soikez
Apr11-11, 01:35 PM
According to the orthogonality property of the associated Legendre function
P_l^{|m|}(cos\theta)
we have that:
\int_{0}^{\pi}P_{l}^{|m|}(cos\theta){\cdot}P_{l'}^ {|m'|}(cos\theta)sin{\theta}d\theta=\frac{2(l+m)!} {(2l+1)(l-m)!}{\delta}_{ll'}
What I am looking for is an orthogonality property for the derivative of the associated Legendre function
P^{'}_{l}^{|m|}(cos\theta):
something like that perhaps:
\int_{0}^{\pi}P^{'}_{l}^{|m|}(cos\theta){\cdot}P^{ '}_{l'}^{|m'|}(cos\theta)sin{\theta}d\theta=?
or even taking into consideration the fact that the derivative of the associated Legendre function is:
P^{'}_{l}^{|m|}(cos\theta)=\frac{lcos{\theta}P_{l} ^{|m|}(cos\theta)-(l+m)P_{l-1}^{|m|}(cos\theta)}{sin\theta}
after some manipulations on my equation an orthogonality property over the sum below:
\sum_{l}^{\infty}\sum_{m=-l}^{m=l}cos{\theta}P_{l}^{|m|}(cos\theta)-(l+m)P_{l-1}^{|m|}(cos\theta)e^{jm\phi}
Thanks in advance
P_l^{|m|}(cos\theta)
we have that:
\int_{0}^{\pi}P_{l}^{|m|}(cos\theta){\cdot}P_{l'}^ {|m'|}(cos\theta)sin{\theta}d\theta=\frac{2(l+m)!} {(2l+1)(l-m)!}{\delta}_{ll'}
What I am looking for is an orthogonality property for the derivative of the associated Legendre function
P^{'}_{l}^{|m|}(cos\theta):
something like that perhaps:
\int_{0}^{\pi}P^{'}_{l}^{|m|}(cos\theta){\cdot}P^{ '}_{l'}^{|m'|}(cos\theta)sin{\theta}d\theta=?
or even taking into consideration the fact that the derivative of the associated Legendre function is:
P^{'}_{l}^{|m|}(cos\theta)=\frac{lcos{\theta}P_{l} ^{|m|}(cos\theta)-(l+m)P_{l-1}^{|m|}(cos\theta)}{sin\theta}
after some manipulations on my equation an orthogonality property over the sum below:
\sum_{l}^{\infty}\sum_{m=-l}^{m=l}cos{\theta}P_{l}^{|m|}(cos\theta)-(l+m)P_{l-1}^{|m|}(cos\theta)e^{jm\phi}
Thanks in advance