Myint Tyn U., Debnath L. Linear Partial Differential Equations for Scientists and Engineers
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A-1 Gamma, Beta, Error, and Airy Functions 751
Figure A-1.1 The gamma function.
d
dx
Γ (x)=Γ
′
(x)=
∞
0
d
dx
(t
x
)
e
−t
t
dt
=
∞
0
d
dx
[exp (x log t)]
e
−t
t
dt
=
∞
0
t
x−1
(log t) e
−t
dt. (A-1.10)
At x =1,thisgives
Γ
′
(1) =
∞
0
e
−t
log tdt= −γ, (A-1.11)
where γ is called the Euler constant and has the value 0.5772.
The graph of the gamma function is shown in Figu re A-1.1.
Several u seful properties of the gamma function ar e recorded below
without proof for reference.
Legendre Duplication Formula
2
2x−1
Γ (x) Γ
x +
1
2
=
√
πΓ(2x) , (A-1.12)
752 Appendix: Some Special Functions and Their Properties
In particular, when x = n (n =0, 1, 2,...)
Γ
n +
1
2
=
√
π (2n)!
2
2n
n!
. (A-1.13)
The following properties also hold for Γ (x):
Γ (x) Γ (1 − x)=π cosec πx, x is a noninteger, (A-1.14)
Γ (x)=p
x
∞
0
exp (−pt) t
x−1
dt, (A-1.15)
Γ (x)=
∞
−∞
exp
xt − e
t
dt. (A-1.16)
Γ (x +1)∼
√
2π exp (−x) x
x+
1
2
for large x, (A-1.17)
n! ∼
√
2π exp (−n) x
n+
1
2
for large n. (A-1.18)
The incomplete gamma function, γ (x, a), is defined by the integral
γ (a, x)=
x
0
e
−t
t
a−1
dt, a > 0. (A-1.19)
The complementary incomplete gamma function, Γ (a, x), is defined by the
integral
Γ (a, x)=
∞
x
e
−t
t
a−1
dt, a > 0. (A-1.20)
Thus, it follows that
γ (a, x)+Γ (a, x)=Γ (a) . (A-1.21)
The beta function, denoted by B (x, y) is defined by the integral
B (x, y)=
t
0
t
x−1
(1 − t)
y −1
dt, x > 0,y>0. (A-1.22)
The beta function B (x, y)issymmetric with respect to its arguments
x and y,thatis,
B (x, y)=B (y, x) . (A-1.23)
This follows from (A-1.22) by the change of variable 1 − t = u,thatis,
B (x, y)=
1
0
u
y −1
(1 − u)
x−1
du = B (y, x) .
If we make the change of variable t = u /(1 + u) in (A-1.22), we obtain
another integral representation of the beta function
A-1 Gamma, Beta, Error, and Airy Functions 753
B (x, y)=
∞
0
u
x−1
(1 + u)
−(x+y )
du =
∞
0
u
y −1
(1 + u)
−(x+y )
du,
(A-1.24)
Putting t =cos
2
θ in (A-1.22), we derive
B (x, y)=2
π/2
0
cos
2x−1
θ sin
2y −1
θdθ. (A-1.25)
Several important results are recorded below without proof for ready
reference.
B (1, 1) = 1,B
1
2
,
1
2
= π, (A-1.26)
B (x, y)=
x − 1
x + y − 1
B (x − 1,y) , (A-1.27)
B (x, y)=
Γ (x) Γ (y)
Γ (x + y)
, (A-1.28)
B
1+x
2
,
1 − x
2
= π sec
πx
2
, 0 <x<1. (A-1.29)
The error function,erf(x) is defined by the integral
erf (x)=
2
√
π
x
0
exp
−t
2
dt, −∞ <x<∞. (A-1.30)
Clearly, it follows from (A-1.30) that
erf (−x)=−erf (x) , (A-1.31)
d
dx
[erf (x)] =
2
√
π
exp
−x
2
, (A-1.32)
erf (0) = 0, erf (∞)=1. (A-1.33)
The complementary error function,erfc(x) is defined by the integral
erfc (x)=
2
√
π
∞
x
exp
−t
2
dt. (A-1.34)
Clearly, it follows that
erfc (x)=1− erf (x) , (A-1.35)
erfc (0) = 1, erfc (∞)=0, (A-1.36)
erfc (x) ∼
1
x
√
π
exp
−x
2
for large x. (A-1.37)
The graphs of erf (x) and erfc (x) are shown in F igure A-1.2.
754 Appendix: Some Special Functions and Their Properties
Figure A-1.2 The error function and the complementary error function.
Closely associated with the error function are the Fresnel integrals,
which are defined by
C (x)=
x
0
cos
πt
2
2
dt and S (x)=
x
0
sin
πt
2
2
dt. (A-1.38)
These integrals arise in diffraction problems in optics, in water waves, in
elasticity, and elsewhere.
Clearly, it follows from (A-1.38) that
C (0) = 0 = S (0) (A-1.39)
C (∞)=S (∞)=
π
2
, (A-1.40)
d
dx
C (x)=cos
πx
2
2
,
d
dx
S (x)=sin
πx
2
2
. (A-1.41)
It also follows f rom (A-1.38) t hat C (x) has extrema at the points where
x
2
=(2n + 1), n =0, 1, 2, 3,..., and S (x) has extrema at the points where
x
2
=2n, n =1, 2, 3,.... The largest maxima occur first and are C (1) =
0.7799 and S
√
2
=0.7139. We also infer that both C (x)andS (x)are
oscillatory about the line y =0.5. The graphs of C (x)andS (x) for non-
negative real x are shown in Figure A-1.3.
The Airy different ial equation
d
2
y
dx
2
− xy = 0 (A-1.42)
has solutions y
1
= Ai (x)andy
2
= Bi(x) which are called Airy functions.
Using the t ransfor mation y (x)=x
α
f
x
β
,whereα and β are constants,
the Airy functions can be expressed in term of Bessel functions. Differenti-
ating y (x) with respect to x gives
A-1 Gamma, Beta, Error, and Airy Functions 755
Figure A-1.3 The Fresnel integrals C (x) and S (x).
y
′
(x)=αx
α−1
f + βx
α+β−1
f
′
,
y
′′
(x)=α (α − 1) x
α−2
f +
2αβ + β
2
− β
x
α+β−2
f
′
+ β
2
x
α+2β−2
f
′′
.
Substituting y and y
′′
into the Airy equation (A-1.42) gives
β
2
x
α+2β−2
f
′′
+
2αβ + β
2
− β
x
α+β−2
f
′
+ x
α−2
α (α − 1) − x
3
!
f =0.
Multiplying this equation by
x
2−α
β
−2
yields
x
2β
f
′′
+(2α + β − 1) β
−1
x
β
f
′
+ β
−2
α (α − 1) − x
3
!
f =0. (A-1.43)
Considering the coefficient of f, we require that x
3
be proportional to
x
2β
so that β =3/2. The coefficient of f
′
gives α =
1
2
. Hence, y (x)=
x
1
2
f
x
3/2
. Consequently, equation (A-1.43) becomes
x
3
f
′′
+ x
3/2
f
′
−
2
3
x
3/2
2
−
1
9
=0. (A-1.44)
This equation admits solutions in terms of Bessel functions K
1/3
(ξ)and
I
1/3
(ξ)whereξ =
2
3
x
3/2
. The general solution of the Airy equation in terms
of Bessel functions is given by
y (x)=
√
x
AK
1/3
(ξ)+BI
1/3
(ξ)
!
. (A-1.45)
In fact, the Airy functions can be expressed as
Ai (x)=
1
π
x
3
K
1/3
(ξ) , (A-1.46)
Bi(x)=
x
3
I
1/3
(ξ)+I
−1/3
(ξ)
!
. (A-1.47)
756 Appendix: Some Special Functions and Their Properties
The Airy equation (A-1.42) can also be solved by the method of Laplace,
that is, by seeking a solution in integral form
y (x)=
C
e
xz
u (z) dz, (A-1.48)
where the path of integration C is chosen such that u vanishes on the
boundary. It follows that u satisfies the first-order differential equation
du
dz
+ z
2
u =0.
The solution of this equation can be obtained in an integral form except
for a normalization factor as
y (x)=
1
2πi
i∞
−i∞
exp
xz −
1
3
z
3
dz = Ai (x) , (A-1.49)
or, equivalently,
Ai (x)=
1
2π
∞
−∞
exp
i
xz +
1
3
z
3
dz =
1
π
∞
0
cos
xz +
1
3
z
3
dz.
(A-1.50)
More generally,
Ai (ax)=
1
2πa
∞
−∞
exp
i
xz +
z
3
3a
3
dz
=
1
πa
∞
0
cos
xz +
z
3
3a
3
dz. (A-1.51)
The graph of y = Ai (x) is shown in Figure A-1.4.
Finally, the method of power series can be used to solve the Airy equa-
tion to obtain the solution as
y (x)=a
0
1+
∞
n=1
x
3n
(3n)(3n − 1) (3n − 3) (3n − 4) ...3.2
+a
1
x +
∞
n=1
x
3n+1
(3n +1)(3n)(3n − 2) (3n − 3) ...4.3
, (A-1.52)
where a
0
and a
1
are constants.
The series solution (A-1.52) converges for all x rapidly due to the rapid
decay of the coefficients as n →∞.
A-2 Hermite Polynomials and Weber–Hermite Functions 757
Figure A-1.4 Graph of the Airy function.
A-2 Hermite Polynomials and Weber–Hermite
Functions
The Hermite polynomials H
n
(x) are defined by the Rodrigues formula
H
n
(x)=(−1)
n
exp
x
2
d
n
dx
n
exp
−x
2
!
, (A-2.1)
where n =0, 1, 2, 3,....
The first seven Hermite polynomials are
H
0
(x)=1
H
1
(x)=2x
H
2
(x)=4x
2
− 2
H
3
(x)=8x
3
− 12x
H
4
(x)=16x
4
− 48x
2
+12
H
5
(x)=32x
5
− 16x
3
+ 120x
H
6
(x)=64x
6
− 480x
4
+ 720x
2
− 120.
The generating function of H
n
(x)is
exp
2xt− t
2
=
∞
n=0
t
n
n!
H
n
(x) . (A-2.2)
It follows from (A-2.2) that H
n
(x) satisfies th e parity relation
H
n
(−x)=(−1)
n
H
n
(x) . (A-2.3)
Also, it follows from (A-2.2) that
758 Appendix: Some Special Functions and Their Properties
H
2n+1
(0) = 0,H
2n
(0) = (−1)
n
(2n)!
n!
. (A-2.4)
The recurrence relations for Hermite polynomials are
H
n+1
(x) − 2xH
n
(x)+2nH
n−1
(x)=0, (A-2.5)
H
′
n
(x)=2xH
n−1
(x) . (A-2.6)
The Hermite polynomials, y = H
n
(x), are solutions of the Hermite differ-
ential equation
y
′′
− 2xy
′
+2ny =0. (A-2.7)
The orthogonal property of Hermite polynomials is
∞
−∞
exp
−x
2
H
n
(x) H
m
(x) dx =2
n
n!
√
πδ
mn
. (A-2.8)
With repeated use of integration by parts, it follows from (A-2.1) that
∞
−∞
exp
−x
2
H
n
(x) x
m
dx =0,m=0, 1,...,(n − 1) , (A-2.9)
∞
−∞
exp
−x
2
H
n
(x) x
n
dx =
√
πn!. (A-2.10)
The Weber–Hermite or, simply the Hermite fu nction
y = h
n
(x)=exp
−
x
2
2
H
n
(x) (A-2.11)
satisfies the differential equation
h
′′
n
(x)+
λ − x
2
h
n
(x)=0, (A-2.12)
where λ =2n +1.If λ =2n +1, thenh
n
(x) is not finite as |x|→∞.
The Hermite functions {h
n
(x)}
∞
0
are an orthogonal basis for L
2
(R)
with weight function one. They satisfy the following relations:
h
′
n
(x)=−xh
n
(x)+2nh
n−1
(x) , (A-2.13)
h
′
n
(x)=xh
n
(x) − h
n+1
(x) , (A-2.14)
h
′′
n
(x) − x
2
h
n
(x)+(2n +1)h
n
(x)=0. (A-2.15)
The normalized Weber–Hermite functions are given by
ψ
n
(x)=
h
n
(x)
(2
n
n!
√
π)
exp
−
x
2
2
H
n
(x) . (A-2.16)
The functions {ψ
n
(x)} form a orthornormal set in (−∞, ∞), that is,
A-2 Hermite Polynomials and Weber–Hermite Functions 759
∞
−∞
ψ
m
(x) ψ
n
(x) dx = δ
mn
. (A-2.17)
Physically, they represent quantum mechanical oscillator wave functions.
Some graphs of these functions are shown in Figure A-2.1.
The Fourier transform of h
n
(x)is(−i)
n
h
n
(x), that is,
F{h
n
(x)} =(−i)
n
h
n
(x) . (A-2.18)
Figure A-2.1 The normalized Weber–Hermite functions.
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interest in future advanced study and research.
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