Spherical harmonics/Addendum

First few spherical harmonics

The following functions are normalized to unity and have the Condon & Shortley phase. The functions are listed first in θ and φ and then in x, y, z, and r. These parameters are connected by

$x=r\sin \theta \cos \phi ,\quad y=r\sin \theta \sin \phi ,\quad z=r\cos \theta ,\quad r={\sqrt {x^{2}+y^{2}+z^{2}}}$

---

${\begin{aligned}Y_{0}^{0}(\theta ,\varphi )&={\sqrt {1 \over 4\pi }}\\&\\Y_{1}^{0}(\theta ,\varphi )&={\sqrt {3 \over 4\pi }}\,\cos \theta ={\sqrt {3 \over 4\pi }}\,{\frac {z}{r}}\\Y_{1}^{\pm 1}(\theta ,\varphi )&=\mp {\sqrt {3 \over 4\pi }}{\sqrt {\frac {1}{2}}}\,\sin \theta \,e^{\pm i\varphi }=\mp {\sqrt {3 \over 4\pi }}{\sqrt {\frac {1}{2}}}\,{\frac {x\pm iy}{r}}\\&\\Y_{2}^{0}(\theta ,\varphi )&={\sqrt {5 \over 4\pi }}\,{\frac {1}{2}}(3\cos ^{2}\theta -1)={\sqrt {5 \over 4\pi }}\,{\frac {1}{2}}{\frac {3z^{2}-r^{2}}{r^{2}}}\\Y_{2}^{\pm 1}(\theta ,\varphi )&=\mp {\sqrt {5 \over 4\pi }}{\sqrt {\frac {3}{2}}}\,\sin \theta \,\cos \theta \,e^{\pm i\varphi }=\mp {\sqrt {5 \over 4\pi }}{\sqrt {\frac {3}{2}}}\,{\frac {z(x\pm iy)}{r^{2}}}\\Y_{2}^{\pm 2}(\theta ,\varphi )&={\sqrt {5 \over 4\pi }}\,{\sqrt {\frac {3}{8}}}\sin ^{2}\theta \,e^{\pm 2i\varphi }={\sqrt {5 \over 4\pi }}\,{\sqrt {\frac {3}{8}}}{\frac {(x\pm iy)^{2}}{r^{2}}}\\&\\Y_{3}^{0}(\theta ,\varphi )&={\sqrt {\frac {7}{4\pi }}}\,{\frac {1}{2}}\,(5\cos ^{3}\theta -3\cos \theta )={\sqrt {\frac {7}{4\pi }}}\,{\frac {1}{2}}\,{\frac {5z^{3}-3zr^{2}}{r^{3}}}\\Y_{3}^{\pm 1}(\theta ,\varphi )&=\mp {\sqrt {\frac {7}{4\pi }}}\,{\sqrt {\frac {3}{16}}}\,\sin \theta (5\cos ^{2}\theta -1)e^{\pm i\varphi }=\mp {\sqrt {\frac {7}{4\pi }}}\,{\sqrt {\frac {3}{16}}}\,{\frac {(x\pm iy)(5z^{2}-r^{2})}{r^{3}}}\\Y_{3}^{\pm 2}(\theta ,\varphi )&={\sqrt {\frac {7}{4\pi }}}\,{\sqrt {\frac {15}{8}}}\,\sin ^{2}\theta \cos \theta e^{\pm 2i\varphi }={\sqrt {\frac {7}{4\pi }}}\,{\sqrt {\frac {15}{8}}}\,{\frac {z(x\pm iy)^{2}}{r^{3}}}\\Y_{3}^{\pm 3}(\theta ,\varphi )&=\mp {\sqrt {\frac {7}{4\pi }}}\,{\sqrt {\frac {5}{16}}}\,\sin ^{3}\theta e^{\pm 3i\varphi }=\mp {\sqrt {\frac {7}{4\pi }}}\,{\sqrt {\frac {5}{16}}}\,{\frac {(x\pm iy)^{3}}{r^{3}}}\\&\\Y_{4}^{0}(\theta ,\varphi )&={\sqrt {\frac {9}{4\pi }}}\,{\frac {1}{8}}\,(35\cos ^{4}\theta -30\cos ^{2}\theta +3)={\sqrt {\frac {9}{4\pi }}}\,{\frac {1}{8}}\,{\frac {35z^{4}-30z^{2}r^{2}+3r^{4}}{r^{4}}}\\Y_{4}^{\pm 1}(\theta ,\varphi )&=\mp {\sqrt {\frac {9}{4\pi }}}\,{\sqrt {\frac {5}{16}}}\,\sin \theta (7\cos ^{3}\theta -3\cos \theta )e^{\pm i\varphi }=\mp {\sqrt {\frac {9}{4\pi }}}\,{\sqrt {\frac {5}{16}}}\,{\frac {(x\pm iy)(7z^{3}-3zr^{2})}{r^{4}}}\\Y_{4}^{\pm 2}(\theta ,\varphi )&={\sqrt {\frac {9}{4\pi }}}\,{\sqrt {\frac {5}{32}}}\,\sin ^{2}\theta (7\cos ^{2}\theta -1)e^{\pm 2i\varphi }={\sqrt {\frac {9}{4\pi }}}\,{\sqrt {\frac {5}{32}}}\,{\frac {(x\pm iy)^{2}(7z^{2}-r^{2})}{r^{4}}}\\Y_{4}^{\pm 3}(\theta ,\varphi )&=\mp {\sqrt {\frac {9}{4\pi }}}\,{\sqrt {\frac {35}{16}}}\,\sin ^{3}\theta \cos \theta e^{\pm 3i\varphi }=\mp {\sqrt {\frac {9}{4\pi }}}\,{\sqrt {\frac {35}{16}}}\,{\frac {z(x\pm iy)^{3}}{r^{4}}}\\Y_{4}^{\pm 4}(\theta ,\varphi )&={\sqrt {\frac {9}{4\pi }}}\,{\sqrt {\frac {35}{128}}}\,\sin ^{4}e^{\pm 4i\varphi }={\sqrt {\frac {9}{4\pi }}}\,{\sqrt {\frac {35}{128}}}\,{\frac {(x\pm iy)^{4}}{r^{4}}}\\\end{aligned}}$