Masă armonică sferică

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Element principal: armonici sferice .

Primele armonici sferice pentru l = 0, 1, ..., 10 și m = - l , ..., -1, 0, 1, ..., l sunt: ^[1]

Armonice sferice cu l = 0

Y_{0}^{0}(x)={\frac {1}{2}}{\sqrt {1 \over \pi }}

{\ displaystyle Y_ {0} ^ {0} (x) = {\ frac {1} {2}} {\ sqrt {1 \ over \ pi}}}

{\ displaystyle Y_ {0} ^ {0} (x) = {\ frac {1} {2}} {\ sqrt {1 \ over \ pi}}}

Armonice sferice cu l = 1

{\begin{aligned}Y_{1}^{-1}(\theta ,\varphi )&=&&{\frac {1}{2}}{\sqrt {3 \over 2\pi }}\,e^{-i\varphi }\,\sin \theta &&=&&{\frac {1}{2}}{\sqrt {3 \over 2\pi }}\,{(x-iy) \over r}\\Y_{1}^{0}(\theta ,\varphi )&=&&{\frac {1}{2}}{\sqrt {3 \over \pi }}\,\cos \theta &&=&&{\frac {1}{2}}{\sqrt {3 \over \pi }}\,{z \over r}\\Y_{1}^{1}(\theta ,\varphi )&=&-&{\frac {1}{2}}{\sqrt {3 \over 2\pi }}\,e^{i\varphi }\,\sin \theta &&=&-&{\frac {1}{2}}{\sqrt {3 \over 2\pi }}\,{(x+iy) \over r}\end{aligned}}

{\ displaystyle {\ begin {align} Y_ {1} ^ {- 1} (\ theta, \ varphi) & = && {\ frac {1} {2}} {\ sqrt {3 \ over 2 \ pi}} \, e ^ {- i \ varphi} \, \ sin \ theta && = && {\ frac {1} {2}} {\ sqrt {3 \ over 2 \ pi}} \, {(x-iy) \ peste r} \\ Y_ {1} ^ {0} (\ theta, \ varphi) & = && {\ frac {1} {2}} {\ sqrt {3 \ over \ pi}} \, \ cos \ theta && = && {\ frac {1} {2}} {\ sqrt {3 \ over \ pi}} \, {z \ over r} \\ Y_ {1} ^ {1} (\ theta, \ varphi) & = & - & {\ frac {1} {2}} {\ sqrt {3 \ over 2 \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta && = & - & {\ frac { 1} {2}} {\ sqrt {3 \ over 2 \ pi}} \, {(x + iy) \ over r} \ end {align}}}

{\ displaystyle {\ begin {align} Y_ {1} ^ {- 1} (\ theta, \ varphi) & = && {\ frac {1} {2}} {\ sqrt {3 \ over 2 \ pi}} \, e ^ {- i \ varphi} \, \ sin \ theta && = && {\ frac {1} {2}} {\ sqrt {3 \ over 2 \ pi}} \, {(x-iy) \ peste r} \\ Y_ {1} ^ {0} (\ theta, \ varphi) & = && {\ frac {1} {2}} {\ sqrt {3 \ over \ pi}} \, \ cos \ theta && = && {\ frac {1} {2}} {\ sqrt {3 \ over \ pi}} \, {z \ over r} \\ Y_ {1} ^ {1} (\ theta, \ varphi) & = & - & {\ frac {1} {2}} {\ sqrt {3 \ over 2 \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta && = & - & {\ frac { 1} {2}} {\ sqrt {3 \ over 2 \ pi}} \, {(x + iy) \ over r} \ end {align}}}

Armonice sferice cu l = 2

{\begin{aligned}Y_{2}^{-2}(\theta ,\varphi )&=&&{\frac {1}{4}}{\sqrt {15 \over 2\pi }}\,e^{-2i\varphi }\,\sin ^{2}\theta \quad &&=&&{\frac {1}{4}}{\sqrt {15 \over 2\pi }}\,{(x-iy)^{2} \over r^{2}}&\\Y_{2}^{-1}(\theta ,\varphi )&=&&{\frac {1}{2}}{\sqrt {15 \over 2\pi }}\,e^{-i\varphi }\,\sin \theta \,\cos \theta \quad &&=&&{\frac {1}{2}}{\sqrt {15 \over 2\pi }}\,{(x-iy)z \over r^{2}}&\\Y_{2}^{0}(\theta ,\varphi )&=&&{\frac {1}{4}}{\sqrt {5 \over \pi }}\,(3\cos ^{2}\theta -1)\quad &&=&&{\frac {1}{4}}{\sqrt {5 \over \pi }}\,{(2z^{2}-x^{2}-y^{2}) \over r^{2}}&\\Y_{2}^{1}(\theta ,\varphi )&=&-&{\frac {1}{2}}{\sqrt {15 \over 2\pi }}\,e^{i\varphi }\,\sin \theta \,\cos \theta \quad &&=&-&{\frac {1}{2}}{\sqrt {15 \over 2\pi }}\,{(x+iy)z \over r^{2}}&\\Y_{2}^{2}(\theta ,\varphi )&=&&{\frac {1}{4}}{\sqrt {15 \over 2\pi }}\,e^{2i\varphi }\,\sin ^{2}\theta \quad &&=&&{\frac {1}{4}}{\sqrt {15 \over 2\pi }}\,{(x+iy)^{2} \over r^{2}}&\end{aligned}}

{\ displaystyle {\ begin {align} Y_ {2} ^ {- 2} (\ theta, \ varphi) & = && {\ frac {1} {4}} {\ sqrt {15 \ over 2 \ pi}} \, e ^ {- 2i \ varphi} \, \ sin ^ {2} \ theta \ quad && = && {\ frac {1} {4}} {\ sqrt {15 \ over 2 \ pi}} \, { (x-iy) ^ {2} \ over r ^ {2}} & \\ Y_ {2} ^ {- 1} (\ theta, \ varphi) & = && {\ frac {1} {2}} { \ sqrt {15 \ over 2 \ pi}} \, e ^ {- i \ varphi} \, \ sin \ theta \, \ cos \ theta \ quad && = && {\ frac {1} {2}} {\ sqrt {15 \ over 2 \ pi}} \, {(x-iy) z \ over r ^ {2}} & \\ Y_ {2} ^ {0} (\ theta, \ varphi) & = && {\ frac {1} {4}} {\ sqrt {5 \ over \ pi}} \, (3 \ cos ^ {2} \ theta -1) \ quad && = && {\ frac {1} {4}} { \ sqrt {5 \ over \ pi}} \, {(2z ^ {2} -x ^ {2} -y ^ {2}) \ over r ^ {2}} & \\ Y_ {2} ^ {1 } (\ theta, \ varphi) & = & - & {\ frac {1} {2}} {\ sqrt {15 \ over 2 \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta \, \ cos \ theta \ quad && = & - & {\ frac {1} {2}} {\ sqrt {15 \ over 2 \ pi}} \, {(x + iy) z \ over r ^ {2 }} & \\ Y_ {2} ^ {2} (\ theta, \ varphi) & = && {\ frac {1} {4}} {\ sqrt {15 \ over 2 \ pi}} \ și ^ { 2i \ varphi} \, \ sin ^ {2} \ theta \ quad && = && {\ frac {1} {4}} {\ sqrt {15 \ over 2 \ pi}} \, {(x + iy) ^ {2} \ over r ^ {2}} și \ end {align}}}

{\ displaystyle {\ begin {align} Y_ {2} ^ {- 2} (\ theta, \ varphi) & = && {\ frac {1} {4}} {\ sqrt {15 \ over 2 \ pi}} \, e ^ {- 2i \ varphi} \, \ sin ^ {2} \ theta \ quad && = && {\ frac {1} {4}} {\ sqrt {15 \ over 2 \ pi}} \, { (x-iy) ^ {2} \ over r ^ {2}} & \\ Y_ {2} ^ {- 1} (\ theta, \ varphi) & = && {\ frac {1} {2}} { \ sqrt {15 \ over 2 \ pi}} \, e ^ {- i \ varphi} \, \ sin \ theta \, \ cos \ theta \ quad && = && {\ frac {1} {2}} {\ sqrt {15 \ over 2 \ pi}} \, {(x-iy) z \ over r ^ {2}} & \\ Y_ {2} ^ {0} (\ theta, \ varphi) & = && {\ frac {1} {4}} {\ sqrt {5 \ over \ pi}} \, (3 \ cos ^ {2} \ theta -1) \ quad && = && {\ frac {1} {4}} { \ sqrt {5 \ over \ pi}} \, {(2z ^ {2} -x ^ {2} -y ^ {2}) \ over r ^ {2}} & \\ Y_ {2} ^ {1 } (\ theta, \ varphi) & = & - & {\ frac {1} {2}} {\ sqrt {15 \ over 2 \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta \, \ cos \ theta \ quad && = & - & {\ frac {1} {2}} {\ sqrt {15 \ over 2 \ pi}} \, {(x + iy) z \ over r ^ {2 }} & \\ Y_ {2} ^ {2} (\ theta, \ varphi) & = && {\ frac {1} {4}} {\ sqrt {15 \ over 2 \ pi}} \ și ^ { 2i \ varphi} \, \ sin ^ {2} \ theta \ quad && = && {\ frac {1} {4}} {\ sqrt {15 \ over 2 \ pi}} \, {(x + iy) ^ {2} \ over r ^ {2}} și \ end {align}}}

Armonice sferice cu l = 3

{\begin{aligned}Y_{3}^{-3}(\theta ,\varphi )&=&&{1 \over 8}{\sqrt {35 \over \pi }}\,e^{-3i\varphi }\,\sin ^{3}\theta \quad &&=&&{1 \over 8}{\sqrt {35 \over \pi }}\,{(x-iy)^{3} \over r^{3}}&\\Y_{3}^{-2}(\theta ,\varphi )&=&&{\frac {1}{4}}{\sqrt {105 \over 2\pi }}\,e^{-2i\varphi }\,\sin ^{2}\theta \,\cos \theta \quad &&=&&{\frac {1}{4}}{\sqrt {105 \over 2\pi }}\,{(x-iy)^{2}z \over r^{3}}&\\Y_{3}^{-1}(\theta ,\varphi )&=&&{1 \over 8}{\sqrt {21 \over \pi }}\,e^{-i\varphi }\,\sin \theta \,(5\cos ^{2}\theta -1)\quad &&=&&{1 \over 8}{\sqrt {21 \over \pi }}\,{(x-iy)(5z^{2}-r^{2}) \over r^{3}}&\\Y_{3}^{0}(\theta ,\varphi )&=&&{\frac {1}{4}}{\sqrt {7 \over \pi }}\,(5\cos ^{3}\theta -3\cos \theta )\quad &&=&&{\frac {1}{4}}{\sqrt {7 \over \pi }}\,{z(5z^{2}-3r^{2}) \over r^{3}}&\\Y_{3}^{1}(\theta ,\varphi )&=&-&{1 \over 8}{\sqrt {21 \over \pi }}\,e^{i\varphi }\,\sin \theta \,(5\cos ^{2}\theta -1)\quad &&=&&{-1 \over 8}{\sqrt {21 \over \pi }}\,{(x+iy)(5z^{2}-r^{2}) \over r^{3}}&\\Y_{3}^{2}(\theta ,\varphi )&=&&{\frac {1}{4}}{\sqrt {105 \over 2\pi }}\,e^{2i\varphi }\,\sin ^{2}\theta \,\cos \theta \quad &&=&&{\frac {1}{4}}{\sqrt {105 \over 2\pi }}\,{(x+iy)^{2}z \over r^{3}}&\\Y_{3}^{3}(\theta ,\varphi )&=&-&{1 \over 8}{\sqrt {35 \over \pi }}\,e^{3i\varphi }\,\sin ^{3}\theta \quad &&=&&{-1 \over 8}{\sqrt {35 \over \pi }}\,{(x+iy)^{3} \over r^{3}}&\end{aligned}}

{\ displaystyle {\ begin {align} Y_ {3} ^ {- 3} (\ theta, \ varphi) & = && {1 \ over 8} {\ sqrt {35 \ over \ pi}} \, e ^ { -3i \ varphi} \, \ sin ^ {3} \ theta \ quad && = && {1 \ over 8} {\ sqrt {35 \ over \ pi}} \, {(x-iy) ^ {3} \ peste r ^ {3}} & \\ Y_ {3} ^ {- 2} (\ theta, \ varphi) & = && {\ frac {1} {4}} {\ sqrt {105 \ over 2 \ pi} } \, e ^ {- 2i \ varphi} \, \ sin ^ {2} \ theta \, \ cos \ theta \ quad && = && {\ frac {1} {4}} {\ sqrt {105 \ over 2 \ pi}} \, {(x-iy) ^ {2} z \ over r ^ {3}} & \\ Y_ {3} ^ {- 1} (\ theta, \ varphi) & = && {1 \ peste 8} {\ sqrt {21 \ over \ pi}} \, e ^ {- i \ varphi} \, \ sin \ theta \, (5 \ cos ^ {2} \ theta -1) \ quad && = && {1 \ over 8} {\ sqrt {21 \ over \ pi}} \, {(x-iy) (5z ^ {2} -r ^ {2}) \ over r ^ {3}} & \\ Y_ {3} ^ {0} (\ theta, \ varphi) & = && {\ frac {1} {4}} {\ sqrt {7 \ over \ pi}} \, (5 \ cos ^ {3} \ theta -3 \ cos \ theta) \ quad && = && {\ frac {1} {4}} {\ sqrt {7 \ over \ pi}} \, {z (5z ^ {2} -3r ^ {2}) \ over r ^ {3}} & \\ Y_ {3} ^ {1} (\ theta, \ varphi) & = & - & {1 \ over 8} {\ sqrt {21 \ over \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta \, (5 \ cos ^ {2} \ theta -1) \ quad && = && {- 1 \ over 8} {\ sqrt {21 \ over \ pi} } \, {(x + iy) (5z ^ {2} -r ^ {2}) \ over r ^ {3}} & \\ Y_ {3 } ^ {2} (\ theta, \ varphi) & = && {\ frac {1} {4}} {\ sqrt {105 \ over 2 \ pi}} \ și ^ {2i \ varphi} \, \ sin ^ {2} \ theta \, \ cos \ theta \ quad && = && {\ frac {1} {4}} {\ sqrt {105 \ over 2 \ pi}} \, {(x + iy) ^ {2 } z \ over r ^ {3}} & \\ Y_ {3} ^ {3} (\ theta, \ varphi) & = & - & {1 \ over 8} {\ sqrt {35 \ over \ pi}} \, e ^ {3i \ varphi} \, \ sin ^ {3} \ theta \ quad && = && {- 1 \ over 8} {\ sqrt {35 \ over \ pi}} \, {(x + iy) ^ {3} \ over r ^ {3}} & \ end {align}}}

{\ displaystyle {\ begin {align} Y_ {3} ^ {- 3} (\ theta, \ varphi) & = && {1 \ over 8} {\ sqrt {35 \ over \ pi}} \, e ^ { -3i \ varphi} \, \ sin ^ {3} \ theta \ quad && = && {1 \ over 8} {\ sqrt {35 \ over \ pi}} \, {(x-iy) ^ {3} \ peste r ^ {3}} & \\ Y_ {3} ^ {- 2} (\ theta, \ varphi) & = && {\ frac {1} {4}} {\ sqrt {105 \ over 2 \ pi} } \, e ^ {- 2i \ varphi} \, \ sin ^ {2} \ theta \, \ cos \ theta \ quad && = && {\ frac {1} {4}} {\ sqrt {105 \ over 2 \ pi}} \, {(x-iy) ^ {2} z \ over r ^ {3}} & \\ Y_ {3} ^ {- 1} (\ theta, \ varphi) & = && {1 \ peste 8} {\ sqrt {21 \ over \ pi}} \, e ^ {- i \ varphi} \, \ sin \ theta \, (5 \ cos ^ {2} \ theta -1) \ quad && = && {1 \ over 8} {\ sqrt {21 \ over \ pi}} \, {(x-iy) (5z ^ {2} -r ^ {2}) \ over r ^ {3}} & \\ Y_ {3} ^ {0} (\ theta, \ varphi) & = && {\ frac {1} {4}} {\ sqrt {7 \ over \ pi}} \, (5 \ cos ^ {3} \ theta -3 \ cos \ theta) \ quad && = && {\ frac {1} {4}} {\ sqrt {7 \ over \ pi}} \, {z (5z ^ {2} -3r ^ {2}) \ over r ^ {3}} & \\ Y_ {3} ^ {1} (\ theta, \ varphi) & = & - & {1 \ over 8} {\ sqrt {21 \ over \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta \, (5 \ cos ^ {2} \ theta -1) \ quad && = && {- 1 \ over 8} {\ sqrt {21 \ over \ pi} } \, {(x + iy) (5z ^ {2} -r ^ {2}) \ over r ^ {3}} & \\ Y_ {3 } ^ {2} (\ theta, \ varphi) & = && {\ frac {1} {4}} {\ sqrt {105 \ over 2 \ pi}} \ și ^ {2i \ varphi} \, \ sin ^ {2} \ theta \, \ cos \ theta \ quad && = && {\ frac {1} {4}} {\ sqrt {105 \ over 2 \ pi}} \, {(x + iy) ^ {2 } z \ over r ^ {3}} & \\ Y_ {3} ^ {3} (\ theta, \ varphi) & = & - & {1 \ over 8} {\ sqrt {35 \ over \ pi}} \, e ^ {3i \ varphi} \, \ sin ^ {3} \ theta \ quad && = && {- 1 \ over 8} {\ sqrt {35 \ over \ pi}} \, {(x + iy) ^ {3} \ over r ^ {3}} & \ end {align}}}

Armonice sferice cu l = 4

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

{\ displaystyle {\ begin {align} Y_ {4} ^ {- 4} (\ theta, \ varphi) & = && {\ frac {3} {16}} {\ sqrt {35 \ over 2 \ pi}} \, e ^ {- 4i \ varphi} \, \ sin ^ {4} \ theta && = && {\ frac {3} {16}} {\ sqrt {\ frac {35} {2 \ pi}}} \ , {\ frac {(x-iy) ^ {4}} {r ^ {4}}} & \\ Y_ {4} ^ {- 3} (\ theta, \ varphi) & = && {\ frac {3 } {8}} {\ sqrt {35 \ over \ pi}} \, e ^ {- 3i \ varphi} \, \ sin ^ {3} \ theta \, \ cos \ theta && = && {\ frac {3 } {8}} {\ sqrt {\ frac {35} {\ pi}}} \, {\ frac {(x-iy) ^ {3} z} {r ^ {4}}} & \\ Y_ { 4} ^ {- 2} (\ theta, \ varphi) & = && {\ frac {3} {8}} {\ sqrt {5 \ over 2 \ pi}} \ și ^ {- 2i \ varphi} \ , \ sin ^ {2} \ theta \, (7 \ cos ^ {2} \ theta -1) && = && {\ frac {3} {8}} {\ sqrt {\ frac {5} {2 \ pi }}} \, {\ frac {(x-iy) ^ {2} \, (7z ^ {2} -r ^ {2})} {r ^ {4}}} & \\ Y_ {4} ^ {-1} (\ theta, \ varphi) & = && {\ frac {3} {8}} {\ sqrt {5 \ over \ pi}} \, e ^ {- i \ varphi} \, \ sin \ theta \, (7 \ cos ^ {3} \ theta -3 \ cos \ theta) && = && {\ frac {3} {8}} {\ sqrt {\ frac {5} {\ pi}}} \, {\ frac {(x-iy) \, z \, (7z ^ {2} -3r ^ {2})} {r ^ {4}}} & \\ Y_ {4} ^ {0} (\ theta , \ varphi) & = && {\ frac {3} {16}} {\ sqrt {1 \ over \ pi}} \, (35 \ cos ^ {4} \ theta -30 \ cos ^ {2} \ theta +3) && = && {\ frac {3} {16}} {\ sqrt {\ fra c {1} {\ pi}}} \, {\ frac {(35z ^ {4} -30z ^ {2} r ^ {2} + 3r ^ {4})} {r ^ {4}}} & \\ Y_ {4} ^ {1} (\ theta, \ varphi) & = && {\ frac {-3} {8}} {\ sqrt {5 \ over \ pi}} \, e ^ {i \ varphi } \, \ sin \ theta \, (7 \ cos ^ {3} \ theta -3 \ cos \ theta) && = && {\ frac {-3} {8}} {\ sqrt {\ frac {5} { \ pi}}} \, {\ frac {(x + iy) \, z \, (7z ^ {2} -3r ^ {2})} {r ^ {4}}} & \\ Y_ {4} ^ {2} (\ theta, \ varphi) & = && {\ frac {3} {8}} {\ sqrt {5 \ over 2 \ pi}} \, e ^ {2i \ varphi} \, \ sin ^ {2} \ theta \, (7 \ cos ^ {2} \ theta -1) && = && {\ frac {3} {8}} {\ sqrt {\ frac {5} {2 \ pi}}} \ , {\ frac {(x + iy) ^ {2} \, (7z ^ {2} -r ^ {2})} {r ^ {4}}} & \\ Y_ {4} ^ {3} ( \ theta, \ varphi) & = && {\ frac {-3} {8}} {\ sqrt {35 \ over \ pi}} \, e ^ {3i \ varphi} \, \ sin ^ {3} \ theta \, \ cos \ theta && = && {\ frac {-3} {8}} {\ sqrt {\ frac {35} {\ pi}}} \, {\ frac {(x + iy) ^ {3} z} {r ^ {4}}} & \\ Y_ {4} ^ {4} (\ theta, \ varphi) & = && {\ frac {3} {16}} {\ sqrt {35 \ over 2 \ pi}} \, e ^ {4i \ varphi} \, \ sin ^ {4} \ theta && = && {\ frac {3} {16}} {\ sqrt {\ frac {35} {2 \ pi}} } \, {\ frac {(x + iy) ^ {4}} {r ^ {4}}} & \\\ end {align}}}

{\ displaystyle {\ begin {align} Y_ {4} ^ {- 4} (\ theta, \ varphi) & = && {\ frac {3} {16}} {\ sqrt {35 \ over 2 \ pi}} \, e ^ {- 4i \ varphi} \, \ sin ^ {4} \ theta && = && {\ frac {3} {16}} {\ sqrt {\ frac {35} {2 \ pi}}} \ , {\ frac {(x-iy) ^ {4}} {r ^ {4}}} & \\ Y_ {4} ^ {- 3} (\ theta, \ varphi) & = && {\ frac {3 } {8}} {\ sqrt {35 \ over \ pi}} \, e ^ {- 3i \ varphi} \, \ sin ^ {3} \ theta \, \ cos \ theta && = && {\ frac {3 } {8}} {\ sqrt {\ frac {35} {\ pi}}} \, {\ frac {(x-iy) ^ {3} z} {r ^ {4}}} & \\ Y_ { 4} ^ {- 2} (\ theta, \ varphi) & = && {\ frac {3} {8}} {\ sqrt {5 \ over 2 \ pi}} \ și ^ {- 2i \ varphi} \ , \ sin ^ {2} \ theta \, (7 \ cos ^ {2} \ theta -1) && = && {\ frac {3} {8}} {\ sqrt {\ frac {5} {2 \ pi }}} \, {\ frac {(x-iy) ^ {2} \, (7z ^ {2} -r ^ {2})} {r ^ {4}}} & \\ Y_ {4} ^ {-1} (\ theta, \ varphi) & = && {\ frac {3} {8}} {\ sqrt {5 \ over \ pi}} \, e ^ {- i \ varphi} \, \ sin \ theta \, (7 \ cos ^ {3} \ theta -3 \ cos \ theta) && = && {\ frac {3} {8}} {\ sqrt {\ frac {5} {\ pi}}} \, {\ frac {(x-iy) \, z \, (7z ^ {2} -3r ^ {2})} {r ^ {4}}} & \\ Y_ {4} ^ {0} (\ theta , \ varphi) & = && {\ frac {3} {16}} {\ sqrt {1 \ over \ pi}} \, (35 \ cos ^ {4} \ theta -30 \ cos ^ {2} \ theta +3) && = && {\ frac {3} {16}} {\ sqrt {\ fra c {1} {\ pi}}} \, {\ frac {(35z ^ {4} -30z ^ {2} r ^ {2} + 3r ^ {4})} {r ^ {4}}} & \\ Y_ {4} ^ {1} (\ theta, \ varphi) & = && {\ frac {-3} {8}} {\ sqrt {5 \ over \ pi}} \, e ^ {i \ varphi } \, \ sin \ theta \, (7 \ cos ^ {3} \ theta -3 \ cos \ theta) && = && {\ frac {-3} {8}} {\ sqrt {\ frac {5} { \ pi}}} \, {\ frac {(x + iy) \, z \, (7z ^ {2} -3r ^ {2})} {r ^ {4}}} & \\ Y_ {4} ^ {2} (\ theta, \ varphi) & = && {\ frac {3} {8}} {\ sqrt {5 \ over 2 \ pi}} \, e ^ {2i \ varphi} \, \ sin ^ {2} \ theta \, (7 \ cos ^ {2} \ theta -1) && = && {\ frac {3} {8}} {\ sqrt {\ frac {5} {2 \ pi}}} \ , {\ frac {(x + iy) ^ {2} \, (7z ^ {2} -r ^ {2})} {r ^ {4}}} & \\ Y_ {4} ^ {3} ( \ theta, \ varphi) & = && {\ frac {-3} {8}} {\ sqrt {35 \ over \ pi}} \, e ^ {3i \ varphi} \, \ sin ^ {3} \ theta \, \ cos \ theta && = && {\ frac {-3} {8}} {\ sqrt {\ frac {35} {\ pi}}} \, {\ frac {(x + iy) ^ {3} z} {r ^ {4}}} & \\ Y_ {4} ^ {4} (\ theta, \ varphi) & = && {\ frac {3} {16}} {\ sqrt {35 \ over 2 \ pi}} \, e ^ {4i \ varphi} \, \ sin ^ {4} \ theta && = && {\ frac {3} {16}} {\ sqrt {\ frac {35} {2 \ pi}} } \, {\ frac {(x + iy) ^ {4}} {r ^ {4}}} & \\\ end {align}}}

Armonice sferice cu l = 5

{\begin{aligned}Y_{5}^{-5}(\theta ,\varphi )&={3 \over 32}{\sqrt {77 \over \pi }}\,e^{-5i\varphi }\,\sin ^{5}\theta \\Y_{5}^{-4}(\theta ,\varphi )&={3 \over 16}{\sqrt {385 \over 2\pi }}\,e^{-4i\varphi }\,\sin ^{4}\theta \,\cos \theta \\Y_{5}^{-3}(\theta ,\varphi )&={1 \over 32}{\sqrt {385 \over \pi }}\,e^{-3i\varphi }\,\sin ^{3}\theta \,(9\cos ^{2}\theta -1)\\Y_{5}^{-2}(\theta ,\varphi )&={1 \over 8}{\sqrt {1155 \over 2\pi }}\,e^{-2i\varphi }\,\sin ^{2}\theta \,(3\cos ^{3}\theta -\cos \theta )\\Y_{5}^{-1}(\theta ,\varphi )&={1 \over 16}{\sqrt {165 \over 2\pi }}\,e^{-i\varphi }\,\sin \theta \,(21\cos ^{4}\theta -14\cos ^{2}\theta +1)\\Y_{5}^{0}(\theta ,\varphi )&={1 \over 16}{\sqrt {11 \over \pi }}\,(63\cos ^{5}\theta -70\cos ^{3}\theta +15\cos \theta )\\Y_{5}^{1}(\theta ,\varphi )&={-1 \over 16}{\sqrt {165 \over 2\pi }}\,e^{i\varphi }\,\sin \theta \,(21\cos ^{4}\theta -14\cos ^{2}\theta +1)\\Y_{5}^{2}(\theta ,\varphi )&={1 \over 8}{\sqrt {1155 \over 2\pi }}\,e^{2i\varphi }\,\sin ^{2}\theta \,(3\cos ^{3}\theta -\cos \theta )\\Y_{5}^{3}(\theta ,\varphi )&={-1 \over 32}{\sqrt {385 \over \pi }}\,e^{3i\varphi }\,\sin ^{3}\theta \,(9\cos ^{2}\theta -1)\\Y_{5}^{4}(\theta ,\varphi )&={3 \over 16}{\sqrt {385 \over 2\pi }}\,e^{4i\varphi }\,\sin ^{4}\theta \,\cos \theta \\Y_{5}^{5}(\theta ,\varphi )&={-3 \over 32}{\sqrt {77 \over \pi }}\,e^{5i\varphi }\,\sin ^{5}\theta \end{aligned}}

{\ displaystyle {\ begin {align} Y_ {5} ^ {- 5} (\ theta, \ varphi) & = {3 \ over 32} {\ sqrt {77 \ over \ pi}} \, e ^ {- 5i \ varphi} \, \ sin ^ {5} \ theta \\ Y_ {5} ^ {- 4} (\ theta, \ varphi) & = {3 \ over 16} {\ sqrt {385 \ over 2 \ pi }} \, e ^ {- 4i \ varphi} \, \ sin ^ {4} \ theta \, \ cos \ theta \\ Y_ {5} ^ {- 3} (\ theta, \ varphi) & = {1 \ over 32} {\ sqrt {385 \ over \ pi}} \, e ^ {- 3i \ varphi} \, \ sin ^ {3} \ theta \, (9 \ cos ^ {2} \ theta -1) \\ Y_ {5} ^ {- 2} (\ theta, \ varphi) & = {1 \ over 8} {\ sqrt {1155 \ over 2 \ pi}} \ și ^ {- 2i \ varphi} \, \ sin ^ {2} \ theta \, (3 \ cos ^ {3} \ theta - \ cos \ theta) \\ Y_ {5} ^ {- 1} (\ theta, \ varphi) & = {1 \ over 16} {\ sqrt {165 \ over 2 \ pi}} \, e ^ {- i \ varphi} \, \ sin \ theta \, (21 \ cos ^ {4} \ theta -14 \ cos ^ {2} \ theta +1) \\ Y_ {5} ^ {0} (\ theta, \ varphi) & = {1 \ over 16} {\ sqrt {11 \ over \ pi}} \, (63 \ cos ^ {5 } \ theta -70 \ cos ^ {3} \ theta +15 \ cos \ theta) \\ Y_ {5} ^ {1} (\ theta, \ varphi) & = {- 1 \ over 16} {\ sqrt { 165 \ over 2 \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta \, (21 \ cos ^ {4} \ theta -14 \ cos ^ {2} \ theta +1) \\ Y_ {5} ^ {2} (\ theta, \ varphi) & = {1 \ over 8} {\ sqrt {1155 \ over 2 \ pi}} \ și ^ {2i \ varphi } \, \ sin ^ {2} \ theta \, (3 \ cos ^ {3} \ theta - \ cos \ theta) \\ Y_ {5} ^ {3} (\ theta, \ varphi) & = {- 1 \ over 32} {\ sqrt {385 \ over \ pi}} \, e ^ {3i \ varphi} \, \ sin ^ {3} \ theta \, (9 \ cos ^ {2} \ theta -1) \\ Y_ {5} ^ {4} (\ theta, \ varphi) & = {3 \ over 16} {\ sqrt {385 \ over 2 \ pi}} \ și ^ {4i \ varphi} \, \ sin ^ {4} \ theta \, \ cos \ theta \\ Y_ {5} ^ {5} (\ theta, \ varphi) & = {- 3 \ over 32} {\ sqrt {77 \ over \ pi}} \ , e ^ {5i \ varphi} \, \ sin ^ {5} \ theta \ end {align}}}

{\ displaystyle {\ begin {align} Y_ {5} ^ {- 5} (\ theta, \ varphi) & = {3 \ over 32} {\ sqrt {77 \ over \ pi}} \, e ^ {- 5i \ varphi} \, \ sin ^ {5} \ theta \\ Y_ {5} ^ {- 4} (\ theta, \ varphi) & = {3 \ over 16} {\ sqrt {385 \ over 2 \ pi }} \, e ^ {- 4i \ varphi} \, \ sin ^ {4} \ theta \, \ cos \ theta \\ Y_ {5} ^ {- 3} (\ theta, \ varphi) & = {1 \ over 32} {\ sqrt {385 \ over \ pi}} \, e ^ {- 3i \ varphi} \, \ sin ^ {3} \ theta \, (9 \ cos ^ {2} \ theta -1) \\ Y_ {5} ^ {- 2} (\ theta, \ varphi) & = {1 \ over 8} {\ sqrt {1155 \ over 2 \ pi}} \ și ^ {- 2i \ varphi} \, \ sin ^ {2} \ theta \, (3 \ cos ^ {3} \ theta - \ cos \ theta) \\ Y_ {5} ^ {- 1} (\ theta, \ varphi) & = {1 \ over 16} {\ sqrt {165 \ over 2 \ pi}} \, e ^ {- i \ varphi} \, \ sin \ theta \, (21 \ cos ^ {4} \ theta -14 \ cos ^ {2} \ theta +1) \\ Y_ {5} ^ {0} (\ theta, \ varphi) & = {1 \ over 16} {\ sqrt {11 \ over \ pi}} \, (63 \ cos ^ {5 } \ theta -70 \ cos ^ {3} \ theta +15 \ cos \ theta) \\ Y_ {5} ^ {1} (\ theta, \ varphi) & = {- 1 \ over 16} {\ sqrt { 165 \ over 2 \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta \, (21 \ cos ^ {4} \ theta -14 \ cos ^ {2} \ theta +1) \\ Y_ {5} ^ {2} (\ theta, \ varphi) & = {1 \ over 8} {\ sqrt {1155 \ over 2 \ pi}} \ și ^ {2i \ varphi } \, \ sin ^ {2} \ theta \, (3 \ cos ^ {3} \ theta - \ cos \ theta) \\ Y_ {5} ^ {3} (\ theta, \ varphi) & = {- 1 \ over 32} {\ sqrt {385 \ over \ pi}} \, e ^ {3i \ varphi} \, \ sin ^ {3} \ theta \, (9 \ cos ^ {2} \ theta -1) \\ Y_ {5} ^ {4} (\ theta, \ varphi) & = {3 \ over 16} {\ sqrt {385 \ over 2 \ pi}} \ și ^ {4i \ varphi} \, \ sin ^ {4} \ theta \, \ cos \ theta \\ Y_ {5} ^ {5} (\ theta, \ varphi) & = {- 3 \ over 32} {\ sqrt {77 \ over \ pi}} \ , e ^ {5i \ varphi} \, \ sin ^ {5} \ theta \ end {align}}}

Armonice sferice cu l = 6

{\begin{aligned}Y_{6}^{-6}(\theta ,\varphi )&={1 \over 64}{\sqrt {3003 \over \pi }}\,e^{-6i\varphi }\,\sin ^{6}\theta \\Y_{6}^{-5}(\theta ,\varphi )&={3 \over 32}{\sqrt {1001 \over \pi }}\,e^{-5i\varphi }\,\sin ^{5}\theta \,\cos \theta \\Y_{6}^{-4}(\theta ,\varphi )&={3 \over 32}{\sqrt {91 \over 2\pi }}\,e^{-4i\varphi }\,\sin ^{4}\theta \,(11\cos ^{2}\theta -1)\\Y_{6}^{-3}(\theta ,\varphi )&={1 \over 32}{\sqrt {1365 \over \pi }}\,e^{-3i\varphi }\,\sin ^{3}\theta \,(11\cos ^{3}\theta -3\cos \theta )\\Y_{6}^{-2}(\theta ,\varphi )&={1 \over 64}{\sqrt {1365 \over \pi }}\,e^{-2i\varphi }\,\sin ^{2}\theta \,(33\cos ^{4}\theta -18\cos ^{2}\theta +1)\\Y_{6}^{-1}(\theta ,\varphi )&={1 \over 16}{\sqrt {273 \over 2\pi }}\,e^{-i\varphi }\,\sin \theta \,(33\cos ^{5}\theta -30\cos ^{3}\theta +5\cos \theta )\\Y_{6}^{0}(\theta ,\varphi )&={1 \over 32}{\sqrt {13 \over \pi }}\,(231\cos ^{6}\theta -315\cos ^{4}\theta +105\cos ^{2}\theta -5)\\Y_{6}^{1}(\theta ,\varphi )&=-{1 \over 16}{\sqrt {273 \over 2\pi }}\,e^{i\varphi }\,\sin \theta \,(33\cos ^{5}\theta -30\cos ^{3}\theta +5\cos \theta )\\Y_{6}^{2}(\theta ,\varphi )&={1 \over 64}{\sqrt {1365 \over \pi }}\,e^{2i\varphi }\,\sin ^{2}\theta \,(33\cos ^{4}\theta -18\cos ^{2}\theta +1)\\Y_{6}^{3}(\theta ,\varphi )&=-{1 \over 32}{\sqrt {1365 \over \pi }}\,e^{3i\varphi }\,\sin ^{3}\theta \,(11\cos ^{3}\theta -3\cos \theta )\\Y_{6}^{4}(\theta ,\varphi )&={3 \over 32}{\sqrt {91 \over 2\pi }}\,e^{4i\varphi }\,\sin ^{4}\theta \,(11\cos ^{2}\theta -1)\\Y_{6}^{5}(\theta ,\varphi )&=-{3 \over 32}{\sqrt {1001 \over \pi }}\,e^{5i\varphi }\,\sin ^{5}\theta \,\cos \theta \\Y_{6}^{6}(\theta ,\varphi )&={1 \over 64}{\sqrt {3003 \over \pi }}\,e^{6i\varphi }\,\sin ^{6}\theta \end{aligned}}

{\ displaystyle {\ begin {align} Y_ {6} ^ {- 6} (\ theta, \ varphi) & = {1 \ over 64} {\ sqrt {3003 \ over \ pi}} \, e ^ {- 6i \ varphi} \, \ sin ^ {6} \ theta \\ Y_ {6} ^ {- 5} (\ theta, \ varphi) & = {3 \ over 32} {\ sqrt {1001 \ over \ pi} } \, e ^ {- 5i \ varphi} \, \ sin ^ {5} \ theta \, \ cos \ theta \\ Y_ {6} ^ {- 4} (\ theta, \ varphi) & = {3 \ peste 32} {\ sqrt {91 \ peste 2 \ pi}} \, e ^ {- 4i \ varphi} \, \ sin ^ {4} \ theta \, (11 \ cos ^ {2} \ theta -1) \\ Y_ {6} ^ {- 3} (\ theta, \ varphi) & = {1 \ over 32} {\ sqrt {1365 \ over \ pi}} \ și ^ {- 3i \ varphi} \, \ sin ^ {3} \ theta \, (11 \ cos ^ {3} \ theta -3 \ cos \ theta) \\ Y_ {6} ^ {- 2} (\ theta, \ varphi) & = {1 \ over 64} {\ sqrt {1365 \ over \ pi}} \, e ^ {- 2i \ varphi} \, \ sin ^ {2} \ theta \, (33 \ cos ^ {4} \ theta -18 \ cos ^ {2} \ theta +1) \\ Y_ {6} ^ {- 1} (\ theta, \ varphi) & = {1 \ over 16} {\ sqrt {273 \ over 2 \ pi}} \ și ^ {-i \ varphi} \, \ sin \ theta \, (33 \ cos ^ {5} \ theta -30 \ cos ^ {3} \ theta +5 \ cos \ theta) \\ Y_ {6} ^ {0 } (\ theta, \ varphi) & = {1 \ over 32} {\ sqrt {13 \ over \ pi}} \, (231 \ cos ^ {6} \ theta -315 \ cos ^ {4} \ theta + 105 \ cos ^ {2} \ theta -5) \\ Y_ {6} ^ {1} (\ theta, \ varphi) & = - {1 \ over 16} {\ sqrt {273 \ over 2 \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta \, (33 \ cos ^ {5} \ theta -30 \ cos ^ {3} \ theta +5 \ cos \ theta) \\ Y_ {6} ^ {2} (\ theta, \ varphi) & = {1 \ over 64} {\ sqrt {1365 \ over \ pi}} \ și ^ {2i \ varphi} \, \ sin ^ {2} \ theta \, (33 \ cos ^ {4} \ theta -18 \ cos ^ {2} \ theta +1) \\ Y_ {6} ^ {3} (\ theta, \ varphi) & = - {1 \ over 32} {\ sqrt {1365 \ over \ pi}} \, e ^ {3i \ varphi} \, \ sin ^ {3} \ theta \, (11 \ cos ^ {3} \ theta -3 \ cos \ theta) \\ Y_ {6} ^ {4} (\ theta, \ varphi) & = {3 \ over 32} {\ sqrt {91 \ over 2 \ pi}} \, e ^ { 4i \ varphi} \, \ sin ^ {4} \ theta \, (11 \ cos ^ {2} \ theta -1) \\ Y_ {6} ^ {5} (\ theta, \ varphi) & = - { 3 \ over 32} {\ sqrt {1001 \ over \ pi}} \, e ^ {5i \ varphi} \, \ sin ^ {5} \ theta \, \ cos \ theta \\ Y_ {6} ^ {6 } (\ theta, \ varphi) & = {1 \ over 64} {\ sqrt {3003 \ over \ pi}} \, e ^ {6i \ varphi} \, \ sin ^ {6} \ theta \ end {align }}}

{\ displaystyle {\ begin {align} Y_ {6} ^ {- 6} (\ theta, \ varphi) & = {1 \ over 64} {\ sqrt {3003 \ over \ pi}} \, e ^ {- 6i \ varphi} \, \ sin ^ {6} \ theta \\ Y_ {6} ^ {- 5} (\ theta, \ varphi) & = {3 \ over 32} {\ sqrt {1001 \ over \ pi} } \, e ^ {- 5i \ varphi} \, \ sin ^ {5} \ theta \, \ cos \ theta \\ Y_ {6} ^ {- 4} (\ theta, \ varphi) & = {3 \ peste 32} {\ sqrt {91 \ peste 2 \ pi}} \, e ^ {- 4i \ varphi} \, \ sin ^ {4} \ theta \, (11 \ cos ^ {2} \ theta -1) \\ Y_ {6} ^ {- 3} (\ theta, \ varphi) & = {1 \ over 32} {\ sqrt {1365 \ over \ pi}} \ și ^ {- 3i \ varphi} \, \ sin ^ {3} \ theta \, (11 \ cos ^ {3} \ theta -3 \ cos \ theta) \\ Y_ {6} ^ {- 2} (\ theta, \ varphi) & = {1 \ over 64} {\ sqrt {1365 \ over \ pi}} \, e ^ {- 2i \ varphi} \, \ sin ^ {2} \ theta \, (33 \ cos ^ {4} \ theta -18 \ cos ^ {2} \ theta +1) \\ Y_ {6} ^ {- 1} (\ theta, \ varphi) & = {1 \ over 16} {\ sqrt {273 \ over 2 \ pi}} \ și ^ {-i \ varphi} \, \ sin \ theta \, (33 \ cos ^ {5} \ theta -30 \ cos ^ {3} \ theta +5 \ cos \ theta) \\ Y_ {6} ^ {0 } (\ theta, \ varphi) & = {1 \ over 32} {\ sqrt {13 \ over \ pi}} \, (231 \ cos ^ {6} \ theta -315 \ cos ^ {4} \ theta + 105 \ cos ^ {2} \ theta -5) \\ Y_ {6} ^ {1} (\ theta, \ varphi) & = - {1 \ over 16} {\ sqrt {273 \ over 2 \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta \, (33 \ cos ^ {5} \ theta -30 \ cos ^ {3} \ theta +5 \ cos \ theta) \\ Y_ {6} ^ {2} (\ theta, \ varphi) & = {1 \ over 64} {\ sqrt {1365 \ over \ pi}} \ și ^ {2i \ varphi} \, \ sin ^ {2} \ theta \, (33 \ cos ^ {4} \ theta -18 \ cos ^ {2} \ theta +1) \\ Y_ {6} ^ {3} (\ theta, \ varphi) & = - {1 \ over 32} {\ sqrt {1365 \ over \ pi}} \, e ^ {3i \ varphi} \, \ sin ^ {3} \ theta \, (11 \ cos ^ {3} \ theta -3 \ cos \ theta) \\ Y_ {6} ^ {4} (\ theta, \ varphi) & = {3 \ over 32} {\ sqrt {91 \ over 2 \ pi}} \, e ^ { 4i \ varphi} \, \ sin ^ {4} \ theta \, (11 \ cos ^ {2} \ theta -1) \\ Y_ {6} ^ {5} (\ theta, \ varphi) & = - { 3 \ over 32} {\ sqrt {1001 \ over \ pi}} \, e ^ {5i \ varphi} \, \ sin ^ {5} \ theta \, \ cos \ theta \\ Y_ {6} ^ {6 } (\ theta, \ varphi) & = {1 \ over 64} {\ sqrt {3003 \ over \ pi}} \, e ^ {6i \ varphi} \, \ sin ^ {6} \ theta \ end {align }}}

Armonice sferice cu l = 7

{\begin{aligned}Y_{7}^{-7}(\theta ,\varphi )&={3 \over 64}{\sqrt {715 \over 2\pi }}\,e^{-7i\varphi }\,\sin ^{7}\theta \\Y_{7}^{-6}(\theta ,\varphi )&={3 \over 64}{\sqrt {5005 \over \pi }}\,e^{-6i\varphi }\,\sin ^{6}\theta \,\cos \theta \\Y_{7}^{-5}(\theta ,\varphi )&={3 \over 64}{\sqrt {385 \over 2\pi }}\,e^{-5i\varphi }\,\sin ^{5}\theta \,(13\cos ^{2}\theta -1)\\Y_{7}^{-4}(\theta ,\varphi )&={3 \over 32}{\sqrt {385 \over 2\pi }}\,e^{-4i\varphi }\,\sin ^{4}\theta \,(13\cos ^{3}\theta -3\cos \theta )\\Y_{7}^{-3}(\theta ,\varphi )&={3 \over 64}{\sqrt {35 \over 2\pi }}\,e^{-3i\varphi }\,\sin ^{3}\theta \,(143\cos ^{4}\theta -66\cos ^{2}\theta +3)\\Y_{7}^{-2}(\theta ,\varphi )&={3 \over 64}{\sqrt {35 \over \pi }}\,e^{-2i\varphi }\,\sin ^{2}\theta \,(143\cos ^{5}\theta -110\cos ^{3}\theta +15\cos \theta )\\Y_{7}^{-1}(\theta ,\varphi )&={1 \over 64}{\sqrt {105 \over 2\pi }}\,e^{-i\varphi }\,\sin \theta \,(429\cos ^{6}\theta -495\cos ^{4}\theta +135\cos ^{2}\theta -5)\\Y_{7}^{0}(\theta ,\varphi )&={1 \over 32}{\sqrt {15 \over \pi }}\,(429\cos ^{7}\theta -693\cos ^{5}\theta +315\cos ^{3}\theta -35\cos \theta )\\Y_{7}^{1}(\theta ,\varphi )&=-{1 \over 64}{\sqrt {105 \over 2\pi }}\,e^{i\varphi }\,\sin \theta \,(429\cos ^{6}\theta -495\cos ^{4}\theta +135\cos ^{2}\theta -5)\\Y_{7}^{2}(\theta ,\varphi )&={3 \over 64}{\sqrt {35 \over \pi }}\,e^{2i\varphi }\,\sin ^{2}\theta \,(143\cos ^{5}\theta -110\cos ^{3}\theta +15\cos \theta )\\Y_{7}^{3}(\theta ,\varphi )&=-{3 \over 64}{\sqrt {35 \over 2\pi }}\,e^{3i\varphi }\,\sin ^{3}\theta \,(143\cos ^{4}\theta -66\cos ^{2}\theta +3)\\Y_{7}^{4}(\theta ,\varphi )&={3 \over 32}{\sqrt {385 \over 2\pi }}\,e^{4i\varphi }\,\sin ^{4}\theta \,(13\cos ^{3}\theta -3\cos \theta )\\Y_{7}^{5}(\theta ,\varphi )&=-{3 \over 64}{\sqrt {385 \over 2\pi }}\,e^{5i\varphi }\,\sin ^{5}\theta \,(13\cos ^{2}\theta -1)\\Y_{7}^{6}(\theta ,\varphi )&={3 \over 64}{\sqrt {5005 \over \pi }}\,e^{6i\varphi }\,\sin ^{6}\theta \,\cos \theta \\Y_{7}^{7}(\theta ,\varphi )&=-{3 \over 64}{\sqrt {715 \over 2\pi }}\,e^{7i\varphi }\,\sin ^{7}\theta \end{aligned}}

{\ displaystyle {\ begin {align} Y_ {7} ^ {- 7} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {715 \ over 2 \ pi}} \ și ^ { -7i \ varphi} \, \ sin ^ {7} \ theta \\ Y_ {7} ^ {- 6} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {5005 \ over \ pi }} \, e ^ {- 6i \ varphi} \, \ sin ^ {6} \ theta \, \ cos \ theta \\ Y_ {7} ^ {- 5} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {385 \ over 2 \ pi}} \, e ^ {- 5i \ varphi} \, \ sin ^ {5} \ theta \, (13 \ cos ^ {2} \ theta -1 ) \\ Y_ {7} ^ {- 4} (\ theta, \ varphi) & = {3 \ over 32} {\ sqrt {385 \ over 2 \ pi}} \ și ^ {- 4i \ varphi} \ , \ sin ^ {4} \ theta \, (13 \ cos ^ {3} \ theta -3 \ cos \ theta) \\ Y_ {7} ^ {- 3} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {35 \ over 2 \ pi}} \, e ^ {- 3i \ varphi} \, \ sin ^ {3} \ theta \, (143 \ cos ^ {4} \ theta -66 \ cos ^ {2} \ theta +3) \\ Y_ {7} ^ {- 2} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {35 \ over \ pi}} \, e ^ {- 2i \ varphi} \, \ sin ^ {2} \ theta \, (143 \ cos ^ {5} \ theta -110 \ cos ^ {3} \ theta +15 \ cos \ theta) \\ Y_ {7} ^ {- 1} (\ theta, \ varphi) & = {1 \ peste 64} {\ sqrt {105 \ peste 2 \ pi}} \, e ^ {- i \ varphi} \, \ sin \ theta \, (429 \ cos ^ {6} \ theta -495 \ cos ^ {4} \ theta +135 \ cos ^ {2} \ theta -5) \\ Y_ {7} ^ {0 } (\ theta, \ varphi) & = {1 \ over 32} {\ sqrt {15 \ over \ pi}} \, (429 \ cos ^ {7} \ theta -693 \ cos ^ {5} \ theta + 315 \ cos ^ {3} \ theta -35 \ cos \ theta) \\ Y_ {7} ^ {1} (\ theta, \ varphi) & = - {1 \ over 64} {\ sqrt {105 \ over 2 \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta \, (429 \ cos ^ {6} \ theta -495 \ cos ^ {4} \ theta +135 \ cos ^ {2} \ theta -5) \\ Y_ {7} ^ {2} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {35 \ over \ pi}} \ și ^ {2i \ varphi} \ , \ sin ^ {2} \ theta \, (143 \ cos ^ {5} \ theta -110 \ cos ^ {3} \ theta +15 \ cos \ theta) \\ Y_ {7} ^ {3} (\ theta, \ varphi) & = - {3 \ over 64} {\ sqrt {35 \ over 2 \ pi}} \, e ^ {3i \ varphi} \, \ sin ^ {3} \ theta \, (143 \ cos ^ {4} \ theta -66 \ cos ^ {2} \ theta +3) \\ Y_ {7} ^ {4} (\ theta, \ varphi) & = {3 \ over 32} {\ sqrt {385 \ over 2 \ pi}} \, e ^ {4i \ varphi} \, \ sin ^ {4} \ theta \, (13 \ cos ^ {3} \ theta -3 \ cos \ theta) \\ Y_ {7 } ^ {5} (\ theta, \ varphi) & = - {3 \ over 64} {\ sqrt {385 \ over 2 \ pi}} \, e ^ {5i \ varphi} \, \ sin ^ {5} \ theta \, (13 \ cos ^ {2} \ theta -1) \\ Y_ {7} ^ {6} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {5005 \ over \ pi}} \, e ^ {6i \ varphi} \, \ sin ^ {6} \ theta \, \ cos \ theta \\ Y_ {7} ^ {7} (\ theta , \ varphi) & = - {3 \ over 64} {\ sqrt {715 \ over 2 \ pi}} \, e ^ {7i \ varphi} \, \ sin ^ {7} \ theta \ end {align}} }

{\ displaystyle {\ begin {align} Y_ {7} ^ {- 7} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {715 \ over 2 \ pi}} \ și ^ { -7i \ varphi} \, \ sin ^ {7} \ theta \\ Y_ {7} ^ {- 6} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {5005 \ over \ pi }} \, e ^ {- 6i \ varphi} \, \ sin ^ {6} \ theta \, \ cos \ theta \\ Y_ {7} ^ {- 5} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {385 \ over 2 \ pi}} \, e ^ {- 5i \ varphi} \, \ sin ^ {5} \ theta \, (13 \ cos ^ {2} \ theta -1 ) \\ Y_ {7} ^ {- 4} (\ theta, \ varphi) & = {3 \ over 32} {\ sqrt {385 \ over 2 \ pi}} \ și ^ {- 4i \ varphi} \ , \ sin ^ {4} \ theta \, (13 \ cos ^ {3} \ theta -3 \ cos \ theta) \\ Y_ {7} ^ {- 3} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {35 \ over 2 \ pi}} \, e ^ {- 3i \ varphi} \, \ sin ^ {3} \ theta \, (143 \ cos ^ {4} \ theta -66 \ cos ^ {2} \ theta +3) \\ Y_ {7} ^ {- 2} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {35 \ over \ pi}} \, e ^ {- 2i \ varphi} \, \ sin ^ {2} \ theta \, (143 \ cos ^ {5} \ theta -110 \ cos ^ {3} \ theta +15 \ cos \ theta) \\ Y_ {7} ^ {- 1} (\ theta, \ varphi) & = {1 \ peste 64} {\ sqrt {105 \ peste 2 \ pi}} \, e ^ {- i \ varphi} \, \ sin \ theta \, (429 \ cos ^ {6} \ theta -495 \ cos ^ {4} \ theta +135 \ cos ^ {2} \ theta -5) \\ Y_ {7} ^ {0 } (\ theta, \ varphi) & = {1 \ over 32} {\ sqrt {15 \ over \ pi}} \, (429 \ cos ^ {7} \ theta -693 \ cos ^ {5} \ theta + 315 \ cos ^ {3} \ theta -35 \ cos \ theta) \\ Y_ {7} ^ {1} (\ theta, \ varphi) & = - {1 \ over 64} {\ sqrt {105 \ over 2 \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta \, (429 \ cos ^ {6} \ theta -495 \ cos ^ {4} \ theta +135 \ cos ^ {2} \ theta -5) \\ Y_ {7} ^ {2} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {35 \ over \ pi}} \ și ^ {2i \ varphi} \ , \ sin ^ {2} \ theta \, (143 \ cos ^ {5} \ theta -110 \ cos ^ {3} \ theta +15 \ cos \ theta) \\ Y_ {7} ^ {3} (\ theta, \ varphi) & = - {3 \ over 64} {\ sqrt {35 \ over 2 \ pi}} \, e ^ {3i \ varphi} \, \ sin ^ {3} \ theta \, (143 \ cos ^ {4} \ theta -66 \ cos ^ {2} \ theta +3) \\ Y_ {7} ^ {4} (\ theta, \ varphi) & = {3 \ over 32} {\ sqrt {385 \ over 2 \ pi}} \, e ^ {4i \ varphi} \, \ sin ^ {4} \ theta \, (13 \ cos ^ {3} \ theta -3 \ cos \ theta) \\ Y_ {7 } ^ {5} (\ theta, \ varphi) & = - {3 \ over 64} {\ sqrt {385 \ over 2 \ pi}} \, e ^ {5i \ varphi} \, \ sin ^ {5} \ theta \, (13 \ cos ^ {2} \ theta -1) \\ Y_ {7} ^ {6} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {5005 \ over \ pi}} \, e ^ {6i \ varphi} \, \ sin ^ {6} \ theta \, \ cos \ theta \\ Y_ {7} ^ {7} (\ theta , \ varphi) & = - {3 \ over 64} {\ sqrt {715 \ over 2 \ pi}} \, e ^ {7i \ varphi} \, \ sin ^ {7} \ theta \ end {align}} }

Armonice sferice cu l = 8

{\begin{aligned}Y_{8}^{-8}(\theta ,\varphi )&={3 \over 256}{\sqrt {12155 \over 2\pi }}\,e^{-8i\varphi }\,\sin ^{8}\theta \\Y_{8}^{-7}(\theta ,\varphi )&={3 \over 64}{\sqrt {12155 \over 2\pi }}\,e^{-7i\varphi }\,\sin ^{7}\theta \,\cos \theta \\Y_{8}^{-6}(\theta ,\varphi )&={1 \over 128}{\sqrt {7293 \over \pi }}\,e^{-6i\varphi }\,\sin ^{6}\theta \,(15\cos ^{2}\theta -1)\\Y_{8}^{-5}(\theta ,\varphi )&={3 \over 64}{\sqrt {17017 \over 2\pi }}\,e^{-5i\varphi }\,\sin ^{5}\theta \,(5\cos ^{3}\theta -\cos \theta )\\Y_{8}^{-4}(\theta ,\varphi )&={3 \over 128}{\sqrt {1309 \over 2\pi }}\,e^{-4i\varphi }\,\sin ^{4}\theta \,(65\cos ^{4}\theta -26\cos ^{2}\theta +1)\\Y_{8}^{-3}(\theta ,\varphi )&={1 \over 64}{\sqrt {19635 \over 2\pi }}\,e^{-3i\varphi }\,\sin ^{3}\theta \,(39\cos ^{5}\theta -26\cos ^{3}\theta +3\cos \theta )\\Y_{8}^{-2}(\theta ,\varphi )&={3 \over 128}{\sqrt {595 \over \pi }}\,e^{-2i\varphi }\,\sin ^{2}\theta \,(143\cos ^{6}\theta -143\cos ^{4}\theta +33\cos ^{2}\theta -1)\\Y_{8}^{-1}(\theta ,\varphi )&={3 \over 64}{\sqrt {17 \over 2\pi }}\,e^{-i\varphi }\,\sin \theta \,(715\cos ^{7}\theta -1001\cos ^{5}\theta +385\cos ^{3}\theta -35\cos \theta )\\Y_{8}^{0}(\theta ,\varphi )&={1 \over 256}{\sqrt {17 \over \pi }}\,(6435\cos ^{8}\theta -12012\cos ^{6}\theta +6930\cos ^{4}\theta -1260\cos ^{2}\theta +35)\\Y_{8}^{1}(\theta ,\varphi )&={-3 \over 64}{\sqrt {17 \over 2\pi }}\,e^{i\varphi }\,\sin \theta \,(715\cos ^{7}\theta -1001\cos ^{5}\theta +385\cos ^{3}\theta -35\cos \theta )\\Y_{8}^{2}(\theta ,\varphi )&={3 \over 128}{\sqrt {595 \over \pi }}\,e^{2i\varphi }\,\sin ^{2}\theta \,(143\cos ^{6}\theta -143\cos ^{4}\theta +33\cos ^{2}\theta -1)\\Y_{8}^{3}(\theta ,\varphi )&={-1 \over 64}{\sqrt {19635 \over 2\pi }}\,e^{3i\varphi }\,\sin ^{3}\theta \,(39\cos ^{5}\theta -26\cos ^{3}\theta +3\cos \theta )\\Y_{8}^{4}(\theta ,\varphi )&={3 \over 128}{\sqrt {1309 \over 2\pi }}\,e^{4i\varphi }\,\sin ^{4}\theta \,(65\cos ^{4}\theta -26\cos ^{2}\theta +1)\\Y_{8}^{5}(\theta ,\varphi )&={-3 \over 64}{\sqrt {17017 \over 2\pi }}\,e^{5i\varphi }\,\sin ^{5}\theta \,(5\cos ^{3}\theta -\cos \theta )\\Y_{8}^{6}(\theta ,\varphi )&={1 \over 128}{\sqrt {7293 \over \pi }}\,e^{6i\varphi }\,\sin ^{6}\theta \,(15\cos ^{2}\theta -1)\\Y_{8}^{7}(\theta ,\varphi )&={-3 \over 64}{\sqrt {12155 \over 2\pi }}\,e^{7i\varphi }\,\sin ^{7}\theta \,\cos \theta \\Y_{8}^{8}(\theta ,\varphi )&={3 \over 256}{\sqrt {12155 \over 2\pi }}\,e^{8i\varphi }\,\sin ^{8}\theta \end{aligned}}

{\ displaystyle {\ begin {align} Y_ {8} ^ {- 8} (\ theta, \ varphi) & = {3 \ over 256} {\ sqrt {12155 \ over 2 \ pi}} \, e ^ { -8i \ varphi} \, \ sin ^ {8} \ theta \\ Y_ {8} ^ {- 7} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {12155 \ over 2 \ pi}} \, e ^ {- 7i \ varphi} \, \ sin ^ {7} \ theta \, \ cos \ theta \\ Y_ {8} ^ {- 6} (\ theta, \ varphi) & = { 1 \ over 128} {\ sqrt {7293 \ over \ pi}} \, e ^ {- 6i \ varphi} \, \ sin ^ {6} \ theta \, (15 \ cos ^ {2} \ theta -1 ) \\ Y_ {8} ^ {- 5} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {17017 \ over 2 \ pi}} \ și ^ {- 5i \ varphi} \ , \ sin ^ {5} \ theta \, (5 \ cos ^ {3} \ theta - \ cos \ theta) \\ Y_ {8} ^ {- 4} (\ theta, \ varphi) & = {3 \ peste 128} {\ sqrt {1309 \ peste 2 \ pi}} \, e ^ {- 4i \ varphi} \, \ sin ^ {4} \ theta \, (65 \ cos ^ {4} \ theta -26 \ cos ^ {2} \ theta +1) \\ Y_ {8} ^ {- 3} (\ theta, \ varphi) & = {1 \ over 64} {\ sqrt {19635 \ over 2 \ pi}} \, e ^ {- 3i \ varphi} \, \ sin ^ {3} \ theta \, (39 \ cos ^ {5} \ theta -26 \ cos ^ {3} \ theta +3 \ cos \ theta) \\ Y_ {8} ^ {- 2} (\ theta, \ varphi) & = {3 \ peste 128} {\ sqrt {595 \ peste \ pi}} \, e ^ {- 2i \ varphi} \, \ sin ^ { 2} \ theta \, (143 \ cos ^ {6} \ theta -143 \ cos ^ {4} \ theta +33 \ cos ^ {2} \ theta -1) \\ Y_ {8} ^ {- 1} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {17 \ over 2 \ pi}} \, e ^ {- i \ varphi } \, \ sin \ theta \, (715 \ cos ^ {7} \ theta -1001 \ cos ^ {5} \ theta +385 \ cos ^ {3} \ theta -35 \ cos \ theta) \\ Y_ { 8} ^ {0} (\ theta, \ varphi) & = {1 \ peste 256} {\ sqrt {17 \ peste \ pi}} \, (6435 \ cos ^ {8} \ theta -12012 \ cos ^ { 6} \ theta +6930 \ cos ^ {4} \ theta -1260 \ cos ^ {2} \ theta +35) \\ Y_ {8} ^ {1} (\ theta, \ varphi) & = {- 3 \ peste 64} {\ sqrt {17 \ peste 2 \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta \, (715 \ cos ^ {7} \ theta -1001 \ cos ^ {5} \ theta +385 \ cos ^ {3} \ theta -35 \ cos \ theta) \\ Y_ {8} ^ {2} (\ theta, \ varphi) & = {3 \ over 128} {\ sqrt {595 \ peste \ pi}} \, e ^ {2i \ varphi} \, \ sin ^ {2} \ theta \, (143 \ cos ^ {6} \ theta -143 \ cos ^ {4} \ theta +33 \ cos ^ {2} \ theta -1) \\ Y_ {8} ^ {3} (\ theta, \ varphi) & = {- 1 \ over 64} {\ sqrt {19635 \ over 2 \ pi}} \ și ^ {3i \ varphi} \, \ sin ^ {3} \ theta \, (39 \ cos ^ {5} \ theta -26 \ cos ^ {3} \ theta +3 \ cos \ theta) \\ Y_ {8 } ^ {4} (\ theta, \ varphi) & = {3 \ over 128} {\ sqrt {1309 \ over 2 \ pi}} \, e ^ {4i \ varphi} \, \ sin ^ {4} \ theta \, (65 \ cos ^ {4} \ theta -26 \ cos ^ {2} \ theta +1) \\ Y_ {8} ^ {5} (\ theta, \ varphi) & = {- 3 \ over 64} {\ sqrt {17017 \ over 2 \ pi}} \, e ^ {5i \ varphi} \, \ sin ^ {5} \ theta \, (5 \ cos ^ {3} \ theta - \ cos \ theta) \\ Y_ {8} ^ {6} (\ theta, \ varphi) & = {1 \ over 128} {\ sqrt {7293 \ over \ pi}} \ , e ^ {6i \ varphi} \, \ sin ^ {6} \ theta \, (15 \ cos ^ {2} \ theta -1) \\ Y_ {8} ^ {7} (\ theta, \ varphi) & = {- 3 \ over 64} {\ sqrt {12155 \ over 2 \ pi}} \, e ^ {7i \ varphi} \, \ sin ^ {7} \ theta \, \ cos \ theta \\ Y_ { 8} ^ {8} (\ theta, \ varphi) & = {3 \ peste 256} {\ sqrt {12155 \ peste 2 \ pi}} \, e ^ {8i \ varphi} \, \ sin ^ {8} \ theta \ end {align}}}

{\ displaystyle {\ begin {align} Y_ {8} ^ {- 8} (\ theta, \ varphi) & = {3 \ over 256} {\ sqrt {12155 \ over 2 \ pi}} \, e ^ { -8i \ varphi} \, \ sin ^ {8} \ theta \\ Y_ {8} ^ {- 7} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {12155 \ over 2 \ pi}} \, e ^ {- 7i \ varphi} \, \ sin ^ {7} \ theta \, \ cos \ theta \\ Y_ {8} ^ {- 6} (\ theta, \ varphi) & = { 1 \ over 128} {\ sqrt {7293 \ over \ pi}} \, e ^ {- 6i \ varphi} \, \ sin ^ {6} \ theta \, (15 \ cos ^ {2} \ theta -1 ) \\ Y_ {8} ^ {- 5} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {17017 \ over 2 \ pi}} \ și ^ {- 5i \ varphi} \ , \ sin ^ {5} \ theta \, (5 \ cos ^ {3} \ theta - \ cos \ theta) \\ Y_ {8} ^ {- 4} (\ theta, \ varphi) & = {3 \ peste 128} {\ sqrt {1309 \ peste 2 \ pi}} \, e ^ {- 4i \ varphi} \, \ sin ^ {4} \ theta \, (65 \ cos ^ {4} \ theta -26 \ cos ^ {2} \ theta +1) \\ Y_ {8} ^ {- 3} (\ theta, \ varphi) & = {1 \ over 64} {\ sqrt {19635 \ over 2 \ pi}} \, e ^ {- 3i \ varphi} \, \ sin ^ {3} \ theta \, (39 \ cos ^ {5} \ theta -26 \ cos ^ {3} \ theta +3 \ cos \ theta) \\ Y_ {8} ^ {- 2} (\ theta, \ varphi) & = {3 \ peste 128} {\ sqrt {595 \ peste \ pi}} \, e ^ {- 2i \ varphi} \, \ sin ^ { 2} \ theta \, (143 \ cos ^ {6} \ theta -143 \ cos ^ {4} \ theta +33 \ cos ^ {2} \ theta -1) \\ Y_ {8} ^ {- 1} (\ theta, \ varphi) & = {3 \ over 64} {\ sqrt {17 \ over 2 \ pi}} \, e ^ {- i \ varphi } \, \ sin \ theta \, (715 \ cos ^ {7} \ theta -1001 \ cos ^ {5} \ theta +385 \ cos ^ {3} \ theta -35 \ cos \ theta) \\ Y_ { 8} ^ {0} (\ theta, \ varphi) & = {1 \ over 256} {\ sqrt {17 \ over \ pi}} \, (6435 \ cos ^ {8} \ theta -12012 \ cos ^ { 6} \ theta +6930 \ cos ^ {4} \ theta -1260 \ cos ^ {2} \ theta +35) \\ Y_ {8} ^ {1} (\ theta, \ varphi) & = {- 3 \ over 64} {\ sqrt {17 \ over 2 \ pi}} \, e ^ {i \ varphi} \, \ sin \ theta \, (715 \ cos ^ {7} \ theta -1001 \ cos ^ {5} \ theta +385 \ cos ^ {3} \ theta -35 \ cos \ theta) \\ Y_ {8} ^ {2} (\ theta, \ varphi) & = {3 \ over 128} {\ sqrt {595 \ over \ pi}} \, e ^ {2i \ varphi} \, \ sin ^ {2} \ theta \, (143 \ cos ^ {6} \ theta -143 \ cos ^ {4} \ theta +33 \ cos ^ {2} \ theta -1) \\ Y_ {8} ^ {3} (\ theta, \ varphi) & = {- 1 \ over 64} {\ sqrt {19635 \ over 2 \ pi}} \, and ^ {3i \ varphi} \, \ sin ^ {3} \ theta \, (39 \ cos ^ {5} \ theta -26 \ cos ^ {3} \ theta +3 \ cos \ theta) \\ Y_ {8 } ^ {4} (\ theta, \ varphi) & = {3 \ over 128} {\ sqrt {1309 \ over 2 \ pi}} \, e ^ {4i \ varphi} \, \ sin ^ {4} \ theta \, (65 \ cos ^ {4} \ theta -26 \ cos ^ {2} \ theta +1) \\ Y_ {8} ^ {5} (\ theta ,\varphi )&={-3 \over 64}{\sqrt {17017 \over 2\pi }}\,e^{5i\varphi }\,\sin ^{5}\theta \,(5\cos ^{3}\theta -\cos \theta )\\Y_{8}^{6}(\theta ,\varphi )&={1 \over 128}{\sqrt {7293 \over \pi }}\,e^{6i\varphi }\,\sin ^{6}\theta \,(15\cos ^{2}\theta -1)\\Y_{8}^{7}(\theta ,\varphi )&={-3 \over 64}{\sqrt {12155 \over 2\pi }}\,e^{7i\varphi }\,\sin ^{7}\theta \,\cos \theta \\Y_{8}^{8}(\theta ,\varphi )&={3 \over 256}{\sqrt {12155 \over 2\pi }}\,e^{8i\varphi }\,\sin ^{8}\theta \end{aligned}}}

Armoniche sferiche con l = 9

{\begin{aligned}Y_{9}^{-9}(\theta ,\varphi )&={1 \over 512}{\sqrt {230945 \over \pi }}\,e^{-9i\varphi }\,\sin ^{9}\theta \\Y_{9}^{-8}(\theta ,\varphi )&={3 \over 256}{\sqrt {230945 \over 2\pi }}\,e^{-8i\varphi }\,\sin ^{8}\theta \,\cos \theta \\Y_{9}^{-7}(\theta ,\varphi )&={3 \over 512}{\sqrt {13585 \over \pi }}\,e^{-7i\varphi }\,\sin ^{7}\theta \,(17\cos ^{2}\theta -1)\\Y_{9}^{-6}(\theta ,\varphi )&={1 \over 128}{\sqrt {40755 \over \pi }}\,e^{-6i\varphi }\,\sin ^{6}\theta \,(17\cos ^{3}\theta -3\cos \theta )\\Y_{9}^{-5}(\theta ,\varphi )&={3 \over 256}{\sqrt {2717 \over \pi }}\,e^{-5i\varphi }\,\sin ^{5}\theta \,(85\cos ^{4}\theta -30\cos ^{2}\theta +1)\\Y_{9}^{-4}(\theta ,\varphi )&={3 \over 128}{\sqrt {95095 \over 2\pi }}\,e^{-4i\varphi }\,\sin ^{4}\theta \,(17\cos ^{5}\theta -10\cos ^{3}\theta +\cos \theta )\\Y_{9}^{-3}(\theta ,\varphi )&={1 \over 256}{\sqrt {21945 \over \pi }}\,e^{-3i\varphi }\,\sin ^{3}\theta \,(221\cos ^{6}\theta -195\cos ^{4}\theta +39\cos ^{2}\theta -1)\\Y_{9}^{-2}(\theta ,\varphi )&={3 \over 128}{\sqrt {1045 \over \pi }}\,e^{-2i\varphi }\,\sin ^{2}\theta \,(221\cos ^{7}\theta -273\cos ^{5}\theta +91\cos ^{3}\theta -7\cos \theta )\\Y_{9}^{-1}(\theta ,\varphi )&={3 \over 256}{\sqrt {95 \over 2\pi }}\,e^{-i\varphi }\,\sin \theta \,(2431\cos ^{8}\theta -4004\cos ^{6}\theta +2002\cos ^{4}\theta -308\cos ^{2}\theta +7)\\Y_{9}^{0}(\theta ,\varphi )&={1 \over 256}{\sqrt {19 \over \pi }}\,(12155\cos ^{9}\theta -25740\cos ^{7}\theta +18018\cos ^{5}\theta -4620\cos ^{3}\theta +315\cos \theta )\\Y_{9}^{1}(\theta ,\varphi )&={-3 \over 256}{\sqrt {95 \over 2\pi }}\,e^{i\varphi }\,\sin \theta \,(2431\cos ^{8}\theta -4004\cos ^{6}\theta +2002\cos ^{4}\theta -308\cos ^{2}\theta +7)\\Y_{9}^{2}(\theta ,\varphi )&={3 \over 128}{\sqrt {1045 \over \pi }}\,e^{2i\varphi }\,\sin ^{2}\theta \,(221\cos ^{7}\theta -273\cos ^{5}\theta +91\cos ^{3}\theta -7\cos \theta )\\Y_{9}^{3}(\theta ,\varphi )&={-1 \over 256}{\sqrt {21945 \over \pi }}\,e^{3i\varphi }\,\sin ^{3}\theta \,(221\cos ^{6}\theta -195\cos ^{4}\theta +39\cos ^{2}\theta -1)\\Y_{9}^{4}(\theta ,\varphi )&={3 \over 128}{\sqrt {95095 \over 2\pi }}\,e^{4i\varphi }\,\sin ^{4}\theta \,(17\cos ^{5}\theta -10\cos ^{3}\theta +\cos \theta )\\Y_{9}^{5}(\theta ,\varphi )&={-3 \over 256}{\sqrt {2717 \over \pi }}\,e^{5i\varphi }\,\sin ^{5}\theta \,(85\cos ^{4}\theta -30\cos ^{2}\theta +1)\\Y_{9}^{6}(\theta ,\varphi )&={1 \over 128}{\sqrt {40755 \over \pi }}\,e^{6i\varphi }\,\sin ^{6}\theta \,(17\cos ^{3}\theta -3\cos \theta )\\Y_{9}^{7}(\theta ,\varphi )&={-3 \over 512}{\sqrt {13585 \over \pi }}\,e^{7i\varphi }\,\sin ^{7}\theta \,(17\cos ^{2}\theta -1)\\Y_{9}^{8}(\theta ,\varphi )&={3 \over 256}{\sqrt {230945 \over 2\pi }}\,e^{8i\varphi }\,\sin ^{8}\theta \,\cos \theta \\Y_{9}^{9}(\theta ,\varphi )&={-1 \over 512}{\sqrt {230945 \over \pi }}\,e^{9i\varphi }\,\sin ^{9}\theta \end{aligned}}

{\begin{aligned}Y_{9}^{-9}(\theta ,\varphi )&={1 \over 512}{\sqrt {230945 \over \pi }}\,e^{-9i\varphi }\,\sin ^{9}\theta \\Y_{9}^{-8}(\theta ,\varphi )&={3 \over 256}{\sqrt {230945 \over 2\pi }}\,e^{-8i\varphi }\,\sin ^{8}\theta \,\cos \theta \\Y_{9}^{-7}(\theta ,\varphi )&={3 \over 512}{\sqrt {13585 \over \pi }}\,e^{-7i\varphi }\,\sin ^{7}\theta \,(17\cos ^{2}\theta -1)\\Y_{9}^{-6}(\theta ,\varphi )&={1 \over 128}{\sqrt {40755 \over \pi }}\,e^{-6i\varphi }\,\sin ^{6}\theta \,(17\cos ^{3}\theta -3\cos \theta )\\Y_{9}^{-5}(\theta ,\varphi )&={3 \over 256}{\sqrt {2717 \over \pi }}\,e^{-5i\varphi }\,\sin ^{5}\theta \,(85\cos ^{4}\theta -30\cos ^{2}\theta +1)\\Y_{9}^{-4}(\theta ,\varphi )&={3 \over 128}{\sqrt {95095 \over 2\pi }}\,e^{-4i\varphi }\,\sin ^{4}\theta \,(17\cos ^{5}\theta -10\cos ^{3}\theta +\cos \theta )\\Y_{9}^{-3}(\theta ,\varphi )&={1 \over 256}{\sqrt {21945 \over \pi }}\,e^{-3i\varphi }\,\sin ^{3}\theta \,(221\cos ^{6}\theta -195\cos ^{4}\theta +39\cos ^{2}\theta -1)\\Y_{9}^{-2}(\theta ,\varphi )&={3 \over 128}{\sqrt {1045 \over \pi }}\,e^{-2i\varphi }\,\sin ^{2}\theta \,(221\cos ^{7}\theta -273\cos ^{5}\theta +91\cos ^{3}\theta -7\cos \theta )\\Y_{9}^{-1}(\theta ,\varphi )&={3 \over 256}{\sqrt {95 \over 2\pi }}\,e^{-i\varphi }\,\sin \theta \,(2431\cos ^{8}\theta -4004\cos ^{6}\theta +2002\cos ^{4}\theta -308\cos ^{2}\theta +7)\\Y_{9}^{0}(\theta ,\varphi )&={1 \over 256}{\sqrt {19 \over \pi }}\,(12155\cos ^{9}\theta -25740\cos ^{7}\theta +18018\cos ^{5}\theta -4620\cos ^{3}\theta +315\cos \theta )\\Y_{9}^{1}(\theta ,\varphi )&={-3 \over 256}{\sqrt {95 \over 2\pi }}\,e^{i\varphi }\,\sin \theta \,(2431\cos ^{8}\theta -4004\cos ^{6}\theta +2002\cos ^{4}\theta -308\cos ^{2}\theta +7)\\Y_{9}^{2}(\theta ,\varphi )&={3 \over 128}{\sqrt {1045 \over \pi }}\,e^{2i\varphi }\,\sin ^{2}\theta \,(221\cos ^{7}\theta -273\cos ^{5}\theta +91\cos ^{3}\theta -7\cos \theta )\\Y_{9}^{3}(\theta ,\varphi )&={-1 \over 256}{\sqrt {21945 \over \pi }}\,e^{3i\varphi }\,\sin ^{3}\theta \,(221\cos ^{6}\theta -195\cos ^{4}\theta +39\cos ^{2}\theta -1)\\Y_{9}^{4}(\theta ,\varphi )&={3 \over 128}{\sqrt {95095 \over 2\pi }}\,e^{4i\varphi }\,\sin ^{4}\theta \,(17\cos ^{5}\theta -10\cos ^{3}\theta +\cos \theta )\\Y_{9}^{5}(\theta ,\varphi )&={-3 \over 256}{\sqrt {2717 \over \pi }}\,e^{5i\varphi }\,\sin ^{5}\theta \,(85\cos ^{4}\theta -30\cos ^{2}\theta +1)\\Y_{9}^{6}(\theta ,\varphi )&={1 \over 128}{\sqrt {40755 \over \pi }}\,e^{6i\varphi }\,\sin ^{6}\theta \,(17\cos ^{3}\theta -3\cos \theta )\\Y_{9}^{7}(\theta ,\varphi )&={-3 \over 512}{\sqrt {13585 \over \pi }}\,e^{7i\varphi }\,\sin ^{7}\theta \,(17\cos ^{2}\theta -1)\\Y_{9}^{8}(\theta ,\varphi )&={3 \over 256}{\sqrt {230945 \over 2\pi }}\,e^{8i\varphi }\,\sin ^{8}\theta \,\cos \theta \\Y_{9}^{9}(\theta ,\varphi )&={-1 \over 512}{\sqrt {230945 \over \pi }}\,e^{9i\varphi }\,\sin ^{9}\theta \end{aligned}}

{\begin{aligned}Y_{9}^{-9}(\theta ,\varphi )&={1 \over 512}{\sqrt {230945 \over \pi }}\,e^{-9i\varphi }\,\sin ^{9}\theta \\Y_{9}^{-8}(\theta ,\varphi )&={3 \over 256}{\sqrt {230945 \over 2\pi }}\,e^{-8i\varphi }\,\sin ^{8}\theta \,\cos \theta \\Y_{9}^{-7}(\theta ,\varphi )&={3 \over 512}{\sqrt {13585 \over \pi }}\,e^{-7i\varphi }\,\sin ^{7}\theta \,(17\cos ^{2}\theta -1)\\Y_{9}^{-6}(\theta ,\varphi )&={1 \over 128}{\sqrt {40755 \over \pi }}\,e^{-6i\varphi }\,\sin ^{6}\theta \,(17\cos ^{3}\theta -3\cos \theta )\\Y_{9}^{-5}(\theta ,\varphi )&={3 \over 256}{\sqrt {2717 \over \pi }}\,e^{-5i\varphi }\,\sin ^{5}\theta \,(85\cos ^{4}\theta -30\cos ^{2}\theta +1)\\Y_{9}^{-4}(\theta ,\varphi )&={3 \over 128}{\sqrt {95095 \over 2\pi }}\,e^{-4i\varphi }\,\sin ^{4}\theta \,(17\cos ^{5}\theta -10\cos ^{3}\theta +\cos \theta )\\Y_{9}^{-3}(\theta ,\varphi )&={1 \over 256}{\sqrt {21945 \over \pi }}\,e^{-3i\varphi }\,\sin ^{3}\theta \,(221\cos ^{6}\theta -195\cos ^{4}\theta +39\cos ^{2}\theta -1)\\Y_{9}^{-2}(\theta ,\varphi )&={3 \over 128}{\sqrt {1045 \over \pi }}\,e^{-2i\varphi }\,\sin ^{2}\theta \,(221\cos ^{7}\theta -273\cos ^{5}\theta +91\cos ^{3}\theta -7\cos \theta )\\Y_{9}^{-1}(\theta ,\varphi )&={3 \over 256}{\sqrt {95 \over 2\pi }}\,e^{-i\varphi }\,\sin \theta \,(2431\cos ^{8}\theta -4004\cos ^{6}\theta +2002\cos ^{4}\theta -308\cos ^{2}\theta +7)\\Y_{9}^{0}(\theta ,\varphi )&={1 \over 256}{\sqrt {19 \over \pi }}\,(12155\cos ^{9}\theta -25740\cos ^{7}\theta +18018\cos ^{5}\theta -4620\cos ^{3}\theta +315\cos \theta )\\Y_{9}^{1}(\theta ,\varphi )&={-3 \over 256}{\sqrt {95 \over 2\pi }}\,e^{i\varphi }\,\sin \theta \,(2431\cos ^{8}\theta -4004\cos ^{6}\theta +2002\cos ^{4}\theta -308\cos ^{2}\theta +7)\\Y_{9}^{2}(\theta ,\varphi )&={3 \over 128}{\sqrt {1045 \over \pi }}\,e^{2i\varphi }\,\sin ^{2}\theta \,(221\cos ^{7}\theta -273\cos ^{5}\theta +91\cos ^{3}\theta -7\cos \theta )\\Y_{9}^{3}(\theta ,\varphi )&={-1 \over 256}{\sqrt {21945 \over \pi }}\,e^{3i\varphi }\,\sin ^{3}\theta \,(221\cos ^{6}\theta -195\cos ^{4}\theta +39\cos ^{2}\theta -1)\\Y_{9}^{4}(\theta ,\varphi )&={3 \over 128}{\sqrt {95095 \over 2\pi }}\,e^{4i\varphi }\,\sin ^{4}\theta \,(17\cos ^{5}\theta -10\cos ^{3}\theta +\cos \theta )\\Y_{9}^{5}(\theta ,\varphi )&={-3 \over 256}{\sqrt {2717 \over \pi }}\,e^{5i\varphi }\,\sin ^{5}\theta \,(85\cos ^{4}\theta -30\cos ^{2}\theta +1)\\Y_{9}^{6}(\theta ,\varphi )&={1 \over 128}{\sqrt {40755 \over \pi }}\,e^{6i\varphi }\,\sin ^{6}\theta \,(17\cos ^{3}\theta -3\cos \theta )\\Y_{9}^{7}(\theta ,\varphi )&={-3 \over 512}{\sqrt {13585 \over \pi }}\,e^{7i\varphi }\,\sin ^{7}\theta \,(17\cos ^{2}\theta -1)\\Y_{9}^{8}(\theta ,\varphi )&={3 \over 256}{\sqrt {230945 \over 2\pi }}\,e^{8i\varphi }\,\sin ^{8}\theta \,\cos \theta \\Y_{9}^{9}(\theta ,\varphi )&={-1 \over 512}{\sqrt {230945 \over \pi }}\,e^{9i\varphi }\,\sin ^{9}\theta \end{aligned}}

Armoniche sferiche con l = 10

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

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

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

Note

^ DA Varshalovich, AN Moskalev e VK Khersonskii, Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols , prima ristampa, Singapore, World Scientific Pub., 1988, pp. 155–156, ISBN 9971-50-107-4 .

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Armoniche sferiche

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Estratto da " https://it.wikipedia.org/w/index.php?title=Tavola_delle_armoniche_sferiche&oldid=112977118 "

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