Table of spherical harmonics

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


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Spherical harmonics — In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplace s equation represented in a system of spherical coordinates. Spherical harmonics are important in many theoretical and practical… …   Wikipedia

  • Cubic harmonic — Cubic harmonics In fields like computational chemistry and solid state and condensed matter physics the so called atomic orbitals, or spin orbitals, as they appear in textbooks[1] …   Wikipedia

  • List of mathematics articles (T) — NOTOC T T duality T group T group (mathematics) T integration T norm T norm fuzzy logics T schema T square (fractal) T symmetry T table T theory T.C. Mits T1 space Table of bases Table of Clebsch Gordan coefficients Table of divisors Table of Lie …   Wikipedia

  • Atomic orbital — The shapes of the first five atomic orbitals: 1s, 2s, 2px, 2py, and 2pz. The colors show the wave function phase. These are graphs of ψ(x,y,z) functions which depend on the coordinates of one electron. To see the elongated shape of ψ(x,y,z)2… …   Wikipedia

  • Legendre wavelet — Legendre wavelets: spherical harmonic wavelets = Compactly supported wavelets derived from Legendre polynomials are termed spherical harmonic or Legendre wavelets [1] . Legendre functions have widespread applications in which spherical coordinate …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • geoid — geoidal, adj. /jee oyd/, n. 1. an imaginary surface that coincides with mean sea level in the ocean and its extension through the continents. 2. the geometric figure formed by this surface, an ellipsoid flattened at the poles. [1880 85; < Gk… …   Universalium

  • Fourier transform — Fourier transforms Continuous Fourier transform Fourier series Discrete Fourier transform Discrete time Fourier transform Related transforms The Fourier transform is a mathematical operation that decomposes a function into its constituent… …   Wikipedia

  • gravitation — gravitational, adj. gravitationally, adv. /grav i tay sheuhn/, n. 1. Physics. a. the force of attraction between any two masses. Cf. law of gravitation. b. an act or process caused by this force. 2. a sinking or falling …   Universalium

  • Clebsch–Gordan coefficients — In physics, the Clebsch–Gordan coefficients are sets of numbers that arise in angular momentum coupling under the laws of quantum mechanics. In more mathematical terms, the CG coefficients are used in representation theory, particularly of… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”