Exercises

SPECIAL FUNCTIONS
which is the required result. We note that is it conventional to define, in addition, the functions
P (a, x) ≡
γ(a, x)
,
Γ(a)
Q(a, x) ≡
Γ(a, x)
,
Γ(a)
which are also often called incomplete gamma functions; it is clear that Q(a, x) =
1 − P (a, x).
18.12.4 The error function
Finally, we mention the error function, which is encountered in probability theory
and in the solutions of some partial differential equations. The error function is
√
related to the incomplete gamma function by erf(x) = γ( 12 , x2 )/ π and is thus
given by
x
∞
2
2
2
2
e−u du = 1 − √
e−u du.
(18.167)
erf(x) = √
π 0
π x
From this definition we can easily see that
erf(∞) = 1,
erf(−x) = −erf(x).
√
By making the substitution y = 2u in (18.167), we find
√2x
2
2
e−y /2 dy.
erf(x) =
π 0
erf(0) = 0,
The cumulative probability function Φ(x) for the standard Gaussian distribution
(discussed in section 30.9.1) may be written in terms of the error function as
follows:
x
1
2
e−y /2 dy
Φ(x) = √
2π −∞
x
1
1
2
e−y /2 dy
= +√
2
2π 0
x
1 1
= + erf √
.
2 2
2
It is also sometimes useful to define the complementary error function
∞
Γ( 12 , x2 )
2
2
erfc(x) = 1 − erf(x) = √
e−u du = √
.
π x
π
18.13 Exercises
18.1
Use the explicit expressions
640
(18.168)
18.13 EXERCISES
Y00 =
Y1±1 = ∓
Y2±1 = ∓
1
,
4π
Y10 =
3
8π
sin θ exp(±iφ),
15
8π
sin θ cos θ exp(±iφ),
Y20 =
Y2±2 =
3
4π
cos θ,
5
(3 cos2
16π
15
32π
θ − 1),
sin2 θ exp(±2iφ),
to verify for = 0, 1, 2 that
|Ym (θ, φ)|2 =
m=−
18.2
2 + 1
,
4π
and so is independent of the values of θ and φ. This is true for any , but
a general proof is more involved. This result helps to reconcile intuition with
the apparently arbitrary choice of polar axis in a general quantum mechanical
system.
Express the function
f(θ, φ) = sin θ[sin2 (θ/2) cos φ + i cos2 (θ/2) sin φ] + sin2 (θ/2)
18.3
as a sum of spherical harmonics.
Use the generating function for the Legendre polynomials Pn (x) to show that
1
(2n)!
P2n+1 (x) dx = (−1)n 2n+1
2
n!(n + 1)!
0
and that, except for the case n = 0,
1
P2n (x) dx = 0.
0
18.4
Carry through the following procedure as a proof of the result
1
2
In =
Pn (z)Pn (z) dz =
.
2n + 1
−1
(a) Square both sides of the generating-function definition of the Legendre
polynomials,
∞
Pn (z)hn .
(1 − 2zh + h2 )−1/2 =
n=0
(b) Express the RHS as a sum of powers of h, obtaining expressions for the
coefficients.
(c) Integrate the RHS from −1 to 1 and use the orthogonality property of the
Legendre polynomials.
(d) Similarly integrate the LHS and expand the result in powers of h.
(e) Compare coefficients.
18.5
The Hermite polynomials Hn (x) may be defined by
Φ(x, h) = exp(2xh − h2 ) =
∞
1
Hn (x)hn .
n!
n=0
Show that
∂Φ
∂Φ
∂2 Φ
− 2x
+ 2h
= 0,
∂x2
∂x
∂h
641
SPECIAL FUNCTIONS
and hence that the Hn (x) satisfy the Hermite equation
y − 2xy + 2ny = 0,
where n is an integer ≥ 0.
Use Φ to prove that
(a) Hn (x) = 2nHn−1 (x),
(b) Hn+1 (x) − 2xHn (x) + 2nHn−1 (x) = 0.
18.6
A charge +2q is situated at the origin and charges of −q are situated at distances
±a from it along the polar axis. By relating it to the generating function for the
Legendre polynomials, show that the electrostatic potential Φ at a point (r, θ, φ)
with r > a is given by
∞
2q a 2s
Φ(r, θ, φ) =
P2s (cos θ).
4π0 r s=1 r
18.7
For the associated Laguerre polynomials, carry through the following exercises.
(a) Prove the Rodrigues’ formula
ex x−m dn n+m −x
(x e ),
n! dxn
taking the polynomials to be defined by
Lmn (x) =
Lmn (x) =
n
(−1)k
k=0
(n + m)!
xk .
k!(n − k)!(k + m)!
(b) Prove the recurrence relations
(n + 1)Lmn+1 (x) = (2n + m + 1 − x)Lmn (x) − (n + m)Lmn−1 (x),
x(Lmn ) (x) = nLmn (x) − (n + m)Lmn−1 (x),
but this time taking the polynomial as defined by
Lmn (x) = (−1)m
dm
Ln+m (x)
dxm
or the generating function.
18.8
The quantum mechanical wavefunction for a one-dimensional simple harmonic
oscillator in its nth energy level is of the form
ψ(x) = exp(−x2 /2)Hn (x),
where Hn (x) is the nth Hermite polynomial. The generating function for the
polynomials is
∞
Hn (x) n
2
G(x, h) = e2hx−h =
h.
n!
n=0
(a) Find Hi (x) for i = 1, 2, 3, 4.
(b) Evaluate by direct calculation
∞
−∞
e−x Hp (x)Hq (x) dx,
2
(i) for p = 2, q = 3; (ii) for p = 2, q = √
4; (iii) for p = q = 3. Check your
answers against the expected values 2p p! π δpq .
642
18.13 EXERCISES
[ You will find it convenient to use
√
∞
(2n)! π
2
x2n e−x dx =
2n
2 n!
−∞
18.9
18.10
for integer n ≥ 0. ]
By initially writing y(x) as x1/2 f(x) and then making subsequent changes of
variable, reduce Stokes’ equation,
d2 y
+ λxy = 0,
dx2
to Bessel’s equation. √
Hence show that a solution that is finite at x = 0 is a
multiple of x1/2 J1/3 ( 23 λx3 ).
By choosing a suitable form for h in their generating function,
∞
1
z
h−
=
Jn (z)hn ,
G(z, h) = exp
2
h
n=−∞
show that integral repesentations of the Bessel functions of the first kind are
given, for integral m, by
(−1)m 2π
J2m (z) =
cos(z cos θ) cos 2mθ dθ
m ≥ 1,
π
0
(−1)m+1 2π
cos(z cos θ) sin(2m + 1)θ dθ
m ≥ 0.
J2m+1 (z) =
π
0
18.11
Identify the series for the following hypergeometric functions, writing them in
terms of better known functions:
(a)
(b)
(c)
(d)
(e)
18.12
F(a, b, b; z),
F(1, 1, 2; −x),
F( 12 , 1, 32 ; −x2 ),
F( 12 , 12 , 32 ; x2 ),
F(−a, a, 12 ; sin2 x); this is a much more difficult exercise.
By making the substitution z = (1 − x)/2 and suitable choices for a, b and c,
convert the hypergeometric equation,
du
d2 u
+ [ c − (a + b + 1)z ] − abu = 0,
dz 2
dz
into the Legendre equation,
z(1 − z)
d2 y
dy
+ ( + 1)y = 0.
− 2x
dx2
dx
Hence, using the hypergeometric series, generate the Legendre polynomials P (x)
for the integer values = 0, 1, 2, 3. Comment on their normalisations.
Find a change of variable that will allow the integral
∞ √
u−1
du
I=
(u + 1)2
1
(1 − x2 )
18.13
18.14
to be expressed in terms of the beta function, and so evaluate it.
Prove that, if m and n are both greater than −1, then
∞
Γ[ 12 (m + 1) ] Γ[ 12 (n + 1) ]
um
.
du = (m+1)/2
I=
2
(m+n+2)/2
(au + b)
2a
b(n+1)/2 Γ[ 12 (m + n + 2) ]
0
643
SPECIAL FUNCTIONS
Deduce the value of
∞
J=
0
18.15
(u + 2)2
du.
(u2 + 4)5/2
The complex function z! is defined by
∞
uz e−u du
z! =
for Re z > −1.
0
For Re z ≤ −1 it is defined by
z! =
(z + n)!
,
(z + n)(z + n − 1) · · · (z + 1)
where n is any (positive) integer > −Re z. Being the ratio of two polynomials, z!
is analytic everywhere in the finite complex plane except at the poles that occur
when z is a negative integer.
(a) Show that the definition of z! for Re z ≤ −1 is independent of the value of
n chosen.
(b) Prove that the residue of z! at the pole z = −m, where m is an integer > 0,
is (−1)m−1 /(m − 1)!.
18.16
18.17
For −1 < Re z < 1, use the definition and value of the beta function to show
that
∞
uz
du.
z! (−z)! =
(1
+
u)2
0
Contour integration gives the value of the integral on the RHS of the above
equation as πz cosec πz. Use this to deduce the value of (− 12 )!.
The integral
∞
2
e−k
I=
dk,
(∗)
2
2
−∞ k + a
in which a > 0, occurs in some statistical mechanics problems. By first considering
the integral
∞
J=
eiu(k+ia) du,
0
18.18
and a suitable variation of it, show that I = (π/a) exp(a2 ) erfc(a), where erfc(x)
is the complementary error function.
Consider two series expansions of the error function as follows.
(a) Obtain a series expansion of the error function erf(x) in ascending powers
of x. How many terms are needed to give a value correct to four significant
figures for erf(1)?
(b) Obtain an asymptotic expansion that can be used to estimate erfc(x) for
large x (> 0) in the form of a series
erfc(x) = R(x) = e−x
2
∞
an
.
n
x
n=0
Consider what bounds can be put on the estimate and at what point the
infinite series should be terminated in a practical estimate. In particular,
estimate erfc(1) and test the answer for compatibility with that in part (a).
18.19
For the functions M(a, c; z) that are the solutions of the confluent hypergeometric
equation,
644
18.13 EXERCISES
(a) use their series representation to prove that
b
d
M(a, c; z) = a M(a + 1, c + 1; z);
dz
(b) use an integral representation to prove that
M(a, c; z) = ez M(c − a, c; −z).
18.20
The Bessel function Jν (z) can be considered as a special case of the solution
M(a, c; z) of the confluent hypergeometric equation, the connection being
lim
a→∞
18.21
√
M(a, ν + 1; −z/a)
= z −ν/2 Jν (2 z).
Γ(ν + 1)
Prove this equality by writing each side in terms of an infinite series and showing
that the series are the same.
Find the differential equation satisfied by the function y(x) defined by
x
y(x) = Ax−n
e−t tn−1 dt ≡ Ax−n γ(n, x),
0
18.22
and, by comparing it with the confluent hypergeometric function, express y as
a multiple of the solution M(a, c; z) of that equation. Determine the value of A
that makes y equal to M.
Show, from its definition, that the Bessel function of the second kind, and of
integral order ν, can be written as
∂J−µ (z)
1 ∂Jµ (z)
Yν (z) =
− (−1)ν
.
π
∂µ
∂µ
µ=ν
Using the explicit series expression for Jµ (z), show that ∂Jµ (z)/∂µ can be written
as
z
+ g(ν, z),
Jν (z) ln
2
and deduce that Yν (z) can be expressed as
z
2
+ h(ν, z),
Yν (z) = Jν (z) ln
π
2
where h(ν, z), like g(ν, z), is a power series in z.
18.23
Prove two of the properties of the incomplete gamma function P (a, x2 ) as follows.
(a) By considering its form for a suitable value of a, show that the error function
can be expressed as a particular case of the incomplete gamma function.
(b) The Fresnel integrals, of importance in the study of the diffraction of light,
are given by
x
x
π π S(x) =
cos
sin
C(x) =
t2 dt,
t2 dt.
2
2
0
0
Show that they can be expressed in terms of the error function by
√
π
(1 − i)x ,
C(x) + iS(x) = A erf
2
where A is a (complex) constant, which you should determine. Hence express
C(x) + iS(x) in terms of the incomplete gamma function.
645