The gamma function and related functions

18.12 THE GAMMA FUNCTION AND RELATED FUNCTIONS
which converges provided c > a > 0.
Prove the result (18.150).
Since F(a, b, c; x) is unchanged by swapping a and b, we may write its integral representation
(18.144) as
1
Γ(c)
F(a, b, c; x) =
ta−1 (1 − t)c−a−1 (1 − tx)−b dt.
Γ(a)Γ(c − a) 0
Setting x = z/b and taking the limit b → ∞, we obtain
−b
1
Γ(c)
tz
M(a, c; z) =
ta−1 (1 − t)c−a−1 lim 1 −
dt.
b→∞
Γ(a)Γ(c − a) 0
b
Since the limit is equal to etz , we obtain result (18.150). Relationships between confluent hypergeometric functions
A large number of relationships exist between confluent hypergeometric functions
with different arguments. These are straightforwardly derived using the integral
representation (18.150) or the series form (18.148). Here, we simply note two
useful examples, which read
M(a, c; x) = ex M(c − a, c; −x),
a
M (a, c; x) = M(a + 1, c + 1; x),
c
(18.151)
(18.152)
where the prime in the second relation denotes d/dx. The first result follows
straightforwardly from the integral representation, and the second result may be
proved from the series expansion (see exercise 18.19).
In an analogous manner to that used for the ordinary hypergeometric functions, one may also derive relationships between M(a, c; x) and any two of the
four ‘contiguous functions’ M(a ± 1, c; x) and M(a, c ± 1; x). These serve as the
recurrence relations for the confluent hypergeometric functions. An example of
such a relationship is
(c − a)M(a − 1, c; x) + (2a − c + x)M(a, c; x) − aM(a + 1, c; x) = 0.
18.12 The gamma function and related functions
Many times in this chapter, and often throughout the rest of the book, we have
made mention of the gamma function and related functions such as the beta and
error functions. Although not derived as the solutions of important second-order
ODEs, these convenient functions appear in a number of contexts, and so here
we gather together some of their properties. This final section should be regarded
merely as a reference containing some useful relations obeyed by these functions;
a minimum of formal proofs is given.
635
SPECIAL FUNCTIONS
18.12.1 The gamma function
The gamma function Γ(n) is defined by
∞
xn−1 e−x dx,
Γ(n) =
(18.153)
0
which converges for n > 0, where in general n is a real number. Replacing n by
n + 1 in (18.153) and integrating the RHS by parts, we find
∞
xn e−x dx
Γ(n + 1) =
0
∞
∞
nxn−1 e−x dx
= −xn e−x 0 +
0
∞
=n
xn−1 e−x dx,
0
from which we obtain the important result
Γ(n + 1) = nΓ(n).
(18.154)
From (18.153), we see that Γ(1) = 1, and so, if n is a positive integer,
Γ(n + 1) = n!.
(18.155)
In fact, equation (18.155) serves as a definition of the factorial function even for
non-integer n. For negative n the factorial function is defined by
n! =
(n + m)!
,
(n + m)(n + m − 1) · · · (n + 1)
(18.156)
where m is any positive integer that makes n + m > 0. Different choices of m
(> −n) do not lead to different values for n!. A plot of the gamma function is
given in figure 18.9, where it can be seen that the function is infinite for negative
integer values of n, in accordance with (18.156). For an extension of the factorial
function to complex arguments, see exercise 18.15.
By letting x = y 2 in (18.153), we immediately obtain another useful representation of the gamma function given by
∞
2
y 2n−1 e−y dy.
(18.157)
Γ(n) = 2
0
Setting n =
1
2
we find the result
∞
2
Γ 12 = 2
e−y dy =
∞
−∞
0
e−y dy =
2
√
π,
where have used the standard integral discussed in section 6.4.2. From this result,
Γ(n) for half-integral n can be found using (18.154). Some immediately derivable
factorial values of half integers are
1
1
3
3
√
√
1√
3√
− 2 ! = π,
− 2 ! = −2 π,
2 ! = 2 π,
2 ! = 4 π.
636
18.12 THE GAMMA FUNCTION AND RELATED FUNCTIONS
Γ(n)
6
4
2
−4 −3
n
−2 −1
1
2
3
4
−2
−4
−6
Figure 18.9 The gamma function Γ(n).
Moreover, it may be shown for non-integral n that the gamma function satisfies
the important identity
π
.
(18.158)
Γ(n)Γ(1 − n) =
sin nπ
This is proved for a restricted range of n in the next section, once the beta
function has been introduced.
It can also be shown that the gamma function is given by
√
1
139
1
+
Γ(n + 1) = 2πn nn e−n 1 +
−
+
.
.
.
= n!,
12n 288n2
51 840n3
(18.159)
which is known as Stirling’s asymptotic series. For large n the first term dominates,
and so
√
(18.160)
n! ≈ 2πn nn e−n ;
this is known as Stirling’s approximation. This approximation is particularly useful
in statistical thermodynamics, when arrangements of a large number of particles
are to be considered.
Prove Stirling’s approximation n! ≈
√
2πn nn e−n for large n.
From (18.153), the extended definition of the factorial function (which is valid for n > −1)
is given by
∞
∞
n! =
xn e−x dx =
en ln x−x dx.
(18.161)
0
0
637
SPECIAL FUNCTIONS
If we let x = n + y, then
y
ln x = ln n + ln 1 +
n
y
y3
y2
= ln n + − 2 + 3 − · · · .
n
2n
3n
Substituting this result into (18.161), we obtain
∞
y2
y
exp n ln n + − 2 + · · · − n − y dy.
n! =
n
2n
−n
Thus, when n is sufficiently large, we may approximate n! by
∞
√
√
2
e−y /(2n) dy = en ln n−n 2πn = 2πn nn e−n ,
n! ≈ en ln n−n
−∞
which is Stirling’s approximation (18.160). 18.12.2 The beta function
The beta function is defined by
1
xm−1 (1 − x)n−1 dx,
B(m, n) =
(18.162)
0
which converges for m > 0, n > 0, where m and n are, in general, real numbers.
By letting x = 1 − y in (18.162) it is easy to show that B(m, n) = B(n, m). Other
useful representations of the beta function may be obtained by suitable changes
of variable. For example, putting x = (1 + y)−1 in (18.162), we find that
∞
y n−1 dy
B(m, n) =
.
(18.163)
m+n
0 (1 + y)
Alternatively, if we let x = sin2 θ in (18.162), we obtain immediately
π/2
B(m, n) = 2
sin2m−1 θ cos2n−1 θ dθ.
(18.164)
0
The beta function may also be written in terms of the gamma function as
B(m, n) =
Γ(m)Γ(n)
.
Γ(m + n)
Prove the result (18.165).
Using (18.157), we have
∞
2
2
x2n−1 e−x dx
y 2m−1 e−y dy
0
0
∞ ∞
2
2
x2n−1 y 2m−1 e−(x +y ) dx dy.
=4
∞
Γ(n)Γ(m) = 4
0
0
638
(18.165)
18.12 THE GAMMA FUNCTION AND RELATED FUNCTIONS
Changing variables to plane polar coordinates (ρ, φ) given by x = ρ cos φ, y = ρ sin φ, we
obtain
π/2 ∞
2
ρ2(m+n−1) e−ρ sin2m−1 φ cos2n−1 φ ρ dρ dφ
Γ(n)Γ(m) = 4
0
0
π/2
∞
sin2m−1 φ cos2n−1 φ dφ
=4
0
ρ2(m+n)−1 e−ρ dρ
2
0
= B(m, n)Γ(m + n),
where in the last line we have used the results (18.157) and (18.164). The result (18.165) is useful in proving the identity (18.158) satisfied by the
gamma function, since
∞ n−1
y dy
,
Γ(n)Γ(1 − n) = B(1 − n, n) =
1+y
0
where, in the second equality, we have used the integral representation (18.163).
For 0 < n < 1 this integral can be evaluated using contour integration and has
the value π/(sin nπ) (see exercise 24.19), thereby proving result (18.158) for this
range of n. Extensions to other ranges require more sophisticated methods.
18.12.3 The incomplete gamma function
In the definition (18.153) of the gamma function, we may divide the range of
integration into two parts and write
∞
x
un−1 e−u du +
un−1 e−u du ≡ γ(n, x) + Γ(n, x),
Γ(n) =
0
x
(18.166)
whereby we have defined the incomplete gamma functions γ(n, x) and Γ(n, x),
respectively. The choice of which of these two functions to use is merely a matter
of convenience.
Show that if n is a positive integer
Γ(n, x) = (n − 1)!e−x
n−1 k
x
.
k!
k=0
From (18.166), on integrating by parts we find
∞
un−1 e−u du = xn−1 e−x + (n − 1)
Γ(n, x) =
x
∞
un−2 e−u du
x
= xn−1 e−x + (n − 1)Γ(n − 1, x),
which is valid for arbitrary n. If n is an integer, however, we obtain
Γ(n, x) = e−x [xn−1 + (n − 1)xn−2 + (n − 1)(n − 2)xn−3 + · · · + (n − 1)!]
= (n − 1)! e−x
n−1 k
x
,
k!
k=0
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