A Partial Table of Integrals
x
cos nx + nx sin nx − 1
for any real n 6= 0
n2
0
Z x
sin nx − nx cos nx
u sin nu du =
for any real n 6= 0
n2
Z0 x
emx (m cos nx + n sin nx) − m
emu cos nu du =
for any real n, m
m2 + n2
Z0 x
emx (−n cos nx + m sin nx) + n
emu sin nu du =
for any real n, m
m2 + n2
Z0 x
m sin nx sin mx + n cos nx cos mx − n
sin nu cos mu du =
for any real numbers m 6= n
m2 − n2
0
Z x
m cos nx sin mx − n sin nx cos mx
cos nu cos mu du =
for any real numbers m 6= n
m2 − n2
0
Z x
n cos nx sin mx − m sin nx cos mx
sin nu sin mu du =
for any real numbers m 6= n
m2 − n2
0
Z
u cos nu du =
Formulas Involving Bessel Functions
• Bessel’s equation: r2 R00 + rR0 + (α2 r2 − n2 )R = 0 – The only solutions of this which are bounded at r = 0 are
R(r) = cJn (αr).
∞
X
(−1)k x n+2k
Jn (x) =
.
k!(k + n)! 2
k=0
J0 (0) = 1, Jn (0) = 0 if n > 0. znm is the mth positive zero of Jn (x).
• Orthogonality relations:
Z
If m 6= k then
1
Z
xJn (znm x)Jn (znk x) dx = 0
0
and
1
x(Jn (znm x))2 dx =
0
1
Jn+1 (znm )2 .
2
• Recursion and differentiation formulas:
d n
(x Jn (x)) = xn Jn−1 (x)
dx
Z
xn Jn−1 (x) dx = xn Jn (x) + C
or
d −n
(x Jn (x)) = −x−n Jn+1 (x) for n ≥ 0
dx
n
Jn0 (x) + Jn (x) = Jn−1 (x)
x
n
0
Jn (x) − Jn (x) = −Jn+1 (x)
x
for n ≥ 1
(1)
(2)
(3)
(4)
2Jn0 (x) = Jn−1 (x) − Jn+1 (x)
(5)
2n
Jn (x) = Jn−1 (x) + Jn+1 (x)
x
(6)
• Modified Bessel’s equation: r2 R00 + rR0 − (α2 r2 + n2 )R = 0 – The only solutions of this which are bounded at r = 0
are R(r) = cIn (αr).
∞
x n+2k
X
1
In (x) = i−n Jn (ix) =
.
k!(k + n)! 2
k=0
1
Formulas Involving Associated Legendre and Spherical Bessel Functions
dg
m2
d
sin φ dφ
+ µ − sin
• Associated Legendre Functions: dφ
φ g = 0. Using the substitution x = cos φ, this equation
dg
d
m2
becomes dx
(1 − x2 ) dx
+ µ − 1−x
g = 0. This equation has bounded solutions only when µ = n(n + 1) and
2
0 ≤ m ≤ n. The solution Pnm (x) is called an associated Legendre function of the first kind.
• Associated Legendre Function Identities:
Pn0 (x) =
m
1 dn 2
n
m
m
2 m/2 d
(x
−
1)
and
P
(x)
=
(−1)
(1
−
x
)
Pn (x) when 1 ≤ m ≤ n
n
2n n! dxn
dxm
• Orthogonality of Associated Legendre Functions: If n and k are both greater than or equal to m,
Z 1
Z 1
2(n + m)!
2
m
m
If n 6= k then
Pn (x)Pk (x)dx = 0 and
(Pnm (x)) dx =
.
(2n + 1)(n − m)!
−1
−1
• Spherical Bessel Functions: (ρ2 f 0 )0 + (α2 ρ2 − n(n + 1))f = 0. If we define the spherical Bessel function jn (ρ) =
1
ρ− 2 Jn+ 21 (ρ), then only solution of this ODE bounded at ρ = 0 is jn (αρ).
• Spherical Bessel Function Identity:
jn (x) = x
2
1 d
−
x dx
n sin x
x
.
• Spherical Bessel Function Orthogonality: Let znm be the m-th positive zero of jm .
Z 1
Z 1
1
2
If m 6= k then
x jn (znm x)jn (znk x)dx = 0 and
x2 (jn (znm x))2 dx = (jn+1 (znm ))2 .
2
0
0
One-Dimensional Fourier Transform
F[u](ω) =
1
2π
Z
∞
F −1 [U ](x) =
u(x)eiωx dx,
−∞
Z
∞
U (ω)e−iωx dω
−∞
Table of Fourier Transform Pairs
Fourier Transform Pairs
Fourier Transform Pairs
(α > 0)
(β > 0)
u(x) = F −1 [U ]
U (ω) = F[u]
u(x) = F −1 [U ]
U (ω) = F[u]
r
2
2
2
2
ω
1
π − x4β
√
e−αx
e− 4α
e
e−βω
β
4πα
1
2β
2α
e−α|x|
e−β|ω|
2
2
2
2
2π
x
+
α
x
+
β
(
(
0 |x| > α
0 |ω| > β
1 sin αω
sin βx
u(x) =
2
U (ω) =
π ω
x
1 |x| < α
1 |ω| < β
1 iωx0
δ(x − x0 )
e
e−iω0 x
δ(ω − ω0 )
2π
2
∂u
∂U
∂ u
∂2U
∂t
∂t
∂t2
∂t2
2
∂u
∂ u
−iωU
(−iω)2 U
∂x
∂x2
∂U
∂2 U
xu
−i
x2 u
(−i)2
∂ω
∂ω 2
Z ∞
1
u(x − x0 )
eiωx0 U
f (s)g(x − s)ds
FG
2π −∞
2