Formula:5664

Digital Library of Mathematical Functions ID 14.30.E5

${\displaystyle{\displaystyle Y_{{l},{m}}\left(\tfrac{1}{2}\pi,\phi\right)=\begin{cases}\dfrac{(-1)^{(l+m)/2}e^{im\phi}}{2^{l}\left(\frac{1}{2}l-\frac{1}{2}m\right)!\left(\frac{1}{2}l+\frac{1}{2}m\right)!}\left(\dfrac{(l-m)!(l+m)!(2l+1)}{4\pi}\right)^{1/2},&\frac{1}{2}l+\frac{1}{2}m\in\mathbb{Z},\\ 0,&\frac{1}{2}l+\frac{1}{2}m\notin\mathbb{Z}.\end{cases}}}$

Symbols List

• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\in}}$:
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\mathbb{Z}}}$:
• ${\displaystyle{\displaystyle\notin}}$:
• ${\displaystyle{\displaystyle Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)}}$: spherical harmonic
• ${\displaystyle{\displaystyle m}}$: integer
• ${\displaystyle{\displaystyle l}}$: nonnegative integer
 

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