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_{0}^{0}( heta,varphi)={1over 2}sqrt{1over pi}


= Spherical harmonics with "l" = 1 =

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


= Spherical harmonics with "l" = 2 =

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

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

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

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

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


= Spherical harmonics with "l" = 3 =

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

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

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

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

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

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

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


= Spherical harmonics with "l" = 4 =

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


= Spherical harmonics with "l" = 5 =

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


= Spherical harmonics with "l" = 6 =

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


= Spherical harmonics with "l" = 7 =

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


= Spherical harmonics with "l" = 8 =

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


= Spherical harmonics with "l" = 9 =

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


= Spherical harmonics with "l" = 10 =

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

ee also

*Spherical harmonics


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