Funciones especiales
e−(t2−2xt)=
∞
∑
n=0
Hn
n!tn
H`(x) =
[`/2]
∑
s=0
(−1)s`!
(`−2s)!s!(2x)`−2s
H`+1(x) = 2x H`(x)−2`H`−1(x)
H0
`(x) = 2`H`−1(x)
Z+∞
−∞[H`(x)]2e−x2dx=√π2`(`)!
L`+1(x)+(x−2`−1)L`(x) + `2L`−1(x) = 0
Ln(x) = ex
n!
dn
dxn[xne−x]
Z∞
0
e−xLm(x)Ln(x)dx=δm,n
e−xz
1−z
(1−z)k+1=
∞
∑
m=0
Lk
m(x)zm,|z|<1
Lk
n(x) = (−1)kdk
dxk[Ln+k(x)]
x[Lk
m(x)]0=m Lk
m(x)−(m+k)Lk
m−1(x)
x[Lk
m(x)]00 + (k+1−x)[Lk
m(x)]0+m Lk
m(x) = 0
Transformada de Laplace
F(s) = R+∞
0f(t)e−stdt
L[f(k)(t)] = sL[f(k−1)(t)]−f(k−1)(0)
L[tnf(t)] = (−1)nF(n)(s)
L[f(t)] = 1
1−e−sT RT
0e−st f(t)dt
L−1[F(s)] = 1
2πiZa+i∞
a−i∞
F(s)estds=
n
∑
i=1
Ress=si{sstF(s)}
L[a] = a
s,L[tn] = n
sL[tn−1],L[tn] = n!
sn+1
................................"................................ ................................"................................
Series de Fourier
f(t) = A0
2++∞
∑
n=1Ancos(2πn
Tt) + BnSen(2πn
Tt)
An=2
TRT/2
−T/2f(t)cos2πn
Ttdt Bn=2
TRT/2
−T/2f(t)sen2πn
Ttdt
f(t) = +∞
∑
n=−∞
Cnei2π
Tnt,Cn=1
TRT/2
−T/2f(t)e−i2π
Tntdt
Tansformada de Fourier
F(w) = R+∞
−∞f(t)e−iwtdt,f(t) = 1
2πR+∞
−∞F(w)eiwtdt
F[f(k)] = (iw)kF[f]F[f(t−t0)] = F(w)e−iwt0
F[f(t)eiwt] = F[w−w0]F[f(at)] = 1
|a|F(w
a)
F(k) = 1
√2πZ+∞
−∞
f(x)eikxdx,f(x) = 1
√2πZ+∞
−∞
F(k)e−ikxdk
I[k] = Z+∞
−∞
r[x]eik µ[x]dx,k→∞
µ[x] = µ[x0]+(x−x0)µ0(x0) + (x−x0)2
2! µ00[x0] + ···
F{δ(x)}=1
√2π
Rf(x0)δ(x0−x)dx0=f(x),Rδ(x)dx=1
δ(bx) = 1
|b|δ(x),R∞
−∞
dmδ(x)
dxmf(x)dx= (−1)mdmf(0)
dxm
δ(x2−b2) = 1
2|b|δ(x+b) + δ(x−b)(b>0)
Funciones de Green
[z−L(~r)]G(~r,~r0;z) = δ(~r−~r0)
G(~r,~r0;z) =<~r|G(z)|~r0>G(z) = ∑n|φn><φn|
z−λn
G∗(~r,~r0;z) = G(~r0,~r;z∗),G−(~r,~r0;λ) = [G+(~r0,~r;λ)]∗
¯
G(λ)≡G+(λ)−G−(λ)l´
ımy→∞1
x±y=P1
x∓iπ δ(x)
Referencias
[1] Métodos Matemáticos de la Física I y II. R. Toribio
Saavedra. FCNM-UNAC. Callao. 2010.
.......................................................................................................................................
FÓRMULAS PARA CIENTÍFICOS
E INGENIEROS
Dr. Richard Toribio Saavedra
13 de junio de 2012
Algebra vectorial y tensorial
~
A×(~
B×~
C) = ~
B(~
A·~
C)−~
C(~
A·~
B)
∇·(ψ~
A) = ∇ψ·~
A+ψ∇·~
A
∇×(ψ~
A) = ∇ψ×~
A+ψ∇×~
A
∇×(~
A×~
B) = (~
B·∇)~
A−(~
A·∇)~
B−~
B(∇·~
A) +~
A(∇·~
B)
~
A×(∇×~
A) = 1
2∇(A2)−(~
A·∇)~
A
∇×~
B=α~
B(campo Beltrami)
3
∑
k=1
a`kxk=x0
`
3
∑
`=1
aj`ak`=δjk
ai j =∂xj
∂x0
i=cos(x0
i,xj)
{∇ψ}α=∂αψ∇·~
F=3
∑
α=1
∂αFα
{∇×~
F}γ=3
∑
α,β=1
εγαβ ∂αFβ∂
∂xα=∂α
RS~
A·d~
σ=RV∇·dτRΓ~
A·d
~
λ=RS(∇×~
A)·d~
σ
RV(u∇·∇v−v∇·∇u)dτ=RS(u∇v−v∇u)·dσ
ds2=∑
i,j
h2
i j dqidqjh2
i j =∑
k
∂xk
∂qi
∂xk
∂qj
ˆai=1
hi
∂~r
∂qi
,d~
F= (d~r.∇)~
F+∂~
F
∂tdt
εipq εj pq =2δij εi jk εpqk =δipδj q −δiqδjp
T0
i j =3
∑
k,`=1
aik aj`Tk`T0i
j=3
∑
k,`=1
aik aj`Tk
`
