- #1
soikez
- 5
- 0
According to the orthogonality property of the associated Legendre function
[tex]P_l^{|m|}(cos\theta)[/tex]
we have that:
[tex]\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'}[/tex]
What I am looking for is an orthogonality property for the derivative of the associated Legendre function
[tex]P^{'}_{l}^{|m|}(cos\theta)[/tex]:
something like that perhaps:
[tex]\int_{0}^{\pi}P^{'}_{l}^{|m|}(cos\theta){\cdot}P^{'}_{l'}^{|m'|}(cos\theta)sin{\theta}d\theta=?[/tex]
or even taking into consideration the fact that the derivative of the associated Legendre function is:
[tex]P^{'}_{l}^{|m|}(cos\theta)=\frac{lcos{\theta}P_{l}^{|m|}(cos\theta)-(l+m)P_{l-1}^{|m|}(cos\theta)}{sin\theta}[/tex]
after some manipulations on my equation an orthogonality property over the sum below:
[tex]\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}[/tex]
Thanks in advance
[tex]P_l^{|m|}(cos\theta)[/tex]
we have that:
[tex]\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'}[/tex]
What I am looking for is an orthogonality property for the derivative of the associated Legendre function
[tex]P^{'}_{l}^{|m|}(cos\theta)[/tex]:
something like that perhaps:
[tex]\int_{0}^{\pi}P^{'}_{l}^{|m|}(cos\theta){\cdot}P^{'}_{l'}^{|m'|}(cos\theta)sin{\theta}d\theta=?[/tex]
or even taking into consideration the fact that the derivative of the associated Legendre function is:
[tex]P^{'}_{l}^{|m|}(cos\theta)=\frac{lcos{\theta}P_{l}^{|m|}(cos\theta)-(l+m)P_{l-1}^{|m|}(cos\theta)}{sin\theta}[/tex]
after some manipulations on my equation an orthogonality property over the sum below:
[tex]\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}[/tex]
Thanks in advance