SphericalHaroicY Notatios Traditioal ae Spherical haroic Traditioal otatio Y, φ Matheatica StadardFor otatio SphericalHaroicY,,, φ Priary defiitio 05.0.0.000.0 Y, φ φ P cos ; 05.0.0.000.0 Y 0 k, φ k k ; k 05.0.0.0003.0 Y, φ 0 ; 05.0.0.0004.0 Y, φ Y, φ ; The followig restrictios apply to all forulas of this fuctio. 05.0.0.0005.0 Specific values Specialized values For fixed,, 05.0.03.000.0 Y, 0 φ Y, φ
http://fuctios.wolfra.co For fixed,, φ 05.0.03.000.0 Y 0, φ 0 ; 0 05.0.03.0003.0 Y, φ 0 ; 0 05.0.03.0004.0 Y k, φ 0 ; 0 k 05.0.03.0005.0 Y od, φ φ 05.0.03.0006.0 Y od, φ φ For fixed,, φ 05.0.03.0007.0 Y 0, φ P cos Y, φ 05.0.03.0008.0 φ cos P cos P cos cos si 05.0.03.0009.0 Y, φ φ si Y, φ 0 Y, φ Y, φ 0 05.0.03.000.0 05.0.03.00.0 φ si 05.0.03.00.0 05.0.03.003.0 Y, φ 0 ; 05.0.03.004.0 Y, φ 0 ; For fixed,, φ
http://fuctios.wolfra.co 3 05.0.03.005.0 φ cos Y si 0, φ 05.0.03.006.0 Y, φ 3 φ cos cos si 05.0.03.007.0 Y, φ 5 φ 3 cos 3 cos cos si 3 3 05.0.03.008.0 Y 3, φ 7 φ 5 cos 3 5 cos 6 9 cos 4 cos si 4 4 05.0.03.009.0 Y 4, φ 3 φ 05 cos 4 05 cos 3 45 cos 5 cos 4 0 9 cos si 5 5 05.0.03.000.0 Y 5, φ φ 945 cos 5 945 cos 4 0 5 cos 3 05 7 cos 5 4 3 5 cos 4 0 64cos si 6 6 05.0.03.00.0 Y 6, φ 3 φ 0 395 cos 6 0 395 cos 5 475 3 cos 4 630 7 cos 3 05 4 3 45 cos 4 5 99 cos 6 35 4 59 5 cos si 7 7 05.0.03.00.0 Y 7, φ 5 φ 35 35 cos 7 35 35 cos 6 3 85 7 cos 5 7 35 0 cos 4 575 4 38 63 cos 3 89 4 60 83 cos 7 4 6 70 4 56 575 cos 6 56 4 784 304 cos si 8 8
http://fuctios.wolfra.co 4 05.0.03.003.0 Y 8, φ 7 φ 07 05 cos 8 07 05 cos 7 945 945 4 cos 6 35 35 3 cos 5 5 975 4 4 cos 4 3465 4 70 383 cos 3 35 6 00 4 043 60 cos 9 4 6 66 4 4396 5 59 cos 8 84 6 974 4 96 05 cos si 9 9 05.0.03.004.0 Y 9, φ 9 φ 34 459 45 cos 9 34 459 45 cos 8 8 08 00 9 cos 7 4 79 75 3 cos 6 945 945 4 5 54 cos 5 35 35 4 40 49 cos 4 6930 6 5 4 373 890 cos 3 495 6 54 4 933 60 cos 45 8 98 6 674 4 0 7 9 845 cos 8 0 6 4368 4 5 480 47 456 cos si 0 0 05.0.03.005.0 Y 0, φ φ 654 79 075 cos 0 654 79 075 cos 9 30 34 85 5 cos 8 45 945 900 9 cos 7 9 459 450 4 56 35 cos 6 837 835 4 45 34 cos 5 35 35 6 65 4 874 350 cos 4 870 6 75 4 3773 6 830 cos 3 485 8 6 3479 4 9 88 33 075 cos 55 8 38 6 5754 4 78 877 5 865 cos 0 65 8 8778 6 7 80 4 057 893 05 cos si 05.0.03.006.0 Y, φ 4 φ cos si k k k k si k k 0 05.0.03.007.0 Y, φ 0 ; 0 05.0.03.008.0 Y, φ φ For fixed, φ si k k k cos cos k ; 0 k k k 0
http://fuctios.wolfra.co 5 05.0.03.009.0 Y 0 0, φ 05.0.03.0030.0 Y 0 k, φ k k ; k For fixed, φ 05.0.03.003.0 Y 0 0, φ 05.0.03.003.0 Y, φ φ 3 si 05.0.03.0033.0 Y 0, φ 3 cos 05.0.03.0034.0 Y, φ φ 3 si 05.0.03.0035.0 Y, φ 4 5 φ si 05.0.03.0036.0 Y, φ φ 5 cos si 05.0.03.0037.0 Y 0, φ 8 5 3 cos 05.0.03.0038.0 Y, φ φ 5 cos si 05.0.03.0039.0 Y, φ 4 5 φ si
http://fuctios.wolfra.co 6 05.0.03.0040.0 Y 3 3, φ 8 3 φ 35 si 3 05.0.03.004.0 Y 3, φ 4 05 φ cos si 05.0.03.004.0 Y 3, φ 6 φ 5 cos 3 si 05.0.03.0043.0 Y 0 3, φ 7 3 cos 5 cos3 6 05.0.03.0044.0 Y 3, φ 6 φ 5 cos 3 si 05.0.03.0045.0 Y 3, φ 4 05 φ cos si 05.0.03.0046.0 Y 3 3, φ 8 3 φ 35 si 3 05.0.03.0047.0 Y 4 4, φ 3 6 35 4 φ si 4 05.0.03.0048.0 Y 4 3, φ 3 8 35 3 φ cos si 3 05.0.03.0049.0 Y 4, φ 3 8 5 φ 7 cos si 05.0.03.0050.0 Y 4, φ 3 8 5 φ cos 7 cos 3 si 05.0.03.005.0 3 Y 0 4, φ 35 cos 4 30 cos 3 6
http://fuctios.wolfra.co 7 05.0.03.005.0 Y 4, φ 3 8 5 φ cos 7 cos 3 si 05.0.03.0053.0 Y 4, φ 3 8 5 φ 7 cos si 05.0.03.0054.0 Y 4 3, φ 3 8 35 3 φ cos si 3 05.0.03.0055.0 Y 4 4, φ 3 6 35 4 φ si 4 05.0.03.0056.0 Y 5 5, φ 3 77 3 5 φ si 5 05.0.03.0057.0 Y 5 4, φ 3 6 385 4 φ cos si 4 05.0.03.0058.0 Y 5 3, φ 3 385 3 φ 9 cos si 3 05.0.03.0059.0 Y 5, φ 8 55 φ cos 3 cos si 05.0.03.0060.0 Y 0 5, φ 6 63 cos5 70 cos 3 5 cos 05.0.03.006.0 Y 5, φ 6 65 φ cos 4 4 cos si 05.0.03.006.0 Y 5, φ 8 55 φ cos 3 cos si
http://fuctios.wolfra.co 8 05.0.03.0063.0 Y 5 3, φ 3 385 3 φ 9 cos si 3 05.0.03.0064.0 Y 5 4, φ 3 6 385 4 φ cos si 4 05.0.03.0065.0 Y 5 5, φ 3 77 3 5 φ si 5 Geeral characteristics Doai ad aalyticity The fuctio Y, φ is defied over. For fixed,, the fuctio Y, φ is a polyoial i si of degree ultiplied o fuctio coa 05.0.04.000.0 φy, φ si. Syetries ad periodicities Parity 05.0.04.000.0 Y, φ φ Y, φ 05.0.04.0003.0 Y, φ Y, φ 05.0.04.0004.0 Y, φ φ Y, φ 05.0.04.0005.0 Y, φ Y, φ 05.0.04.0006.0 Y, φ Y, φ Mirror syetry 05.0.04.0007.0 Y, φ Y, φ ; φ Periodicity Y, φ is a periodic fuctio with respect to ad φ with periods ad respectively.
http://fuctios.wolfra.co 9 05.0.04.0008.0 Y k, φ Y, φ ; k 05.0.04.0009.0 Y, φ k Y, φ ; k Phase shifts 05.0.04.000.0 Y, φ Y, φ 05.0.04.00.0 Y, φ Y, φ 05.0.04.00.0 Y, φ Y, φ Poles ad essetial sigularities With respect to φ For fixed,,, the fuctio Y, φ has oly oe sigular poit at φ. It is a essetial sigular poit. 05.0.04.003.0 ig φ Y, φ, With respect to For fixed, φ, ;, the fuctio Y, φ does ot have poles ad essetial sigularities. 05.0.04.004.0 ig Y, φ ; For iteger, the fuctio Y, φ is polyoial ad has pole of order at cos. 05.0.04.005.0 ig Y, φ, ; Brach poits With respect to φ For fixed,,, the fuctio Y, φ does ot have brach poits. 05.0.04.006.0 φ Y, φ With respect to For fixed geeric, φ, ;, the fuctio Y, φ has the set of brach poits where: cos, cos ad cos. For fixed φ ad itegers, the fuctio Y, φ does ot have brach poits.
http://fuctios.wolfra.co 0 Brach cuts With respect to φ For fixed,,, the fuctio Y, φ does ot have brach cuts. 05.0.04.007.0 φ Y, φ With respect to For fixed geeric, φ, ;, the fuctio Y, φ is a sigle-valued fuctio o the -plae cut alog the itervals cos ad cos. For fixed φ ad itegers, the fuctio Y, φ is a polyoial ad does ot have brach cuts. Series represetatios Geeralized power series Expasios at si 0 Y, φ 05.0.06.000.0 φ si si 4 5 3 4 8 4 4 si 4 8 3 ; si 0 05.0.06.000.0 Y, φ φ si k j j kj j j k j si k ; si k 0 j 0 05.0.06.0003.0 Y, φ φ si F 0 ; ; si F, ; ; si 05.0.06.0004.0 Y, φ φ si O si ; si 0 05.0.06.0005.0 Y, φ φ si cos k k k k si k k 0
http://fuctios.wolfra.co 05.0.06.0006.0 Y, φ φ ta si si k k si k k k 0 k 05.0.06.0007.0 Y, φ φ si k k si k k k 0 k 05.0.06.0008.0 Y, φ φ si k k si k k k 0 k Expasios at cos 0 05.0.06.0009.0 Y, φ φ csc si ta k k cos k 0 k k k 05.0.06.000.0 Y, φ φ csc si cot k k cos k 0 k k k 05.0.06.00.0 Y, φ φ si cos csc si k k cos k k k 0 k 05.0.06.00.0 Y, φ φ csc sec csc si k k cos k k k 0 k Expasios at ta 0 05.0.06.003.0 Y, φ sg φ cos csc si k k ta k k k k k 0 Expasios at cot 0
http://fuctios.wolfra.co 05.0.06.004.0 Y, φ sg φ si csc si k k cot kk k k k 0 Expasios at si 0 05.0.06.005.0 Y, φ Y, φ k k k k Expasios at cos 0 k k si k k k 05.0.06.006.0 Y, φ φ si csc si od k k k k cos k k Expasios at ta 0 05.0.06.007.0 Y, φ φ cos csc si od k k k k ta k k Expasios at cot 0 05.0.06.008.0 Y, φ φ si csc si k od k k k cot k k Expasios at 0 Y, φ 05.0.06.009.0 φ 3 440 30 45 3 7 5 4 36 880 63 4 3 5 945 3 945 6 4 7 5 4 6 O 8 ; 0 Y, φ 05.0.06.000.0 φ O ; 0
http://fuctios.wolfra.co 3 Expasios at cos 05.0.06.00.0 Y, φ φ cos si cos z 05.0.06.00.0 cos cos ; cos cos 3 cos Y φ, φ cos cos k k si k 0 k k cos k 05.0.06.003.0 Y φ, φ cos cos si F, ; ; cos 05.0.06.004.0 Y φ, φ cos cos O si ; cos Cos 05.0.06.005.0 φ Y, φ cos cos k k si k 0 k k cos k 05.0.06.006.0 φ Y, φ 3 cos cos O si cos ; cos I Cartesia coordiates 05.0.06.007.0 Y, φ x y z ijk, ij, x y j x y i z k ; i j k i j i 0 j 0 k 0 x cosφ si y siφ si z cos Itegral represetatios O the real axis
http://fuctios.wolfra.co 4 Of the direct fuctio 05.0.07.000.0 Y, φ cos cost φ si t t 3 3 0 05.0.07.000.0 Y, φ φ 4 csc P t t cos t ; 0 cos Y, φ 05.0.07.0003.0 3 φ 0 cos si cost cos tt Y, φ 05.0.07.0004.0 4 3 0 cos si cost φ t t 05.0.07.0005.0 Y, φ 3 φ si 0 cos si cost sit t ; 0 05.0.07.0006.0 Y, φ φ 4 0 t cos J t si t t ; cos 0 Ivolvig the direct fuctio 05.0.07.0007.0 Y, φ 0 J t si Multiple itegral represetatios 05.0.07.0008.0 Y, φ φ 4 csc J tt ; φ 0 si cos t 3 t P t t t t ; 0 Itegral represetatios of egative iteger order 05.0.07.0009.0 Y, φ r x y r z ; r x y z x r cosφ si y r siφ si z r cos 0
http://fuctios.wolfra.co 5 Y, φ 05.0.07.000.0 φ si z ; z cos z 05.0.07.00.0 Y, φ z φ si ; z cos z 05.0.07.00.0 φ Y, φ cos cos cot si si z z ; z z cos 05.0.07.003.0 Y, φ φ cos cos cot si si z z ; z z cos 05.0.07.004.0 Y, φ 4 φ si P z z ; z cos 0 05.0.07.005.0 Y, φ φ si z z ; z cos 05.0.07.006.0 Y, φ φ si z z ; z cos 05.0.07.007.0 Y, φ φ cos cot cot si z z ; z cos z 05.0.07.008.0 Y, φ φ cos cot si ta z z ; z z cos 05.0.07.009.0 Y, φ φ si P z ; z cos 0 z
http://fuctios.wolfra.co 6 Geeratig fuctios 05.0..000.0 w cos w si w 4 Y, 0 ; 0 w 05.0..000.0 w cos w si w 4 Y, 0 ; 0 w 05.0..0003.0 w cos w w w cos w si w 0 w Y, 0 ; Differetial equatios Ordiary liear differetial equatios ad wroskias For the direct fuctio itself With respect to 05.0.3.000.0 si si Y, φ si Y, φ 0 With respect to φ 05.0.3.000.0 Y, φ Y, φ 0 φ Partial differetial equatios φ 05.0.3.0003.0 Y, φ cot Y, φ φ Y, φ 05.0.3.0004.0 si si Y, φ Y, φ si φ Y, φ 0 Trasforatios
http://fuctios.wolfra.co 7 Trasforatios ad arguet siplificatios Arguet ivolvig basic arithetic operatios 05.0.6.000.0 Y, φ Y, φ ; φ 05.0.6.000.0 Y, φ φ Y, φ 05.0.6.0003.0 Y, φ Y, φ 05.0.6.0004.0 Y, φ φ Y, φ 05.0.6.0005.0 Y, φ Y, φ 05.0.6.0006.0 Y, φ Y, φ Products, sus, ad powers of the direct fuctio Products ivolvig the direct fuctio Clebsch-Gorda series for product of two spherical haroics 05.0.6.0007.0 Y, φ Y, φ k ax, k Y k, φ 0 0 k 0 k ; Clebsch-Gorda double series for product of three spherical haroics 05.0.6.0008.0 Y, φ Y, φ 3 Y 3, φ 3 4 k 3 k ax, k axk 3, 3 k Y k 3, φ 0 0 k 0 k 3 0 0 k 3 k 0 k k 3 3 k 3 k 3 ; j j j j j,, 3 Clebsch-Gorda ultiple series for product of several spherical haroics
http://fuctios.wolfra.co 8 p j 05.0.6.0009.0 j Y j, φ p p j k 3 j k p p k ax,m k axk 3,M 3 k p axk p p,m p k p M p, φk j j 0 0 k j j k j 0 k j j M j j k j j k j M j ; Y kp p j j k p p k k k k k 0 M 0 0 M j k Idetities Recurrece idetities Cosecutive eighbors 05.0.7.000.0 Y, φ 3 cos Y, φ 5 Y, φ 05.0.7.000.0 Y, φ cos Y, φ 3 Y, φ ; 05.0.7.0003.0 Y, φ φ cot Y, φ φ Y, φ 05.0.7.0004.0 Y, φ 0 φ cot Y, φ φ Y, φ ; Fuctioal idetities Relatios betwee cotiguous fuctios 05.0.7.0005.0 Y, φ sec Y, φ 3 Y, φ 05.0.7.0006.0 Y, φ ta φ Y, φ φ Y, φ ; 0
http://fuctios.wolfra.co 9 Additioal relatios betwee cotiguous fuctios Below relatios are correct oly uder soe restrictios o the paraeters 05.0.7.0007.0 Y, φ csc φ 3 Y, φ Y, φ 05.0.7.0008.0 Y, φ csc φ Y, φ 3 Y, φ 05.0.7.0009.0 Y, φ φ csc cos Y, φ Y, φ ; 0 05.0.7.000.0 Y, φ φ csc 3 Y, φ cos Y, φ 05.0.7.00.0 Y, φ φ csc Y, φ cos Y, φ ; 05.0.7.00.0 Y, φ φ csc cos Y, φ 3 Y, φ Relatios of special kid 05.0.7.003.0 Y, 0 φ Y, φ 05.0.7.004.0 Y, 0 Y, p ; p Coplex characteristics Cojugate value 05.0.9.000.0 Y, φ Y, φ ; φ Differetiatio
http://fuctios.wolfra.co 0 Low-order differetiatio With respect to 05.0.0.000.0 Y, φ cot Y, φ φ Y, φ 05.0.0.000.0 Y, φ 4 φ csc cos P cos P cos Y, φ 05.0.0.0003.0 cot csc Y, φ φ cot Y, φ φ Y, φ Y, φ 05.0.0.0004.0 With respect to φ 4 05.0.0.0005.0 Y, φ Y, φ φ Y, φ φ 05.0.0.0006.0 Y, φ φ csc cos P cos si P cos Sybolic differetiatio With respect to φ 05.0.0.0007.0 p Y, φ p Y φ p, φ ; p Fractioal itegro-differetiatio With respect to φ 05.0.0.0008.0 Α Y, φ φ Α φ Α QΑ, 0, φ Y φ Α, φ Itegratio Idefiite itegratio
http://fuctios.wolfra.co Ivolvig oly oe direct fuctio with respect to φ 05.0..000.0 Y, φφ Y, φ Ivolvig oe direct fuctio ad eleetary fuctios with respect to φ Ivolvig power fuctio 05.0..000.0 φ Α Y, φφ φ Α φ Α φ Α, φ Y, φ Ivolvig fuctios of the direct fuctio ad eleetary fuctios with respect to Ivolvig eleetary fuctios of the direct fuctio ad eleetary fuctios Ivolvig products of the direct fuctio ad trigooetric fuctios 05.0..0003.0 k k l si si Y k l, φ Y, φ l cos Y, φ Y k l, φ φ si k l k l k l k l Y, φ Y k l, φ Y, φ Y k l, φ Defiite itegratio Ivolvig the direct fuctio 05.0..0004.0 φ si Y, φ 0 ; 0 05.0..0005.0 φ si Y k, φ Y, φ 0 05.0..0006.0 0 csc Y, φ Y, φ 05.0..0007.0,k ; k 0 4, ; 0 0 0 si cos p Y, φ 4 p φ ; p 0 p p
http://fuctios.wolfra.co a b 05.0..0008.0 Y z cos z, φ Y cos z, φ z b Y cos b, φ Y cos b, φ Y cos b, φ Y cos b, φ a Y cos a, φ Y cos a, φ Y cos a, φ Y cos a, φ ; a b Multiple itegratio 05.0..0009.0 0 0 si Y, φ φ,0,0 05.0..000.0 si Y, φ Y, φ 3 Y 3, φφ 0 0 3 0 0 3 0 3 3 05.0..00.0 si Y, φ Y, φ 3 Y 3, φφ 0 0 3 0 0 3 0 3 3 ; 3 3 3 3 05.0..00.0 si Y, φ Y, φ 3 Y 3, φφ 0 0 3 3 0 0 0 3 3 ; 3 3 3 3 Orthoorality relatios: 05.0..003.0 si Y, φy, φφ,, ; 0 0 05.0..004.0 si Y, φ Y, φφ,, ; 0 0 05.0..005.0 si Y, φ Y, φφ,, ; 0 0 Suatio
http://fuctios.wolfra.co 3 Fiite suatio Ivolvig the direct fuctio 05.0.3.000.0 Y, φ Y, φ P cos cos cosφ φ si si ; k φ k k, 05.0.3.000.0 Y, φ ; φ 05.0.3.0003.0 Y, φ 0 ; φ 05.0.3.0004.0 Y, φ si ; φ 8 05.0.3.0005.0 Y, φ Y, φ 8 3 3 cos ; φ 05.0.3.0006.0 Y, φ cos si cosφ ; φ 05.0.3.0007.0 p l w pl l p l l l l Y l, φ p si φ w cos w p C p p cos w w cos w ; 0 Ivolvig Clebsch-Gorda fuctios The iverse Clebsch-Gorda series: 05.0.3.0008.0 im, k axm, k M k L M Y k, φ Y Mk, φ L M L L M L 4 L 0 0 L 0 Y L M, φ ; Ifiite suatio
http://fuctios.wolfra.co 4 05.0.3.0009.0 Y, φ w 0 φ w si φ 0 F ; ; w cos 0F ; ; w si ; 05.0.3.000.0 p p si φ w cos p F Y, φ w p, p ; ; w si w cos ; 0 p φ w 05.0.3.00.0 p p Y, φ w p p si φ w w cos w F p, p ; ; w F p, p ; ; w cos w ; 0 p p φ w 05.0.3.00.0 L z Y, φ w si φ w cos w z w cos w z w si exp 0F ; ; ; 0 φ w w cos w 4 w cos w 05.0.3.003.0 05.0.3.004.0 J w Y, φ w w si φ w cos ; 0 φ, J w Y, φ w w si φ cosw cos ; 0 φ 05.0.3.005.0, J w Y, φ w w si φ siw cos ; 0 φ
http://fuctios.wolfra.co 5 05.0.3.006.0 J w Y, φ Y, φ 0 k φ k k, w 4 3 J w si si w cos cos φ φ ; Multiple ifiite suatio Copleteess relatio: 05.0.3.007.0 Y, φ Y, φ φ φ cos cos ; k φ k k, 0 Represetatios through ore geeral fuctios Through hypergeoetric fuctios Ivolvig F 05.0.6.000.0 Y, φ cos φ F, ; ; si si Ivolvig F Y, φ 05.0.6.000.0 cos φ F, ; ; si si ; 05.0.6.0003.0 Y, φ φ ta csc si F, ; ; si ; 0 05.0.6.0004.0 Y φ, φ cos cos si F, ; ; cos 05.0.6.0005.0 Y, φ sg φ si F, ; ; si
http://fuctios.wolfra.co 6 05.0.6.0006.0 Y, φ sg φ si si F, ; ; csc ; 0 05.0.6.0007.0 Y, φ sg φ si F, ; ; cos 05.0.6.0008.0 Y, φ sg φ cos si F, ; ; sec ; 0 05.0.6.0009.0 Y, φ sg φ cos si F, ; ; ta 05.0.6.000.0 Y, φ sg φ si si F, ; ; cot Through Meijer G Classical cases 05.0.6.00.0 Y, φ 4 cos φ li si Ν G, si, si Ν Ν, Ν 0, ; Through other fuctios Ivolvig Legedre fuctios 05.0.6.00.0 Y, φ 4 05.0.6.003.0 Y, φ 4 φ P cos φ cos cos cos Ivolvig soe hypergeoetric-type fuctios 05.0.6.004.0 Y, φ φ cos cot si ta P, cos
http://fuctios.wolfra.co 7 05.0.6.005.0 φ Y, φ csc si ta C cos Represetatios through equivalet fuctios With related fuctios Ivolvig Wiger-D fuctios 05.0.7.000.0 Y, φ D 0, 0,, φ 05.0.7.000.0 Y, φ 05.0.7.0003.0 Y, φ D 0, 0,, φ D,0 φ,, 0 05.0.7.0004.0 Y, φ D,0 φ,, 0 Zeros Whe Y, φ is ot idetically zero, it possesses a fiite uber of zeros i the iterval 0, all of which are odegeerate. For itegers ad with, the fuctio Y, φ has zeros i the iterval 0. If 0, there are also two ore zeros at 0,. All of these zeros are syetric about. Theores Eigefuctio to the agular part of the Laplace operator i spherical coordiates The fuctio Y, φ is a eigefuctio to the agular part L of the Laplace operator i spherical coordiates L si φ si si with eigevalue. Eigefuctio of the z-copoet of the quatu echaical agular oetu operator The fuctio Y, φ is a eigefuctio to the z-copoet of the quatu echaical agular oetu operator L z φ with eigevalue.
http://fuctios.wolfra.co 8 Eigefuctios of the curl operator i spherical coordiates The fuctio ur,, φ Λ r Ψr,, φ r Ψr,, φ with Ψr,, φ gry, φ ad gr Λ c J Λ r Λ r ur,, φ Λ ur,, φ. Multiple expasio theore Y Λ r are eigefuctios of the curl operator i spherical coordiates Λ r Ay fuctio f, φ that is square itegrable over 0, 0 φ ca be expaded i a series of spherical haroics Y, φ, with series coefficiets a, called the ultipole oets: f, φ 0 a, Y, φ ; a, 0 0 si Y, φ f, φφ. Expasio of two-poit distaces The power r r of the distace betwee two poits r ad r ca be expaded i the followig way (assuig r r ad Ω is the agle betwe r ad r ) (Sack 964): r r r l 0 l l r r l F l, ; l 3 ; r r P l cosω History A.M. Legedre (785); P.S. Laplace (785) gave the ae spherical haroic ; K.F. Gauss (88). Refereces L.C. Biedehar ad J.D. louck, Agular Moetu i Quatu Physics, Addiso-Wesley, Readig, 98. L.C. Biedehar ad J.D. louck, The Racah-Wiger Algebra i Quatu Theory, Addiso-Wesley, Readig, 98. E.W. Hobso, The Theory of Spherical ad Ellipsoidal Haroics, Cabridge Uiversity Press, Cabridge, 955. M.E. Rose, Eleetary Theory of Agular Moetu, Dover, New York, 995. D.A. Varshalovich, A.N. Moskalev ad V.K. Khersoskii, Quatu Theory of Agular Moetu, World Scietific, Sigapore, 988.
http://fuctios.wolfra.co 9 Copyright This docuet was dowloaded fro fuctios.wolfra.co, a coprehesive olie copediu of forulas ivolvig the special fuctios of atheatics. For a key to the otatios used here, see http://fuctios.wolfra.co/notatios/. Please cite this docuet by referrig to the fuctios.wolfra.co page fro which it was dowloaded, for exaple: http://fuctios.wolfra.co/costats/e/ To refer to a particular forula, cite fuctios.wolfra.co followed by the citatio uber. e.g.: http://fuctios.wolfra.co/0.03.03.000.0 This docuet is curretly i a preliiary for. If you have coets or suggestios, please eail coets@fuctios.wolfra.co. 00-008, Wolfra Research, Ic.