\(\displaystyle{ P_{l}(t)=\frac{1}{2^{l}l!} \frac{d^{l}}{dt^{l}}(t^{2}-1)^{l}}\)
\(\displaystyle{ \int_{-1}^{1} P_{l_{n}}(t)P_{l_{m}}(t)dt= \frac{2}{(2l+1)}*\frac{(l+m)!}{(l-m)!} \delta_{mn}}\)
Trzeba to scałkować...Wykazac, ze funkcje dołączone tworzą ciąg funkcji ortogonalnych.
m==n 0
m=n 1
-l<=m<=l
Czy ktoś wie jak to zrobić?? Pomoże ktoś... ?
funkcja legendre'a
funkcja legendre'a
Skad wzial sie trzeci wiersz? Co oznacza single a co quote...? Na tej stronie nie widac tego dowodu, jest 'zaciemniony' dostepna byla tylko instrukcja texa, ktora tu umiescilam... ale nie wiem skad wynikaja te przeksztalcenia... Czy zna ktos moze inna strone, albo ksiazke, gdzie mozna wyprowadzic?
ortogonalnosc wielomiany legendre'a... PROSZE to dla mnie bardzo wazne...
\(\displaystyle{ P_{l}^{m} (x) = (1-x^2 ) ^{\frac{m}{2}} \frac{d^m}{dx^m} P_l (x)}\)
\(\displaystyle{ P_l (x) = \frac{1}{2^l l!} \frac{d^l}{dx^l} (x^2 -1 )^l}\)
\(\displaystyle{ I = \int_{-1}^{+1} P_l^m (x) P_lsingle-quote^m (x) dx}\)
\(\displaystyle{ I = \int_{-1}^{+1} P_l^m (x) P_lsingle-quote^m (x) dx}\)
\(\displaystyle{ \Leftrightarrow I = \int_{-1}^{+1} (1-x^2)^{\frac{m}{2}} \frac{d^m}{dx^m} P_l(x) (1-x)^2}\)
\(\displaystyle{ \Leftrightarrow I = \int_{-1}^{+1} (1-x^2)^m \frac{d^m}{dx^m} P_l (x) \frac{d^m}{dx^m} P_lsingle-quote (x) dx}\)
\(\displaystyle{ \Leftrightarrow I = \int_{-1}^{+1} \left[(1-x^2)^m \frac{d^m}{dx^m} P_l (x) \right] \frac{d}{dx} \left[\frac{d^m-1}{dx^m-1} P_lsingle-quote (x)\right] dx}\)
\(\displaystyle{ I = \int_{a}^{b} u(x)vsingle-quote(x) dx}\)
\(\displaystyle{ I = \left[( 1-x^2 )^m \frac{d^m}{dx^m} P_l (x) \frac{d^m-1}{dx^m-1} P_lsingle-quote (x) \right]_{-1}^{+1} - \int_{-1}^{+1} \frac{d}{dx} \left[ (1-x^2 )^m \frac{d^m}{dx^m} P_l (x) \right] \frac{d}{dx} \left[ \frac{d^{m-2}}{dx^{m-2}} P_lsingle-quote (x) \right] dx}\)
\(\displaystyle{ \Leftrightarrow I = - \int_{-1}^{+1} \frac{d}{dx} \left[ ( 1-x^2 )^m \frac{d^m}{dx^m} P_l (x) \right] \frac{d}{dx} \left[ \frac{d^{m-2}}{dx^{m-2}} P_lsingle-quote (x) \right] dx}\)
\(\displaystyle{ I = (-1)^m \int_{-1}^{+1} \frac{d^m}{dx^m} \left[ ( 1 -x^2 )^m \frac{d^m}{dx^m} P_l (x) \right] P_lsingle-quote (x) dx}\)
\(\displaystyle{ P_l (x) = \frac{1}{2^l l!} \frac{d^l}{dx^l} (x^2 -1)^l}\)
\(\displaystyle{ (-1)^m \frac{1}{2^l l!} \frac{(2l)!}{l!} \frac{d^m}{dx^m} \left[ x^2m \frac{d^m}{dx^m} x^l \right]}\)
\(\displaystyle{ \frac{1}{2^l l!} \frac{(2l)!}{l!} x^l}\)
\(\displaystyle{ (-1)^m \frac{1}{2^l l!} \frac{(2l)!}{l!} \frac{l!}{(l-m)!} \frac{(l+m)!}{l!} x^l}\)
\(\displaystyle{ = (-1)^m \frac{(l+m)!}{(l-m)!} P_l (x) + \cdots}\)
\(\displaystyle{ I = (-1)^m \int_{-1}^{+1} \frac{d^m}{dx^m} \left[ ( 1 -x^2 )^m \frac{d^m}{dx^m} P_l (x) \right] P_lsingle-quote (x) dx}\)
\(\displaystyle{ I=(-1)^{2m} \int_{-1}^{+1} \frac{(l+m)!}{(l-m)!} P_l (x) P_lsingle-quote (x) dx}\)
\(\displaystyle{ \int_{-1}^{+1} P_l (x) P_lsingle-quote (x) dx = \frac{2}{2l+1} \delta_{llsingle-quote}}\)
\(\displaystyle{ I = \int_{-1}^{+1} P_{l}^{m} (x) P_{lsingle-quote}^{m} (x) dx = \frac{2}{2l+1} \frac{(l+m)!}{(l-m)!} \delta_{llsingle-quote}}\)
ortogonalnosc wielomiany legendre'a... PROSZE to dla mnie bardzo wazne...
\(\displaystyle{ P_{l}^{m} (x) = (1-x^2 ) ^{\frac{m}{2}} \frac{d^m}{dx^m} P_l (x)}\)
\(\displaystyle{ P_l (x) = \frac{1}{2^l l!} \frac{d^l}{dx^l} (x^2 -1 )^l}\)
\(\displaystyle{ I = \int_{-1}^{+1} P_l^m (x) P_lsingle-quote^m (x) dx}\)
\(\displaystyle{ I = \int_{-1}^{+1} P_l^m (x) P_lsingle-quote^m (x) dx}\)
\(\displaystyle{ \Leftrightarrow I = \int_{-1}^{+1} (1-x^2)^{\frac{m}{2}} \frac{d^m}{dx^m} P_l(x) (1-x)^2}\)
\(\displaystyle{ \Leftrightarrow I = \int_{-1}^{+1} (1-x^2)^m \frac{d^m}{dx^m} P_l (x) \frac{d^m}{dx^m} P_lsingle-quote (x) dx}\)
\(\displaystyle{ \Leftrightarrow I = \int_{-1}^{+1} \left[(1-x^2)^m \frac{d^m}{dx^m} P_l (x) \right] \frac{d}{dx} \left[\frac{d^m-1}{dx^m-1} P_lsingle-quote (x)\right] dx}\)
\(\displaystyle{ I = \int_{a}^{b} u(x)vsingle-quote(x) dx}\)
\(\displaystyle{ I = \left[( 1-x^2 )^m \frac{d^m}{dx^m} P_l (x) \frac{d^m-1}{dx^m-1} P_lsingle-quote (x) \right]_{-1}^{+1} - \int_{-1}^{+1} \frac{d}{dx} \left[ (1-x^2 )^m \frac{d^m}{dx^m} P_l (x) \right] \frac{d}{dx} \left[ \frac{d^{m-2}}{dx^{m-2}} P_lsingle-quote (x) \right] dx}\)
\(\displaystyle{ \Leftrightarrow I = - \int_{-1}^{+1} \frac{d}{dx} \left[ ( 1-x^2 )^m \frac{d^m}{dx^m} P_l (x) \right] \frac{d}{dx} \left[ \frac{d^{m-2}}{dx^{m-2}} P_lsingle-quote (x) \right] dx}\)
\(\displaystyle{ I = (-1)^m \int_{-1}^{+1} \frac{d^m}{dx^m} \left[ ( 1 -x^2 )^m \frac{d^m}{dx^m} P_l (x) \right] P_lsingle-quote (x) dx}\)
\(\displaystyle{ P_l (x) = \frac{1}{2^l l!} \frac{d^l}{dx^l} (x^2 -1)^l}\)
\(\displaystyle{ (-1)^m \frac{1}{2^l l!} \frac{(2l)!}{l!} \frac{d^m}{dx^m} \left[ x^2m \frac{d^m}{dx^m} x^l \right]}\)
\(\displaystyle{ \frac{1}{2^l l!} \frac{(2l)!}{l!} x^l}\)
\(\displaystyle{ (-1)^m \frac{1}{2^l l!} \frac{(2l)!}{l!} \frac{l!}{(l-m)!} \frac{(l+m)!}{l!} x^l}\)
\(\displaystyle{ = (-1)^m \frac{(l+m)!}{(l-m)!} P_l (x) + \cdots}\)
\(\displaystyle{ I = (-1)^m \int_{-1}^{+1} \frac{d^m}{dx^m} \left[ ( 1 -x^2 )^m \frac{d^m}{dx^m} P_l (x) \right] P_lsingle-quote (x) dx}\)
\(\displaystyle{ I=(-1)^{2m} \int_{-1}^{+1} \frac{(l+m)!}{(l-m)!} P_l (x) P_lsingle-quote (x) dx}\)
\(\displaystyle{ \int_{-1}^{+1} P_l (x) P_lsingle-quote (x) dx = \frac{2}{2l+1} \delta_{llsingle-quote}}\)
\(\displaystyle{ I = \int_{-1}^{+1} P_{l}^{m} (x) P_{lsingle-quote}^{m} (x) dx = \frac{2}{2l+1} \frac{(l+m)!}{(l-m)!} \delta_{llsingle-quote}}\)