Formulas

Useful Formulas
Orthogonality relations for trig functions (for m and n positive integers):
a

cos
0 a

mπx nπx cos a a mπx nπx sin a a nπx mπx sin a a

dx = dx = dx = 0,

0, if m = n a 2, if m = n 0, if m = n a 2 , if m = n for all m, n

sin
0 a

cos
0

Bessel equation of order m (for m integer): r(rR′ )′ + (α2 r2 − m2 )R = 0, for 0 < r < a Eigenfunctions: Eigenvalues: R(r)= AJm (αr) + BYm (αr) αmn =
jmn a

where Jm (jmn ) = 0

for m = 0, 1, 2, . . . and n = 1, 2, 3, . . . Orthogonality:
a 0

Jm (αmn r) Jm (αmq r) r dr = 0

when n = q

Legendre equation:(sin(ϕ) Φ′ )′ + µ2 sin(ϕ) Φ = 0, for 0 < ϕ < π OR (1 − x2 )y ′′ − 2xy ′ + µ2 y = 0, for −1 < x < 1 (and x = cos ϕ) Eigenfunctions: Φ(ϕ) = Pn (cos ϕ) OR y(x) = Pn (x), where P0 (x) = 1, P1 (x) = x,
1P2 (x) = 3 x2 − 2 , . . . 2

(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x) Eigenvalues: Orthogonality:
π 1

µ2 = µ2 = n(n + 1), n = 0, 1, 2, . . . n

Pn (cos ϕ)Pm (cos ϕ) sin ϕ dϕ =
0 −1Pn (x) Pm (x) dx 0,
2 2n+1 ,

=

if m = n if m = n

Laplacian in cylindrical polar coordinates: ∆u(r, θ) = 1 ∂ r ∂r r ∂u ∂r + 1 ∂2u r2 ∂θ2

Laplacian in spherical polar coordinates: ∆u(r, θ,ϕ) = 1 ∂ r2 ∂r r2 ∂u ∂r + r2 ∂ 1 sin ϕ ∂ϕ sin ϕ ∂u ∂ϕ + r2 ∂2u 1 2 sin ϕ ∂θ2

13

Useful Formulas (cont’d)
Some integral identities: x sin(kx) dx = x cos(kx) dx = x2 sin(kx) dx = x2 cos(kx) dx =eax sin(kx) dx = eax cos(kx) dx = sin(kx) x cos(kx) − k2 k cos(kx) x sin(kx) + k2 k 2x sin(kx) (2 − k 2 x2 ) cos(kx) + k2 k3 2x cos(kx) (k 2 x2 − 2) sin(kx) + k2 k3 eax (a sin(kx) − k cos(kx)) a2 +k 2 eax (a cos(kx) + k sin(kx)) a2 + k 2

Hyperbolic trig identities: sinh x = 1 x (e − e−x ) 2 cosh x = 1 x (e + e−x ) 2

cosh2 x − sinh2 x = 1 d sinh x = cosh x dx d cosh x = sinh x dx

14Useful Formulas (cont’d)
Laplace transform properties:
∞

Definition: Linearity:

L (f (t)) = F (s) =
0

e−st f (t) dt

L (cf (t)) = cF (s) L (f (t) + g(t)) = L (f (t)) + L (g(t))...