Table of spherical harmonics

This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree l=10. Some of these formulas give the "Cartesian" version. This assumes x, y, z, and r are related to $heta,$ and $varphi,$ through the usual spherical-to-Cartesian coordinate transformation::$x = r sin hetacosvarphi,$:$y = r sin hetasinvarphi,$:$z = r cos heta,$

= Spherical harmonics with "l" = 0 =

:$Y_\left\{0\right\}^\left\{0\right\}\left( heta,varphi\right)=\left\{1over 2\right\}sqrt\left\{1over pi\right\}$

= Spherical harmonics with "l" = 1 =

:$Y_\left\{1\right\}^\left\{-1\right\}\left( heta,varphi\right)=\left\{1over 2\right\}sqrt\left\{3over 2pi\right\}cdot e^\left\{-ivarphi\right\}cdotsin hetaquad=\left\{1over 2\right\}sqrt\left\{3over 2pi\right\}cdot\left\{\left(x-iy\right)over r\right\}$:$Y_\left\{1\right\}^\left\{0\right\}\left( heta,varphi\right)=\left\{1over 2\right\}sqrt\left\{3over pi\right\}cdotcos hetaquad=\left\{1over 2\right\}sqrt\left\{3over pi\right\}cdot\left\{zover r\right\}$:$Y_\left\{1\right\}^\left\{1\right\}\left( heta,varphi\right)=\left\{-1over 2\right\}sqrt\left\{3over 2pi\right\}cdot e^\left\{ivarphi\right\}cdotsin hetaquad=\left\{-1over 2\right\}sqrt\left\{3over 2pi\right\}cdot\left\{\left(x+iy\right)over r\right\}$

= Spherical harmonics with "l" = 2 =

:$Y_\left\{2\right\}^\left\{-2\right\}\left( heta,varphi\right)=\left\{1over 4\right\}sqrt\left\{15over 2pi\right\}cdot e^\left\{-2ivarphi\right\}cdotsin^\left\{2\right\} hetaquad=\left\{1over 4\right\}sqrt\left\{15over 2pi\right\}cdot\left\{\left(x - iy\right)^2 over r^\left\{2$

:$Y_\left\{2\right\}^\left\{-1\right\}\left( heta,varphi\right)=\left\{1over 2\right\}sqrt\left\{15over 2pi\right\}cdot e^\left\{-ivarphi\right\}cdotsin hetacdotcos hetaquad=\left\{1over 2\right\}sqrt\left\{15over 2pi\right\}cdot\left\{\left(x - iy\right)z over r^\left\{2$

:$Y_\left\{2\right\}^\left\{0\right\}\left( heta,varphi\right)=\left\{1over 4\right\}sqrt\left\{5over pi\right\}cdot\left(3cos^\left\{2\right\} heta-1\right)quad=\left\{1over 4\right\}sqrt\left\{5over pi\right\}cdot\left\{\left(-x^\left\{2\right\}-y^\left\{2\right\}+2z^\left\{2\right\}\right)over r^\left\{2$

:$Y_\left\{2\right\}^\left\{1\right\}\left( heta,varphi\right)=\left\{-1over 2\right\}sqrt\left\{15over 2pi\right\}cdot e^\left\{ivarphi\right\}cdotsin hetacdotcos hetaquad=\left\{-1over 2\right\}sqrt\left\{15over 2pi\right\}cdot\left\{\left(x + iy\right)z over r^\left\{2$

:$Y_\left\{2\right\}^\left\{2\right\}\left( heta,varphi\right)=\left\{1over 4\right\}sqrt\left\{15over 2pi\right\}cdot e^\left\{2ivarphi\right\}cdotsin^\left\{2\right\} hetaquad=\left\{1over 4\right\}sqrt\left\{15over 2pi\right\}cdot\left\{\left(x + iy\right)^2 over r^\left\{2$

= Spherical harmonics with "l" = 3 =

:$Y_\left\{3\right\}^\left\{-3\right\}\left( heta,varphi\right)= \left\{1over 8\right\}sqrt\left\{35over pi\right\}cdot e^\left\{-3ivarphi\right\}cdotsin^\left\{3\right\} hetaquad= \left\{1over 8\right\}sqrt\left\{35over pi\right\}cdot\left\{\left(x - iy\right)^\left\{3\right\}over r^\left\{3$

:$Y_\left\{3\right\}^\left\{-2\right\}\left( heta,varphi\right)= \left\{1over 4\right\}sqrt\left\{105over 2pi\right\}cdot e^\left\{-2ivarphi\right\}cdotsin^\left\{2\right\} hetacdotcos hetaquad= \left\{1over 4\right\}sqrt\left\{105over 2pi\right\}cdot\left\{\left(x- iy\right)^2 z over r^\left\{3$

:$Y_\left\{3\right\}^\left\{-1\right\}\left( heta,varphi\right)=\left\{1over 8\right\}sqrt\left\{21over pi\right\}cdot e^\left\{-ivarphi\right\}cdotsin hetacdot\left(5cos^\left\{2\right\} heta-1\right)quad=\left\{1over 8\right\}sqrt\left\{21over pi\right\}cdot\left\{\left(x - iy\right)\left(4z^2- x^2 - y^2\right)over r^\left\{3$

:$Y_\left\{3\right\}^\left\{0\right\}\left( heta,varphi\right)=\left\{1over 4\right\}sqrt\left\{7over pi\right\}cdot\left(5cos^\left\{3\right\} heta-3cos heta\right)quad=\left\{1over 4\right\}sqrt\left\{7over pi\right\}cdot\left\{z\left(2z^2 - 3x^2 - 3y^2\right)over r^\left\{3$

:$Y_\left\{3\right\}^\left\{1\right\}\left( heta,varphi\right)=\left\{-1over 8\right\}sqrt\left\{21over pi\right\}cdot e^\left\{ivarphi\right\}cdotsin hetacdot\left(5cos^\left\{2\right\} heta-1\right)quad=\left\{-1over 8\right\}sqrt\left\{21over pi\right\}cdot\left\{\left(x + iy\right) \left(4z^2 - x^2 - y^2\right) over r^\left\{3$

:$Y_\left\{3\right\}^\left\{2\right\}\left( heta,varphi\right)=\left\{1over 4\right\}sqrt\left\{105over 2pi\right\}cdot e^\left\{2ivarphi\right\}cdotsin^\left\{2\right\} hetacdotcos hetaquad=\left\{1over 4\right\}sqrt\left\{105over 2pi\right\}cdot\left\{\left(x + iy\right)^2 z over r^\left\{3$

:$Y_\left\{3\right\}^\left\{3\right\}\left( heta,varphi\right)=\left\{-1over 8\right\}sqrt\left\{35over pi\right\}cdot e^\left\{3ivarphi\right\}cdotsin^\left\{3\right\} hetaquad=\left\{-1over 8\right\}sqrt\left\{35over pi\right\}cdot\left\{\left(x + iy\right)^3over r^\left\{3$

= Spherical harmonics with "l" = 4 =

:$Y_\left\{4\right\}^\left\{-4\right\}\left( heta,varphi\right)=\left\{3over 16\right\}sqrt\left\{35over 2pi\right\}cdot e^\left\{-4ivarphi\right\}cdotsin^\left\{4\right\} heta= frac\left\{3\right\}\left\{16\right\} sqrt\left\{frac\left\{35\right\}\left\{2 pi cdot frac\left\{\left(x - i y\right)^4\right\}\left\{r^4\right\}$:$Y_\left\{4\right\}^\left\{-3\right\}\left( heta,varphi\right)=\left\{3over 8\right\}sqrt\left\{35over pi\right\}cdot e^\left\{-3ivarphi\right\}cdotsin^\left\{3\right\} hetacdotcos heta= frac\left\{3\right\}\left\{8\right\} sqrt\left\{frac\left\{35\right\}\left\{pi cdot frac\left\{\left(x - i y\right)^3 z\right\}\left\{r^4\right\}$:$Y_\left\{4\right\}^\left\{-2\right\}\left( heta,varphi\right)=\left\{3over 8\right\}sqrt\left\{5over 2pi\right\}cdot e^\left\{-2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(7cos^\left\{2\right\} heta-1\right)= frac\left\{3\right\}\left\{8\right\} sqrt\left\{frac\left\{5\right\}\left\{2 pi cdot frac\left\{\left(x - i y\right)^2 cdot \left(7 z^2 - r^2\right)\right\}\left\{r^4\right\}$:$Y_\left\{4\right\}^\left\{-1\right\}\left( heta,varphi\right)=\left\{3over 8\right\}sqrt\left\{5over pi\right\}cdot e^\left\{-ivarphi\right\}cdotsin hetacdot\left(7cos^\left\{3\right\} heta-3cos heta\right)= frac\left\{3\right\}\left\{8\right\} sqrt\left\{frac\left\{5\right\}\left\{pi cdot frac\left\{\left(x - i y\right) cdot z cdot \left(7 z^2 - 3 r^2\right)\right\}\left\{r^4\right\}$:$Y_\left\{4\right\}^\left\{0\right\}\left( heta,varphi\right)=\left\{3over 16\right\}sqrt\left\{1over pi\right\}cdot\left(35cos^\left\{4\right\} heta-30cos^\left\{2\right\} heta+3\right)= frac\left\{3\right\}\left\{16\right\} sqrt\left\{frac\left\{1\right\}\left\{pi cdot frac\left\{\left(35 z^4 - 30 z^2 r^2 + 3 r^4\right)\right\}\left\{r^4\right\}$:$Y_\left\{4\right\}^\left\{1\right\}\left( heta,varphi\right)=\left\{-3over 8\right\}sqrt\left\{5over pi\right\}cdot e^\left\{ivarphi\right\}cdotsin hetacdot\left(7cos^\left\{3\right\} heta-3cos heta\right)= frac\left\{- 3\right\}\left\{8\right\} sqrt\left\{frac\left\{5\right\}\left\{pi cdot frac\left\{\left(x + i y\right) cdot z cdot \left(7 z^2 - 3 r^2\right)\right\}\left\{r^4\right\}$:$Y_\left\{4\right\}^\left\{2\right\}\left( heta,varphi\right)=\left\{3over 8\right\}sqrt\left\{5over 2pi\right\}cdot e^\left\{2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(7cos^\left\{2\right\} heta-1\right)= frac\left\{3\right\}\left\{8\right\} sqrt\left\{frac\left\{5\right\}\left\{2 pi cdot frac\left\{\left(x + i y\right)^2 cdot \left(7 z^2 - r^2\right)\right\}\left\{r^4\right\}$:$Y_\left\{4\right\}^\left\{3\right\}\left( heta,varphi\right)=\left\{-3over 8\right\}sqrt\left\{35over pi\right\}cdot e^\left\{3ivarphi\right\}cdotsin^\left\{3\right\} hetacdotcos heta= frac\left\{- 3\right\}\left\{8\right\} sqrt\left\{frac\left\{35\right\}\left\{pi cdot frac\left\{\left(x + i y\right)^3 z\right\}\left\{r^4\right\}$:$Y_\left\{4\right\}^\left\{4\right\}\left( heta,varphi\right)=\left\{3over 16\right\}sqrt\left\{35over 2pi\right\}cdot e^\left\{4ivarphi\right\}cdotsin^\left\{4\right\} heta= frac\left\{3\right\}\left\{16\right\} sqrt\left\{frac\left\{35\right\}\left\{2 pi cdot frac\left\{\left(x + i y\right)^4\right\}\left\{r^4\right\}$

= Spherical harmonics with "l" = 5 =

:$Y_\left\{5\right\}^\left\{-5\right\}\left( heta,varphi\right)=\left\{3over 32\right\}sqrt\left\{77over pi\right\}cdot e^\left\{-5ivarphi\right\}cdotsin^\left\{5\right\} heta$:$Y_\left\{5\right\}^\left\{-4\right\}\left( heta,varphi\right)=\left\{3over 16\right\}sqrt\left\{385over 2pi\right\}cdot e^\left\{-4ivarphi\right\}cdotsin^\left\{4\right\} hetacdotcos heta$:$Y_\left\{5\right\}^\left\{-3\right\}\left( heta,varphi\right)=\left\{1over 32\right\}sqrt\left\{385over pi\right\}cdot e^\left\{-3ivarphi\right\}cdotsin^\left\{3\right\} hetacdot\left(9cos^\left\{2\right\} heta-1\right)$:$Y_\left\{5\right\}^\left\{-2\right\}\left( heta,varphi\right)=\left\{1over 8\right\}sqrt\left\{1155over 2pi\right\}cdot e^\left\{-2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(3cos^\left\{3\right\} heta-1cos heta\right)$:$Y_\left\{5\right\}^\left\{-1\right\}\left( heta,varphi\right)=\left\{1over 16\right\}sqrt\left\{165over 2pi\right\}cdot e^\left\{-ivarphi\right\}cdotsin hetacdot\left(21cos^\left\{4\right\} heta-14cos^\left\{2\right\} heta+1\right)$:$Y_\left\{5\right\}^\left\{0\right\}\left( heta,varphi\right)=\left\{1over 16\right\}sqrt\left\{11over pi\right\}cdot\left(63cos^\left\{5\right\} heta-70cos^\left\{3\right\} heta+15cos heta\right)$:$Y_\left\{5\right\}^\left\{1\right\}\left( heta,varphi\right)=\left\{-1over 16\right\}sqrt\left\{165over 2pi\right\}cdot e^\left\{ivarphi\right\}cdotsin hetacdot\left(21cos^\left\{4\right\} heta-14cos^\left\{2\right\} heta+1\right)$:$Y_\left\{5\right\}^\left\{2\right\}\left( heta,varphi\right)=\left\{1over 8\right\}sqrt\left\{1155over 2pi\right\}cdot e^\left\{2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(3cos^\left\{3\right\} heta-1cos heta\right)$:$Y_\left\{5\right\}^\left\{3\right\}\left( heta,varphi\right)=\left\{-1over 32\right\}sqrt\left\{385over pi\right\}cdot e^\left\{3ivarphi\right\}cdotsin^\left\{3\right\} hetacdot\left(9cos^\left\{2\right\} heta-1\right)$:$Y_\left\{5\right\}^\left\{4\right\}\left( heta,varphi\right)=\left\{3over 16\right\}sqrt\left\{385over 2pi\right\}cdot e^\left\{4ivarphi\right\}cdotsin^\left\{4\right\} hetacdotcos heta$:$Y_\left\{5\right\}^\left\{5\right\}\left( heta,varphi\right)=\left\{-3over 32\right\}sqrt\left\{77over pi\right\}cdot e^\left\{5ivarphi\right\}cdotsin^\left\{5\right\} heta$

= Spherical harmonics with "l" = 6 =

:$Y_\left\{6\right\}^\left\{-6\right\}\left( heta,varphi\right)=\left\{1over 64\right\}sqrt\left\{3003over pi\right\}cdot e^\left\{-6ivarphi\right\}cdotsin^\left\{6\right\} heta$:$Y_\left\{6\right\}^\left\{-5\right\}\left( heta,varphi\right)=\left\{3over 32\right\}sqrt\left\{1001over pi\right\}cdot e^\left\{-5ivarphi\right\}cdotsin^\left\{5\right\} hetacdotcos heta$:$Y_\left\{6\right\}^\left\{-4\right\}\left( heta,varphi\right)=\left\{3over 32\right\}sqrt\left\{91over 2pi\right\}cdot e^\left\{-4ivarphi\right\}cdotsin^\left\{4\right\} hetacdot\left(11cos^\left\{2\right\} heta-1\right)$:$Y_\left\{6\right\}^\left\{-3\right\}\left( heta,varphi\right)=\left\{1over 32\right\}sqrt\left\{1365over pi\right\}cdot e^\left\{-3ivarphi\right\}cdotsin^\left\{3\right\} hetacdot\left(11cos^\left\{3\right\} heta-3cos heta\right)$:$Y_\left\{6\right\}^\left\{-2\right\}\left( heta,varphi\right)=\left\{1over 64\right\}sqrt\left\{1365over pi\right\}cdot e^\left\{-2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(33cos^\left\{4\right\} heta-18cos^\left\{2\right\} heta+1\right)$:$Y_\left\{6\right\}^\left\{-1\right\}\left( heta,varphi\right)=\left\{1over 16\right\}sqrt\left\{273over 2pi\right\}cdot e^\left\{-ivarphi\right\}cdotsin hetacdot\left(33cos^\left\{5\right\} heta-30cos^\left\{3\right\} heta+5cos heta\right)$:$Y_\left\{6\right\}^\left\{0\right\}\left( heta,varphi\right)=\left\{1over 32\right\}sqrt\left\{13over pi\right\}cdot\left(231cos^\left\{6\right\} heta-315cos^\left\{4\right\} heta+105cos^\left\{2\right\} heta-5\right)$:$Y_\left\{6\right\}^\left\{1\right\}\left( heta,varphi\right)=\left\{-1over 16\right\}sqrt\left\{273over 2pi\right\}cdot e^\left\{ivarphi\right\}cdotsin hetacdot\left(33cos^\left\{5\right\} heta-30cos^\left\{3\right\} heta+5cos heta\right)$:$Y_\left\{6\right\}^\left\{2\right\}\left( heta,varphi\right)=\left\{1over 64\right\}sqrt\left\{1365over pi\right\}cdot e^\left\{2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(33cos^\left\{4\right\} heta-18cos^\left\{2\right\} heta+1\right)$:$Y_\left\{6\right\}^\left\{3\right\}\left( heta,varphi\right)=\left\{-1over 32\right\}sqrt\left\{1365over pi\right\}cdot e^\left\{3ivarphi\right\}cdotsin^\left\{3\right\} hetacdot\left(11cos^\left\{3\right\} heta-3cos heta\right)$:$Y_\left\{6\right\}^\left\{4\right\}\left( heta,varphi\right)=\left\{3over 32\right\}sqrt\left\{91over 2pi\right\}cdot e^\left\{4ivarphi\right\}cdotsin^\left\{4\right\} hetacdot\left(11cos^\left\{2\right\} heta-1\right)$:$Y_\left\{6\right\}^\left\{5\right\}\left( heta,varphi\right)=\left\{-3over 32\right\}sqrt\left\{1001over pi\right\}cdot e^\left\{5ivarphi\right\}cdotsin^\left\{5\right\} hetacdotcos heta$:$Y_\left\{6\right\}^\left\{6\right\}\left( heta,varphi\right)=\left\{1over 64\right\}sqrt\left\{3003over pi\right\}cdot e^\left\{6ivarphi\right\}cdotsin^\left\{6\right\} heta$

= Spherical harmonics with "l" = 7 =

:$Y_\left\{7\right\}^\left\{-7\right\}\left( heta,varphi\right)=\left\{3over 64\right\}sqrt\left\{715over 2pi\right\}cdot e^\left\{-7ivarphi\right\}cdotsin^\left\{7\right\} heta$:$Y_\left\{7\right\}^\left\{-6\right\}\left( heta,varphi\right)=\left\{3over 64\right\}sqrt\left\{5005over pi\right\}cdot e^\left\{-6ivarphi\right\}cdotsin^\left\{6\right\} hetacdotcos heta$:$Y_\left\{7\right\}^\left\{-5\right\}\left( heta,varphi\right)=\left\{3over 64\right\}sqrt\left\{385over 2pi\right\}cdot e^\left\{-5ivarphi\right\}cdotsin^\left\{5\right\} hetacdot\left(13cos^\left\{2\right\} heta-1\right)$:$Y_\left\{7\right\}^\left\{-4\right\}\left( heta,varphi\right)=\left\{3over 32\right\}sqrt\left\{385over 2pi\right\}cdot e^\left\{-4ivarphi\right\}cdotsin^\left\{4\right\} hetacdot\left(13cos^\left\{3\right\} heta-3cos heta\right)$:$Y_\left\{7\right\}^\left\{-3\right\}\left( heta,varphi\right)=\left\{3over 64\right\}sqrt\left\{35over 2pi\right\}cdot e^\left\{-3ivarphi\right\}cdotsin^\left\{3\right\} hetacdot\left(143cos^\left\{4\right\} heta-66cos^\left\{2\right\} heta+3\right)$:$Y_\left\{7\right\}^\left\{-2\right\}\left( heta,varphi\right)=\left\{3over 64\right\}sqrt\left\{35over pi\right\}cdot e^\left\{-2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(143cos^\left\{5\right\} heta-110cos^\left\{3\right\} heta+15cos heta\right)$:$Y_\left\{7\right\}^\left\{-1\right\}\left( heta,varphi\right)=\left\{1over 64\right\}sqrt\left\{105over 2pi\right\}cdot e^\left\{-ivarphi\right\}cdotsin hetacdot\left(429cos^\left\{6\right\} heta-495cos^\left\{4\right\} heta+135cos^\left\{2\right\} heta-5\right)$:$Y_\left\{7\right\}^\left\{0\right\}\left( heta,varphi\right)=\left\{1over 32\right\}sqrt\left\{15over pi\right\}cdot\left(429cos^\left\{7\right\} heta-693cos^\left\{5\right\} heta+315cos^\left\{3\right\} heta-35cos heta\right)$:$Y_\left\{7\right\}^\left\{1\right\}\left( heta,varphi\right)=\left\{-1over 64\right\}sqrt\left\{105over 2pi\right\}cdot e^\left\{ivarphi\right\}cdotsin hetacdot\left(429cos^\left\{6\right\} heta-495cos^\left\{4\right\} heta+135cos^\left\{2\right\} heta-5\right)$:$Y_\left\{7\right\}^\left\{2\right\}\left( heta,varphi\right)=\left\{3over 64\right\}sqrt\left\{35over pi\right\}cdot e^\left\{2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(143cos^\left\{5\right\} heta-110cos^\left\{3\right\} heta+15cos heta\right)$:$Y_\left\{7\right\}^\left\{3\right\}\left( heta,varphi\right)=\left\{-3over 64\right\}sqrt\left\{35over 2pi\right\}cdot e^\left\{3ivarphi\right\}cdotsin^\left\{3\right\} hetacdot\left(143cos^\left\{4\right\} heta-66cos^\left\{2\right\} heta+3\right)$:$Y_\left\{7\right\}^\left\{4\right\}\left( heta,varphi\right)=\left\{3over 32\right\}sqrt\left\{385over 2pi\right\}cdot e^\left\{4ivarphi\right\}cdotsin^\left\{4\right\} hetacdot\left(13cos^\left\{3\right\} heta-3cos heta\right)$:$Y_\left\{7\right\}^\left\{5\right\}\left( heta,varphi\right)=\left\{-3over 64\right\}sqrt\left\{385over 2pi\right\}cdot e^\left\{5ivarphi\right\}cdotsin^\left\{5\right\} hetacdot\left(13cos^\left\{2\right\} heta-1\right)$:$Y_\left\{7\right\}^\left\{6\right\}\left( heta,varphi\right)=\left\{3over 64\right\}sqrt\left\{5005over pi\right\}cdot e^\left\{6ivarphi\right\}cdotsin^\left\{6\right\} hetacdotcos heta$:$Y_\left\{7\right\}^\left\{7\right\}\left( heta,varphi\right)=\left\{-3over 64\right\}sqrt\left\{715over 2pi\right\}cdot e^\left\{7ivarphi\right\}cdotsin^\left\{7\right\} heta$

= Spherical harmonics with "l" = 8 =

:$Y_\left\{8\right\}^\left\{-8\right\}\left( heta,varphi\right)=\left\{3over 256\right\}sqrt\left\{12155over 2pi\right\}cdot e^\left\{-8ivarphi\right\}cdotsin^\left\{8\right\} heta$:$Y_\left\{8\right\}^\left\{-7\right\}\left( heta,varphi\right)=\left\{3over 64\right\}sqrt\left\{12155over 2pi\right\}cdot e^\left\{-7ivarphi\right\}cdotsin^\left\{7\right\} hetacdotcos heta$:$Y_\left\{8\right\}^\left\{-6\right\}\left( heta,varphi\right)=\left\{1over 128\right\}sqrt\left\{7293over pi\right\}cdot e^\left\{-6ivarphi\right\}cdotsin^\left\{6\right\} hetacdot\left(15cos^\left\{2\right\} heta-1\right)$:$Y_\left\{8\right\}^\left\{-5\right\}\left( heta,varphi\right)=\left\{3over 64\right\}sqrt\left\{17017over 2pi\right\}cdot e^\left\{-5ivarphi\right\}cdotsin^\left\{5\right\} hetacdot\left(5cos^\left\{3\right\} heta-1cos heta\right)$:$Y_\left\{8\right\}^\left\{-4\right\}\left( heta,varphi\right)=\left\{3over 128\right\}sqrt\left\{1309over 2pi\right\}cdot e^\left\{-4ivarphi\right\}cdotsin^\left\{4\right\} hetacdot\left(65cos^\left\{4\right\} heta-26cos^\left\{2\right\} heta+1\right)$:$Y_\left\{8\right\}^\left\{-3\right\}\left( heta,varphi\right)=\left\{1over 64\right\}sqrt\left\{19635over 2pi\right\}cdot e^\left\{-3ivarphi\right\}cdotsin^\left\{3\right\} hetacdot\left(39cos^\left\{5\right\} heta-26cos^\left\{3\right\} heta+3cos heta\right)$:$Y_\left\{8\right\}^\left\{-2\right\}\left( heta,varphi\right)=\left\{3over 128\right\}sqrt\left\{595over pi\right\}cdot e^\left\{-2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(143cos^\left\{6\right\} heta-143cos^\left\{4\right\} heta+33cos^\left\{2\right\} heta-1\right)$:$Y_\left\{8\right\}^\left\{-1\right\}\left( heta,varphi\right)=\left\{3over 64\right\}sqrt\left\{17over 2pi\right\}cdot e^\left\{-ivarphi\right\}cdotsin hetacdot\left(715cos^\left\{7\right\} heta-1001cos^\left\{5\right\} heta+385cos^\left\{3\right\} heta-35cos heta\right)$:$Y_\left\{8\right\}^\left\{0\right\}\left( heta,varphi\right)=\left\{1over 256\right\}sqrt\left\{17over pi\right\}cdot\left(6435cos^\left\{8\right\} heta-12012cos^\left\{6\right\} heta+6930cos^\left\{4\right\} heta-1260cos^\left\{2\right\} heta+35\right)$:$Y_\left\{8\right\}^\left\{1\right\}\left( heta,varphi\right)=\left\{-3over 64\right\}sqrt\left\{17over 2pi\right\}cdot e^\left\{ivarphi\right\}cdotsin hetacdot\left(715cos^\left\{7\right\} heta-1001cos^\left\{5\right\} heta+385cos^\left\{3\right\} heta-35cos heta\right)$:$Y_\left\{8\right\}^\left\{2\right\}\left( heta,varphi\right)=\left\{3over 128\right\}sqrt\left\{595over pi\right\}cdot e^\left\{2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(143cos^\left\{6\right\} heta-143cos^\left\{4\right\} heta+33cos^\left\{2\right\} heta-1\right)$:$Y_\left\{8\right\}^\left\{3\right\}\left( heta,varphi\right)=\left\{-1over 64\right\}sqrt\left\{19635over 2pi\right\}cdot e^\left\{3ivarphi\right\}cdotsin^\left\{3\right\} hetacdot\left(39cos^\left\{5\right\} heta-26cos^\left\{3\right\} heta+3cos heta\right)$:$Y_\left\{8\right\}^\left\{4\right\}\left( heta,varphi\right)=\left\{3over 128\right\}sqrt\left\{1309over 2pi\right\}cdot e^\left\{4ivarphi\right\}cdotsin^\left\{4\right\} hetacdot\left(65cos^\left\{4\right\} heta-26cos^\left\{2\right\} heta+1\right)$:$Y_\left\{8\right\}^\left\{5\right\}\left( heta,varphi\right)=\left\{-3over 64\right\}sqrt\left\{17017over 2pi\right\}cdot e^\left\{5ivarphi\right\}cdotsin^\left\{5\right\} hetacdot\left(5cos^\left\{3\right\} heta-1cos heta\right)$:$Y_\left\{8\right\}^\left\{6\right\}\left( heta,varphi\right)=\left\{1over 128\right\}sqrt\left\{7293over pi\right\}cdot e^\left\{6ivarphi\right\}cdotsin^\left\{6\right\} hetacdot\left(15cos^\left\{2\right\} heta-1\right)$:$Y_\left\{8\right\}^\left\{7\right\}\left( heta,varphi\right)=\left\{-3over 64\right\}sqrt\left\{12155over 2pi\right\}cdot e^\left\{7ivarphi\right\}cdotsin^\left\{7\right\} hetacdotcos heta$:$Y_\left\{8\right\}^\left\{8\right\}\left( heta,varphi\right)=\left\{3over 256\right\}sqrt\left\{12155over 2pi\right\}cdot e^\left\{8ivarphi\right\}cdotsin^\left\{8\right\} heta$

= Spherical harmonics with "l" = 9 =

:$Y_\left\{9\right\}^\left\{-9\right\}\left( heta,varphi\right)=\left\{1over 512\right\}sqrt\left\{230945over pi\right\}cdot e^\left\{-9ivarphi\right\}cdotsin^\left\{9\right\} heta$:$Y_\left\{9\right\}^\left\{-8\right\}\left( heta,varphi\right)=\left\{3over 256\right\}sqrt\left\{230945over 2pi\right\}cdot e^\left\{-8ivarphi\right\}cdotsin^\left\{8\right\} hetacdotcos heta$:$Y_\left\{9\right\}^\left\{-7\right\}\left( heta,varphi\right)=\left\{3over 512\right\}sqrt\left\{13585over pi\right\}cdot e^\left\{-7ivarphi\right\}cdotsin^\left\{7\right\} hetacdot\left(17cos^\left\{2\right\} heta-1\right)$:$Y_\left\{9\right\}^\left\{-6\right\}\left( heta,varphi\right)=\left\{1over 128\right\}sqrt\left\{40755over pi\right\}cdot e^\left\{-6ivarphi\right\}cdotsin^\left\{6\right\} hetacdot\left(17cos^\left\{3\right\} heta-3cos heta\right)$:$Y_\left\{9\right\}^\left\{-5\right\}\left( heta,varphi\right)=\left\{3over 256\right\}sqrt\left\{2717over pi\right\}cdot e^\left\{-5ivarphi\right\}cdotsin^\left\{5\right\} hetacdot\left(85cos^\left\{4\right\} heta-30cos^\left\{2\right\} heta+1\right)$:$Y_\left\{9\right\}^\left\{-4\right\}\left( heta,varphi\right)=\left\{3over 128\right\}sqrt\left\{95095over 2pi\right\}cdot e^\left\{-4ivarphi\right\}cdotsin^\left\{4\right\} hetacdot\left(17cos^\left\{5\right\} heta-10cos^\left\{3\right\} heta+1cos heta\right)$:$Y_\left\{9\right\}^\left\{-3\right\}\left( heta,varphi\right)=\left\{1over 256\right\}sqrt\left\{21945over pi\right\}cdot e^\left\{-3ivarphi\right\}cdotsin^\left\{3\right\} hetacdot\left(221cos^\left\{6\right\} heta-195cos^\left\{4\right\} heta+39cos^\left\{2\right\} heta-1\right)$:$Y_\left\{9\right\}^\left\{-2\right\}\left( heta,varphi\right)=\left\{3over 128\right\}sqrt\left\{1045over pi\right\}cdot e^\left\{-2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(221cos^\left\{7\right\} heta-273cos^\left\{5\right\} heta+91cos^\left\{3\right\} heta-7cos heta\right)$:$Y_\left\{9\right\}^\left\{-1\right\}\left( heta,varphi\right)=\left\{3over 256\right\}sqrt\left\{95over 2pi\right\}cdot e^\left\{-ivarphi\right\}cdotsin hetacdot\left(2431cos^\left\{8\right\} heta-4004cos^\left\{6\right\} heta+2002cos^\left\{4\right\} heta-308cos^\left\{2\right\} heta+7\right)$:$Y_\left\{9\right\}^\left\{0\right\}\left( heta,varphi\right)=\left\{1over 256\right\}sqrt\left\{19over pi\right\}cdot\left(12155cos^\left\{9\right\} heta-25740cos^\left\{7\right\} heta+18018cos^\left\{5\right\} heta-4620cos^\left\{3\right\} heta+315cos heta\right)$:$Y_\left\{9\right\}^\left\{1\right\}\left( heta,varphi\right)=\left\{-3over 256\right\}sqrt\left\{95over 2pi\right\}cdot e^\left\{ivarphi\right\}cdotsin hetacdot\left(2431cos^\left\{8\right\} heta-4004cos^\left\{6\right\} heta+2002cos^\left\{4\right\} heta-308cos^\left\{2\right\} heta+7\right)$:$Y_\left\{9\right\}^\left\{2\right\}\left( heta,varphi\right)=\left\{3over 128\right\}sqrt\left\{1045over pi\right\}cdot e^\left\{2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(221cos^\left\{7\right\} heta-273cos^\left\{5\right\} heta+91cos^\left\{3\right\} heta-7cos heta\right)$:$Y_\left\{9\right\}^\left\{3\right\}\left( heta,varphi\right)=\left\{-1over 256\right\}sqrt\left\{21945over pi\right\}cdot e^\left\{3ivarphi\right\}cdotsin^\left\{3\right\} hetacdot\left(221cos^\left\{6\right\} heta-195cos^\left\{4\right\} heta+39cos^\left\{2\right\} heta-1\right)$:$Y_\left\{9\right\}^\left\{4\right\}\left( heta,varphi\right)=\left\{3over 128\right\}sqrt\left\{95095over 2pi\right\}cdot e^\left\{4ivarphi\right\}cdotsin^\left\{4\right\} hetacdot\left(17cos^\left\{5\right\} heta-10cos^\left\{3\right\} heta+1cos heta\right)$:$Y_\left\{9\right\}^\left\{5\right\}\left( heta,varphi\right)=\left\{-3over 256\right\}sqrt\left\{2717over pi\right\}cdot e^\left\{5ivarphi\right\}cdotsin^\left\{5\right\} hetacdot\left(85cos^\left\{4\right\} heta-30cos^\left\{2\right\} heta+1\right)$:$Y_\left\{9\right\}^\left\{6\right\}\left( heta,varphi\right)=\left\{1over 128\right\}sqrt\left\{40755over pi\right\}cdot e^\left\{6ivarphi\right\}cdotsin^\left\{6\right\} hetacdot\left(17cos^\left\{3\right\} heta-3cos heta\right)$:$Y_\left\{9\right\}^\left\{7\right\}\left( heta,varphi\right)=\left\{-3over 512\right\}sqrt\left\{13585over pi\right\}cdot e^\left\{7ivarphi\right\}cdotsin^\left\{7\right\} hetacdot\left(17cos^\left\{2\right\} heta-1\right)$:$Y_\left\{9\right\}^\left\{8\right\}\left( heta,varphi\right)=\left\{3over 256\right\}sqrt\left\{230945over 2pi\right\}cdot e^\left\{8ivarphi\right\}cdotsin^\left\{8\right\} hetacdotcos heta$:$Y_\left\{9\right\}^\left\{9\right\}\left( heta,varphi\right)=\left\{-1over 512\right\}sqrt\left\{230945over pi\right\}cdot e^\left\{9ivarphi\right\}cdotsin^\left\{9\right\} heta$

= Spherical harmonics with "l" = 10 =

:$Y_\left\{10\right\}^\left\{-10\right\}\left( heta,varphi\right)=\left\{1over 1024\right\}sqrt\left\{969969over pi\right\}cdot e^\left\{-10ivarphi\right\}cdotsin^\left\{10\right\} heta$:$Y_\left\{10\right\}^\left\{-9\right\}\left( heta,varphi\right)=\left\{1over 512\right\}sqrt\left\{4849845over pi\right\}cdot e^\left\{-9ivarphi\right\}cdotsin^\left\{9\right\} hetacdotcos heta$:$Y_\left\{10\right\}^\left\{-8\right\}\left( heta,varphi\right)=\left\{1over 512\right\}sqrt\left\{255255over 2pi\right\}cdot e^\left\{-8ivarphi\right\}cdotsin^\left\{8\right\} hetacdot\left(19cos^\left\{2\right\} heta-1\right)$:$Y_\left\{10\right\}^\left\{-7\right\}\left( heta,varphi\right)=\left\{3over 512\right\}sqrt\left\{85085over pi\right\}cdot e^\left\{-7ivarphi\right\}cdotsin^\left\{7\right\} hetacdot\left(19cos^\left\{3\right\} heta-3cos heta\right)$:$Y_\left\{10\right\}^\left\{-6\right\}\left( heta,varphi\right)=\left\{3over 1024\right\}sqrt\left\{5005over pi\right\}cdot e^\left\{-6ivarphi\right\}cdotsin^\left\{6\right\} hetacdot\left(323cos^\left\{4\right\} heta-102cos^\left\{2\right\} heta+3\right)$:$Y_\left\{10\right\}^\left\{-5\right\}\left( heta,varphi\right)=\left\{3over 256\right\}sqrt\left\{1001over pi\right\}cdot e^\left\{-5ivarphi\right\}cdotsin^\left\{5\right\} hetacdot\left(323cos^\left\{5\right\} heta-170cos^\left\{3\right\} heta+15cos heta\right)$:$Y_\left\{10\right\}^\left\{-4\right\}\left( heta,varphi\right)=\left\{3over 256\right\}sqrt\left\{5005over 2pi\right\}cdot e^\left\{-4ivarphi\right\}cdotsin^\left\{4\right\} hetacdot\left(323cos^\left\{6\right\} heta-255cos^\left\{4\right\} heta+45cos^\left\{2\right\} heta-1\right)$:$Y_\left\{10\right\}^\left\{-3\right\}\left( heta,varphi\right)=\left\{3over 256\right\}sqrt\left\{5005over pi\right\}cdot e^\left\{-3ivarphi\right\}cdotsin^\left\{3\right\} hetacdot\left(323cos^\left\{7\right\} heta-357cos^\left\{5\right\} heta+105cos^\left\{3\right\} heta-7cos heta\right)$:$Y_\left\{10\right\}^\left\{-2\right\}\left( heta,varphi\right)=\left\{3over 512\right\}sqrt\left\{385over 2pi\right\}cdot e^\left\{-2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(4199cos^\left\{8\right\} heta-6188cos^\left\{6\right\} heta+2730cos^\left\{4\right\} heta-364cos^\left\{2\right\} heta+7\right)$:$Y_\left\{10\right\}^\left\{-1\right\}\left( heta,varphi\right)=\left\{1over 256\right\}sqrt\left\{1155over 2pi\right\}cdot e^\left\{-ivarphi\right\}cdotsin hetacdot\left(4199cos^\left\{9\right\} heta-7956cos^\left\{7\right\} heta+4914cos^\left\{5\right\} heta-1092cos^\left\{3\right\} heta+63cos heta\right)$:$Y_\left\{10\right\}^\left\{0\right\}\left( heta,varphi\right)=\left\{1over 512\right\}sqrt\left\{21over pi\right\}cdot\left(46189cos^\left\{10\right\} heta-109395cos^\left\{8\right\} heta+90090cos^\left\{6\right\} heta-30030cos^\left\{4\right\} heta+3465cos^\left\{2\right\} heta-63\right)$:$Y_\left\{10\right\}^\left\{1\right\}\left( heta,varphi\right)=\left\{-1over 256\right\}sqrt\left\{1155over 2pi\right\}cdot e^\left\{ivarphi\right\}cdotsin hetacdot\left(4199cos^\left\{9\right\} heta-7956cos^\left\{7\right\} heta+4914cos^\left\{5\right\} heta-1092cos^\left\{3\right\} heta+63cos heta\right)$:$Y_\left\{10\right\}^\left\{2\right\}\left( heta,varphi\right)=\left\{3over 512\right\}sqrt\left\{385over 2pi\right\}cdot e^\left\{2ivarphi\right\}cdotsin^\left\{2\right\} hetacdot\left(4199cos^\left\{8\right\} heta-6188cos^\left\{6\right\} heta+2730cos^\left\{4\right\} heta-364cos^\left\{2\right\} heta+7\right)$:$Y_\left\{10\right\}^\left\{3\right\}\left( heta,varphi\right)=\left\{-3over 256\right\}sqrt\left\{5005over pi\right\}cdot e^\left\{3ivarphi\right\}cdotsin^\left\{3\right\} hetacdot\left(323cos^\left\{7\right\} heta-357cos^\left\{5\right\} heta+105cos^\left\{3\right\} heta-7cos heta\right)$:$Y_\left\{10\right\}^\left\{4\right\}\left( heta,varphi\right)=\left\{3over 256\right\}sqrt\left\{5005over 2pi\right\}cdot e^\left\{4ivarphi\right\}cdotsin^\left\{4\right\} hetacdot\left(323cos^\left\{6\right\} heta-255cos^\left\{4\right\} heta+45cos^\left\{2\right\} heta-1\right)$:$Y_\left\{10\right\}^\left\{5\right\}\left( heta,varphi\right)=\left\{-3over 256\right\}sqrt\left\{1001over pi\right\}cdot e^\left\{5ivarphi\right\}cdotsin^\left\{5\right\} hetacdot\left(323cos^\left\{5\right\} heta-170cos^\left\{3\right\} heta+15cos heta\right)$:$Y_\left\{10\right\}^\left\{6\right\}\left( heta,varphi\right)=\left\{3over 1024\right\}sqrt\left\{5005over pi\right\}cdot e^\left\{6ivarphi\right\}cdotsin^\left\{6\right\} hetacdot\left(323cos^\left\{4\right\} heta-102cos^\left\{2\right\} heta+3\right)$:$Y_\left\{10\right\}^\left\{7\right\}\left( heta,varphi\right)=\left\{-3over 512\right\}sqrt\left\{85085over pi\right\}cdot e^\left\{7ivarphi\right\}cdotsin^\left\{7\right\} hetacdot\left(19cos^\left\{3\right\} heta-3cos heta\right)$:$Y_\left\{10\right\}^\left\{8\right\}\left( heta,varphi\right)=\left\{1over 512\right\}sqrt\left\{255255over 2pi\right\}cdot e^\left\{8ivarphi\right\}cdotsin^\left\{8\right\} hetacdot\left(19cos^\left\{2\right\} heta-1\right)$:$Y_\left\{10\right\}^\left\{9\right\}\left( heta,varphi\right)=\left\{-1over 512\right\}sqrt\left\{4849845over pi\right\}cdot e^\left\{9ivarphi\right\}cdotsin^\left\{9\right\} hetacdotcos heta$:$Y_\left\{10\right\}^\left\{10\right\}\left( heta,varphi\right)=\left\{1over 1024\right\}sqrt\left\{969969over pi\right\}cdot e^\left\{10ivarphi\right\}cdotsin^\left\{10\right\} heta$

ee also

*Spherical harmonics

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