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