Fymm IIb luentojen betaversio

18
voidaan kirjoittaa muodossa L = ϕ(x)L. Kuinka?
ϕ(x)L = −ϕ(x)p(x) d2
dx 2 − ϕ(x)p′ (x) d
dx + ϕ(x)q(x)
=⇒ p′ (x)
p(x) = a 1(x)
a 2 (x)
(∫ x
=⇒ p(x) = exp
=⇒ ϕ(x) = − a 2(x)
p(x) = −a 2(x)exp
a 1 (x ′ )
)
x 0
a 2 (x ′ ) dx′ ( ∫ x
−
a 1 (x ′ )
)
x 0
a 2 (x ′ ) dx′
=⇒ q(x) = a 0(x)
ϕ(x) = −a 0(x)
a 2 (x) exp (∫ x
x 0
a 1 (x ′ )
a 2 (x ′ ) dx′ )
3.1.1 Erikoisfunktiot ja Sturm-Liouville yhtälöt
Legendren polynomit
Legendren liittofunktiot
− d [
(1 − x 2 ) dP ]
l(x)
= l(l + 1)P l (x) (3.9)
dx
dx
a = −1, b = 1, p(x) = 1 − x 2 , q(x) = 0, w(x) = 1, λ = l(l + 1)
− d
dx
[(1 − x 2 ) dP l
m
dx
]
(x)
+ m
1 − x 2 P l
m (x) = l(l + 1)Pl m (x) (3.10)
a = −1, b = 1, p(x) = 1 − x 2 , q(x) = m2
, w(x) = 1, λ = l(l + 1)
1 − x2 Besselin funktiot (R > 0 J ν :n 0-kohta, ξ = x/R ∈ [0, 1])
[
d
ξ dJ ]
ν(ξR)
+ ν2
dξ dξ ξ J ν(ξR) = R 2 ξJ ν (ξR) (3.11)
Laguerren polynomit
Laguerren liittopolynomit
Hermiten polynomit
a = 0, b = 1, p(ξ) = ξ, q(ξ) = ν2
, w(ξ) = ξ, λ = R2
ξ
− d [
xe −x dL ]
n
= ne −x L n (x) (3.12)
dx dx
a = 0, b = ∞, p(x) = xe −x , q(x) = 0, w(x) = e −x , λ = n
− d (
)
x k+1 e −x dLk l
= x k e −x nL k
dx dx
n(x) (3.13)
a = 0, b = ∞, p(x) = x k+1 e −x , q(x) = 0, w(x) = x k e −x , λ = n
− d [
e dH ]
−x2 n
= 2ne −x2 H n (x) (3.14)
dx dx
a = −∞, b = ∞, p(x) = e −x2 , q(x) = 0, w(x) = e −x2 , λ = 2n