Appendix E: Properties of special functions

Appendix E
Properties of special functions
E.1 Bessel functions
Notation
z = complex number; ν, x = real numbers; n = integer
Jν(z) = ordinary Bessel function of the first kind
Nν(z) = ordinary Bessel function of the second kind
Iν(z) = modified Bessel function of the first kind
Kν(z) = modified Bessel function of the second kind
H (1)
ν = Hankel function of the first kind
H (2)
ν = Hankel function of the second kind
jn(z) = ordinary spherical Bessel function of the first kind
nn(z) = ordinary spherical Bessel function of the second kind
h (1)
n (z) = spherical Hankel function of the first kind
h (2)
n (z) = spherical Hankel function of the second kind
f ′ (z) = df(z)/dz = derivative with respect to argument
Differential equations
d 2 Zν(z)
dz 2
1
+
z
dZν(z)
dz +
�
1 − ν2
z2 �
Zν(z) = 0 (E.1)
⎧
⎪⎨
Jν(z)
Nν(z)
Zν(z) =
⎪⎩
H (1)
ν (z)
H (2)
ν (z)
(E.2)
Nν(z) = cos(νπ)Jν(z) − J−ν(z)
, ν �= n, | arg(z)|

d 2 zn(z)
dz 2
d 2 ¯Zν(x)
dz 2
1
+
z
d ¯Zν(z)
dz −
�
1 + ν2
z2 �
¯Zν = 0 (E.6)
�
¯Zν(z)
Iν(z)
=
(E.7)
Kν(z)
�
Iν(z)
L(z) =
e jνπ (E.8)
Kν(z)
Iν(z) = e − jνπ/2 Jν(ze jπ/2 ), −π

� a
Jν
0
Specific examples
� ′ p νm
a ρ
� � ′ p νn
Jν
a ρ
�
a
ρ dρ = δmn
2 �
2
� ∞
0
1 − ν2
p ′2
νm
�
J 2 ν (p′ νm ), ν > −1 (E.23)
Jν(αx)Jν(βx)x dx= 1
δ(α − β) (E.24)
α
� a
0
�
αlm
jl
a r
� �
αln
jl
a r
�
r 2 a
dr = δmn
3
2 j 2 n+1 (αlna) (E.25)
� ∞
−∞
Functional relationships
π
jm(x) jn(x) dx = δmn , m, n ≥ 0 (E.26)
2n + 1
h (1)
0
h (2)
0
Jm(pmn) = 0 (E.27)
J ′ m (p′ mn ) = 0 (E.28)
jm(αmn) = 0 (E.29)
j ′ m (α′ mn ) = 0 (E.30)
sin z
j0(z) = (E.31)
z
cos z
n0(z) =− (E.32)
z
jz
(z) =−j e (E.33)
z
j jz
(z) = e− (E.34)
z
sin z cos z
j1(z) = − (E.35)
z2 z
cos z sin z
n1(z) =− − (E.36)
z2 � � z
3 1
j2(z) = − sin z −
z3 z
3
cos z (E.37)
z2 �
n2(z) = − 3
�
1
+ cos z −
z3 z
3
sin z (E.38)
z2 Jn(−z) = (−1) n Jn(z) (E.39)
In(−z) = (−1) n In(z) (E.40)
jn(−z) = (−1) n jn(z) (E.41)
nn(−z) = (−1) n+1 nn(z) (E.42)
J−n(z) = (−1) n Jn(z) (E.43)

Power series
N−n(z) = (−1) n Nn(z) (E.44)
I−n(z) = In(z) (E.45)
K−n(z) = Kn(z) (E.46)
j−n(z) = (−1) n nn−1(z), n > 0 (E.47)
∞�
k (z/2)n+2k
Jn(z) = (−1)
k!(n + k)!
k=0
In(z) =
∞�
k=0
Small argument approximations |z| ≪1.
(z/2) n+2k
k!(n + k)!
(E.48)
(E.49)
Jn(z) ≈ 1
�
z
�n n! 2
1
�
z
�ν Jν(z) ≈
Ɣ(ν + 1) 2
(E.50)
(E.51)
N0(z) ≈ 2
(ln z + 0.5772157 − ln 2)
π
� �n (n − 1)! 2
Nn(z) ≈− , n > 0
π z
Nν(z) ≈−
(E.52)
(E.53)
Ɣ(ν)
� �ν 2
, ν > 0
π z
In(z) ≈
(E.54)
1
�
z
�n n! 2
1
�
z
�ν Iν(z) ≈
Ɣ(ν + 1) 2
(E.55)
(E.56)
jn(z) ≈ 2n n!
(2n + 1)! zn
nn(z) ≈− (2n)!
2 n n! z−(n+1)
Large argument approximations |z| ≫1.
�
2
Jν(z) ≈
πz cos
�
z − π
�
4
2
Nν(z) ≈
πz sin
�
z − π νπ
−
4 2
H (1)
�
2
ν (z) ≈
πz
H (2)
�
2
ν (z) ≈
�
πz
1
Iν(z) ≈
2πz ez , | arg(z)| < π
2
(E.57)
(E.58)
νπ
�
− , | arg(z)|

Recursion relationships
Integral representations
�
π
Kν(z) ≈
2z e−z , | arg(z)| < 3π
2
jn(z) ≈
(E.64)
1
z sin
�
z − nπ
�
, | arg(z)|

Wronskians and cross products
Jν(z)Nν+1(z) − Jν+1(z)Nν(z) =− 2
πz
H (2) (1)
(1) (2) 4
ν (z)H ν+1 (z) − H ν (z)H ν+1 (z) =
jπz
Iν(z)Kν+1(z) + Iν+1(z)Kν(z) = 1
z
Iν(z)K ′ ν (z) − I ′ ν (z)Kν(z) =− 1
z
Jν(z)H (1) ′ ′
ν (z) − J ν (z)H (1) 2 j
ν (z) =
πz
Jν(z)H (2) ′ ′
ν (z) − J ν (z)H (2) j
ν (z) =−2
πz
(2) ′ (1) ′ (2) j
(z)H ν (z) − H ν (z)H ν (z) =−4
πz
jn(z)nn−1(z) − jn−1(z)nn(z) = 1
z2 2n + 1
jn+1(z)nn−1(z) − jn−1(z)nn+1(z) =
z3 jn(z)n ′ n (z) − j ′ n (z)nn(z) = 1
z2 h (1)
n (z)h(2)
′ (1) ′ (2) j
n (z) − h n (z)h n (z) =−2
z2 H (1)
ν
✁
✁
Summation formulas ✁
✁
✦
r ψ
R
✦
✦✦✦✦✦✦✦ φ
ρ
e jνψ Zν(zR) =
e jnψ Jn(zR) =
e jzρ cos φ =
∞�
k=−∞
∞�
k=−∞
R, r, ρ, φ, ψ as shown.
R = � r 2 + ρ 2 − 2rρ cos φ.
Jk(zρ)Zν+k(zr)e jkφ , ρ < r, 0

Integrals
�
�
x ν+1 Jν(x) dx = x ν+1 Jν+1(x) + C (E.104)
Zν(ax)Zν(bx)x dx= x [bZν(ax)Zν−1(bx) − aZν−1(ax)Zν(bx)]
+ C, a �= b (E.105)
�
xZ 2 2 x
ν (ax) dx =
� ∞
0
2
a 2 − b 2
� Z 2 ν (ax) − Zν−1(ax)Zν+1(ax) � + C (E.106)
Jν(ax) dx = 1
, ν > −1, a > 0 (E.107)
a
Fourier–Bessel expansion of a function
∞�
f (ρ) =
�
ρ
�
,
a
0 ≤ ρ ≤ a, ν > −1 (E.108)
�
ρ
�
f (ρ)Jν pνm ρ dρ
a
(E.109)
am Jν pνm
m=1
2
am =
a2 J 2 ν+1 (pνm)
� a
0
f (ρ) =
bm =
∞�
m=1
Series of Bessel functions
bm Jν
2
�
p ′ νm
a2 �
1 − ν2
p ′ 2 J
νm
2 ν (p′ νm )
e jzcos φ =
∞�
k=−∞
ρ
�
, 0 ≤ ρ ≤ a, ν > −1 (E.110)
a
�
e jzcos φ = J0(z) + 2
sin z = 2
E.2 Legendre functions
Notation
� a
0
j k Jk(z)e jkφ
f (ρ)Jν
� ′ p νm
a ρ
�
ρ dρ (E.111)
(E.112)
∞�
j k Jk(z) cos φ (E.113)
k=1
∞�
(−1) k J2k+1(z) (E.114)
k=0
cos z = J0(z) + 2
x, y,θ = real numbers; l, m, n = integers;
Pm n (cos θ) = associated Legendre function of the first kind
∞�
(−1) k J2k(z) (E.115)
k=1

Qm n
Pn(cos θ) = P0 n
Qn(cos θ) = Q0 n
(cos θ) = associated Legendre function of the second kind
(cos θ) = Legendre polynomial
(cos θ) = Legendre function of the second kind
Differential equation x = cos θ.
(1 − x 2 ) d2 R m n (x)
dx 2
− 2x dRm n (x)
dx +
�
n(n + 1) − m2
1 − x 2
�
R m n (x) = 0, −1 ≤ x ≤ 1 (E.116)
R m n
(x) =
� P m n (x)
Q m n (x)
Orthogonality relationships
� 1
P
−1
m
l (x)Pm 2 (n + m)!
n (x) dx = δln
2n + 1 (n − m)!
� π
P
0
m
l (cos θ)Pm 2 (n + m)!
n (cos θ)sin θ dθ = δln
2n + 1 (n − m)!
� 1 P
−1
m n (x)Pk n (x)
1 − x 2
1 (n + m)!
dx = δmk
m (n − m)!
� π P
0
m n (cos θ)Pk n (cos θ) 1 (n + m)!
dθ = δmk
sin θ
m (n − m)!
� 1
2
Pl(x)Pn(x) dx = δln
−1
2n + 1
� π
2
Pl(cos θ)Pn(cos θ)sin θ dθ = δln
2n + 1
Specific examples
0
(E.117)
(E.118)
(E.119)
(E.120)
(E.121)
(E.122)
(E.123)
P0(x) = 1 (E.124)
P1(x) = x = cos(θ) (E.125)
P2(x) = 1
2 (3x 2 − 1) = 1
(3 cos 2θ + 1)
4
P3(x) =
(E.126)
1
2 (5x 3 − 3x) = 1
(5 cos 3θ + 3 cos θ)
8
P4(x) =
(E.127)
1
8 (35x 4 − 30x 2 + 3) = 1
(35 cos 4θ + 20 cos 2θ + 9)
64
P5(x) =
(E.128)
1
8 (63x 5 − 70x 3 + 15x) = 1
(63 cos 5θ + 35 cos 3θ + 30 cos θ)
128
(E.129)
Q0(x) = 1
2 ln
� � �
1 + x
= ln cot
1 − x
θ
�
(E.130)
2
Q1(x) = x
2 ln
� �
�
1 + x
− 1 = cos θ ln cot
1 − x
θ
�
− 1 (E.131)
2

Q2(x) = 1
4 (3x 2 � �
1 + x
− 1) ln −
1 − x
3
x (E.132)
2
Q3(x) = 1
4 (5x 3 � �
1 + x
− 3x) ln −
1 − x
5
2 x 2 + 2
(E.133)
3
Q4(x) = 1
16 (35x 4 − 30x 2 � �
1 + x
+ 3) ln −
1 − x
35
8 x 3 + 55
x (E.134)
24
P 1 1 (x) =−(1 − x 2 ) 1/2 =−sin θ (E.135)
P 1 2 (x) =−3x(1 − x 2 ) 1/2 =−3cos θ sin θ (E.136)
P 2 2 (x) = 3(1 − x 2 ) = 3 sin 2 θ (E.137)
P 1 3 (x) =−3
2 (5x 2 − 1)(1 − x 2 ) 1/2 =− 3
2 (5 cos2 θ − 1) sin θ (E.138)
P 2 3 (x) = 15x(1 − x 2 ) = 15 cos θ sin 2 θ (E.139)
P 3 3 (x) =−15(1 − x 2 ) 3/2 =−15 sin 3 θ (E.140)
P 1 4 (x) =−5
2 (7x 3 − 3x)(1 − x 2 ) 1/2 =− 5
2 (7 cos3 θ − 3 cos θ)sin θ
P
(E.141)
2 15
4 (x) =
2 (7x 2 − 1)(1 − x 2 ) = 15
2 (7 cos2 θ − 1) sin 2 θ (E.142)
P 3 4 (x) =−105x(1 − x 2 ) 3/2 =−105 cos θ sin 3 θ (E.143)
P 4 4 (x) = 105(1 − x 2 ) 2 = 105 sin 4 θ (E.144)
Functional relationships
�
0, m > n,
(x) =
P m n
(−1) m (1−x2 ) m/2
2 n n!
d n+m (x 2 −1) n
dx n+m , m ≤ n.
Pn(x) = 1
2n d
n!
n (x 2 − 1) n
dxn R m n (x) = (−1)m (1 − x 2 ) m/2 dm Rn(x)
dx m
(E.145)
(E.146)
(E.147)
P −m
m (n − m)!
n (x) = (−1)
(n + m)! Pm n (x) (E.148)
Pn(−x) = (−1) n Pn(x) (E.149)
Qn(−x) = (−1) n+1 Qn(x) (E.150)
P m n (−x) = (−1)n+m P m n (x) (E.151)
Q m n (−x) = (−1)n+m+1Q m n (x) (E.152)
P m n
(1) =
�
1, m = 0,
0, m > 0.
(E.153)
|Pn(x)| ≤Pn(1) = 1 (E.154)

Power series
Pn(x) =
n�
k=0
Recursion relationships
Pn(0) = Ɣ � � n 1
+ 2 2 nπ
cos
+ 1� 2
√ πƔ � n
2
(E.155)
P −m
m (n − m)!
n (x) = (−1)
(n + m)! Pm n (x) (E.156)
(−1) k (n + k)!
(n − k)!(k!) 22k+1 � k n k
(1 − x) + (−1) (1 + x) �
(E.157)
(n + 1 − m)R m n+1 (x) + (n + m)Rm n−1 (x) = (2n + 1)xRm n (x) (E.158)
(1 − x 2 )R m n
Integral representations
√
2
Pn(cos θ) =
π
Pn(x) = 1
π
Addition formula
Summations
′ m
(x) = (n + 1)xRn (x) − (n − m + 1)R m n+1 (x) (E.159)
(2n + 1)xRn(x) = (n + 1)Rn+1(x) + nRn−1(x) (E.160)
(x 2 − 1)R ′ n (x) = (n + 1)[Rn+1(x) − xRn(x)] (E.161)
R ′ n+1 (x) − R′ n−1 (x) = (2n + 1)Rn(x) (E.162)
� π
0 � π
0
sin � n + 1
�
u 2
√ du
cos θ − cos u
(E.163)
� � 2 1/2 n
x + (x − 1) cos θ dθ (E.164)
Pn(cos γ) = Pn(cos θ)Pn(cos θ ′ ) +
n� (n − m)!
+ 2
(n + m)! Pm n (cos θ)Pm n (cos θ ′ ) cos m(φ − φ ′ ), (E.165)
m=1
cos γ = cos θ cos θ ′ + sin θ sin θ ′ cos(φ − φ ′ ) (E.166)
1
|r − r ′ | =
1
� =
r 2 + r ′2 − 2rr ′ cos γ
∞�
n=0
r n <
r n+1 Pn(cos γ) (E.167)
>
cos γ = cos θ cos θ ′ + sin θ sin θ ′ cos(φ − φ ′ ) (E.168)
r< = min � |r|, |r ′ | � , r> = max � |r|, |r ′ | �
(E.169)

Integrals
�
Pn(x) dx = Pn+1(x) − Pn−1(x)
+ C
2n + 1
� 1
x
(E.170)
−1
m Pn(x) dx = 0, m < n (E.171)
� 1
x
−1
n Pn(x) dx = 2n+1 (n!) 2
(2n + 1)!
� 1
x
−1
(E.172)
2k P2n(x) dx = 22n+1 (2k)!(k + n)!
(2k + 2n + 1)!(k − n)!
� 1 Pn(x)
√ dx =
−1 1 − x
(E.173)
2√2 2n + 1
� � � ��
1
1
2
P2n(x) Ɣ n + 2
√ dx =
−1 1 − x 2 n!
� 1
n (2n)! 1
P2n+1(x) dx = (−1)
2n + 2 (2
(E.174)
(E.175)
nn!) 2
(E.176)
0
Fourier–Legendre series expansion of a function
f (x) =
an =
E.3 Spherical harmonics
Notation
∞�
an Pn(x), −1 ≤ x ≤ 1 (E.177)
n=0
2n + 1
2
� 1
θ,φ = real numbers; m, n = integers
Ynm(θ, φ) = spherical harmonic function
Differential equation
1
sin θ
∂
∂θ
�
sin θ
−1
�
∂Y (θ, φ)
+
∂θ
1
sin 2 ∂
θ
2Y (θ, φ)
∂φ2 Ynm(θ, φ) =
�
f (x)Pn(x) dx (E.178)
+ 1
λY (θ, φ) = 0 (E.179)
a2 λ = a 2 n(n + 1) (E.180)
2n + 1 (n − m)!
4π (n + m)! Pm jmθ
n (cos θ)e
(E.181)

Orthogonality relationships
∞�
Specific examples
� π � π
−π
n�
n=0 m=−n
Functional relationships
Addition formulas
0
Y ∗ n ′ m ′(θ, φ)Ynm(θ, φ) sin θ dθ dφ = δn ′ nδm ′ m
(E.182)
Y ∗ nm (θ ′ ,φ ′ )Ynm(θ, φ) = δ(φ − φ ′ )δ(cos θ − cos θ ′ ) (E.183)
�
1
Y00(θ, φ) =
(E.184)
�
4π
3
Y10(θ, φ) = cos θ (E.185)
�
4π
3 jφ
Y11(θ, φ) =− sin θe (E.186)
�
8π
�
5 3
Y20(θ, φ) =
4π 2 cos2 θ − 1
�
(E.187)
2
�
15
jφ
Y21(θ, φ) =− sin θ cos θe (E.188)
�
8π
15
Y22(θ, φ) =
32π sin2 2 jφ
θe (E.189)
� �
7 5
Y30(θ, φ) =
4π 2 cos3 θ − 3
�
cos θ
(E.190)
2
�
21
Y31(θ, φ) =−
64π sin θ � 5 cos 2 θ − 1 � e jφ
(E.191)
�
105
Y32(θ, φ) =
32π sin2 2 jφ
θ cos θe (E.192)
�
35
Y33(θ, φ) =−
64π sin3 3 jφ
θe (E.193)
Yn0(θ, φ) =
� 2n + 1
4π Pn(cos θ) (E.194)
Yn,−m(θ, φ) = (−1) m Y ∗ nm (θ, φ) (E.195)
Pn(cos γ)= 4π
2n + 1
n�
Ynm(θ, φ)Y
m=−n
∗ nm (θ ′ ,φ ′ ) (E.196)
Pn(cos γ) = Pn(cos θ)Pn(cos θ ′ ) +
+
n� (n − m)!
(n + m)! Pm n (cos θ)Pm n (cos θ ′ ) cos � m(φ − φ ′ ) �
m=−n
(E.197)

Series
1
|r − r ′ | =
cos γ = cos θ cos θ ′ + sin θ sin θ ′ cos(φ − φ ′ ) (E.198)
n�
m=−n
|Ynm(θ, φ)| 2 =
1
� r 2 + r ′2 − 2rr ′ cos γ
= 4π
∞�
Series expansion of a function
n�
n=0 m=−n
1
2n + 1
2n + 1
4π
r< = min � |r|, |r ′ | � , r> = max � |r|, |r ′ | �
f (θ, φ) =
� π � π
anm =
−π 0
∞�
n�
n=0 m=−n
(E.199)
r n <
r n+1 Y
>
∗ nm (θ ′ ,φ ′ )Ynm(θ, φ), (E.200)
(E.201)
anmYnm(θ, φ) (E.202)
f (θ, φ)Y ∗ nm (θ, φ) sin θ dθ dφ (E.203)