)
.
Γ( x + y)
d) Hence deduce that

1π
2
0
( cos θ )2n dθ
 2n  π
=   2 n+1 .
 n 2
1 Γ( x)
2
Created by T. Madas
Created by T. Madas
Question 49
a) By using techniques involving the Beta function and the Gamma function,
show that

π
2
( cosθ )
2 k +1
dθ =
0
( k !)2 22k .
( 2k + 1)!
b) Use a suitable substitution in the above integral to deduce an expression for

π
2
( sin θ )2k +1 dθ .
0

π
2
0
( sin θ )
2 k +1
dθ =

π
2
0
Created by T. Madas
( cosθ )
2 k +1
dθ =
( k !)2 22k
( 2k + 1)!
Created by T. Madas
Question 50
f (z) =
z p −1
, z ∈» , p∈ » , 0 < p <1.
z +1
a) By integrating f ( z ) over a keyhole contour with a branch cut along the
positive x axis, show that

∞
0
x p −1
π
.
dx =
1+ x
sin ( pπ )
b) Hence show further that
Γ ( x ) Γ (1 − x ) =

1
u x −1 (1 − u )
−x
du .
0
c) Use the substitution u =
t
to deduce that
t +1
Γ ( x ) Γ (1 − x ) =
π
, 0 < x < 1.
sin (π x )
V , MM3P , proof
[solution overleaf]
Created by T. Madas
Created by T. Madas
Created by T. Madas
Created by T. Madas
Question 51
B ( m, n ) ≡

1
x m−1 (1 − x )
n −1
dx .
0
a) Use a suitable substitution to show that
B(k, k ) =
1
2
2 k −1

1
(1 + u )k −1 (1 − u )k −1
du .
−1
b) Hence show further
Γ ( 2k ) π
.
Γ k + 1 = 2 k −1
2
2
Γ(k )
(
)
proof
Created by T. Madas
Created by T. Madas
Question 52
I ( x) ≡

1
0
1
1 2
t
−
−
 4
2 
(
)
x −1
dt .
a) Express I ( x ) in terms of Gamma functions.
b) By considering the symmetry of the integrand about the line t = 1 and using
2
2
the substitution y = 4 t − 1 , show that
2
(
)
(
)
Γ ( x ) Γ x + 1 = 21−2 x π Γ ( 2k ) .
2
I ( x) =
Created by T. Madas
Γ( x) Γ( x)
Γ (2x)
Created by T. Madas
Question 53
I (α ) =

∞
0
( ) dx ,
arctan xα
α < −1 .
By using integration by parts, and justifying all steps, show that
 π 
1 1 1 1 1
I (α ) = 1 π sec 
 = 2 B 2 − 2α , 2 + 2α .
2
2
α


(
You may assume that

∞
0
)
x p −1
π
dx =
, 0 < p <1.
sin pπ
( x + 1)
proof
Created by T. Madas
Created by T. Madas
Question 54
It is given that for any real constants x and y
B ( x + 1, y + 1) =
Γ ( x + 1) Γ ( y + 1)
.
Γ ( x + y + 2)
a) Use the integral definition of B ( x + 1, y + 1) , with a suitable substitution to
derive Gauss’ definition of the gamma function


n x n!
Γ ( x ) = lim 
.
n→∞  x ( x + 1)( x + 2 ) ... ( x + n − 1)( x + n ) 


b) Hence show

1
Γ′ ( x ) = Γ ( x )  −γ − +

x

∞

k =1

x
.
k (x + k)

proof
Created by T. Madas
Created by T. Madas
Question 55
1
I=

0
1− x
1
xy (1 − x − y ) 2 dydx .
0
Determine the exact value of I by transforming it into an expression involving Beta
functions.
16
945
Created by T. Madas
Created by T. Madas
Question 56
A finite region R defined by the inequalities
x3 + y 3 + z 3 ≤ 1 , x ≥ 0 , y ≥ 0 , z ≥ 0 .
Show that the volume of R is
( )
k
1 Γ 1  ,
 k k 
where k is a positive integer to be found.
k =3
Created by T. Madas