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3 . y B K K φ θ A F O F A x F K A F, F O A F K A S FK A OK A S OK A OKA S OKA OB OA S OK A OB OA S OKA OB OA φoa OB OA φ OK F S OK F OKF S OKF OB OA S FK A OF OA S OK F OB OA S OKF OB OA OF OA sin φ OB OF sin φ S FK A S OK A S OK F OB OA φ OF OA sin θ ε S FK A OB OA φ ε sin φ

4 T πoa OB S FK A t φ ε sin φ π T t πoa OB T π T O π T t O π T t mean anomaly φ eccentric anomaly. φ t nt ε φ ε sin φ nt Kepler s equation fnt φ nt ε sin φ fnt + π ε sin nt + π + ε sin φ ε sin nt + ε sin φ f nt ε sin nt + ε sin φ ε sin nt ε sin φ ε sin nt + ε sin nt ε sin φ fnt π fnt φ nt + fnt φ fnt fnt a k π a k sin knt k a k ε sin φ sin kntnt π 0 π k [ε sin φ cos knt]π 0 + π ε sin φ cos kntnt k 0 nt π φ nt cos kntnt πk 0 nt π π cos kntφ cos kntnt πk 0 0 π cos kφ ε sin φφ πk. 0 k N x R J k x π π 0 cos kφ x sin φφ J k x Bessel functions of the first kin Bessel function

5 3 φ φ nt + fnt nt + k k J kkε sin knt.. k N x R J k x J k x π e ikφ x sin φ φ π e ix sin φ kφ φ π π π π J k x π π π π π π π π 0 π cos kφ x sin φφ e kφ x sin φ + e kφ x sin φ φ 0 π π e kφ x sin φ φ + e kφ x sin φ φ 0 π e kφ x sin φ φ + 0 π e kφ x sin φ φ + 0 π π π π e ikφ x sin φ φ e ix sin φ kφ φ 0 0 π 0 π e kφ x sin φ φ e kφ x sin φ φ. k N x R J k x J k x m0 m x m+k m + k!

6 4 e ix sin φ e ix sin φ ix sin φ m m0 m0 m0 m0 ix m x m m x e iφ e iφ m l0 i m l m m l0 m e iφl e iφm l m l m m l e iφl m l. J k x π e ix sin φ kφ φ π π x m m m π m l π l m0 l0 x m m m π m l π l m0 π π l0 e iφl m k φ e iφl m e ikφ φ π e iφl m k φ π { π l m k 0 0 other l l m+k J k x π m0 m0 x m m m+k x r m k m r + k J k x r0 r0 r0 m m m+k m+k m π m k x r+k r + k r r + k! r + k x r+k r + k! r + k! r!r + k! r r x r+k r!r + k!.3 k N x R J k x x xk J k x x k J k x, x x k J k x x k J k+ x

7 5 J k x J k+ x J kx, J k x + J k+ x k x J kx x xk J k x m m+k x m+k x m + k! m0 m m+k m + kx m+k m + k! m0 x k m m0 m m + k! m+k x m+k x k J k x x x k J k x m m+k x m x m + k! m0 m m+k mx m m + k! x k m x k m0 x k J k+ x m m!m + k! m+ m + k +! m+k x m+k m+k+ x m+k+ x xk J k x kx k J k x + x k J kx x k J k x k x J kx + J kx J k x x x k J k x kx k J k x + x k J kx x k J k+ x k x J kx + J kx J k+ x J k x J k+ x J kx, J k x + J k+ x k x J kx x xk J k x x k J k x, x x k J k x x k J k+ x.3.3 k N x R J k x x xk J k x x k J k x, x x k+ J k x x k+ J k x

8 6 J k x x k+ x x xk J k x x k+ J k x x k+ x x xk k+ J k x x x xk J k x + k + x k x xk J k x y fx x k+ kk x k J k x + kx k J kx + x k J k x + k + x k kx k J k x + x k J kx kk x k J k x + kx k J kx + x k+ J k x + k + kx k J k x + x k J kx x k xj k x + J kx x k J k x x k xj k x + J kx x k J k x x k+ J k x x J k x + xj kx k J k x x J k x x J k x + xj kx + x k J k x 0 x y x + x y x + x k y 0 J k x.3 z C w fz ν C z w z + z w z + z ν w 0 c 0, c, Cc 0 0 fz z ρ c c 0 k0 c k z k z ν w z ρ c k ν c k z k z w z zρ z w z zρ k0 k + ρc k z k k0 k + ρk + ρ c k z k z ρ+k k0 c k ν c k + k + ρc k + k + ρk + ρ c k 0 c k ν c k + k + ρc k + k + ρ c k k + ρc k 0 k + ρ ν c k c k

9 7 c k 0 k, c k 0 k 0 k + ρ ν c k c k other c 0, c, c 0 0 k 0 ρ ν 0 ρ ±ν ρ ν k 0 k + ν ν 0 c k+ 0 ρ ν k mm Z c m m + ν ν c m m + ν ν m + 4mν mm + ν m + ν ν m + 4m ν + ν ν c 0 + 4ν c 0 m m + ν + ν c 0 m m m + νm + ν + ν c 0 c 0 c m c 0 ν Γν + m ν+m Γν + + m ρ µ c m ν ν f ν z m0 m z ν+m, f ν z Γν + + m n Z ν n lim ν n m0 0 ν + + m 0 Γ ν + + m f n z n + + m 0 0 f n z mn m0 n m z n+m Γ n + + m m+n m + n!γ n + + m + n z m0 n f n z m z m+n Γm + + n m z ν+m Γ ν + + m n+m+n ν x 0 m 0 c m+ c m Γν + + m z m +!Γν + + m + z m + ν + + m c m+ lim 0 m c m

10 8 0 C.4 ν C z C J ν z m0 m z m+ν Γν + + m J ν z.4 n Z z C J n z J n z n J n z.5 ν C z C J ν z 0 C ν z 0 Re ν Re ν > 0 z ν J ν z ν m z m Γν + + m m0 z 0 Re ν.3 f ν z J ν z, f ν z J ν z W J ν z, J ν z J νz J νz z J ν z J νz z W J νz, J ν z J νzj νz + J ν zj νz J νzj νz J ν zj ν z J ν z z J νz ν J ν z J ν z z J νz ν J ν z z J νzj νz J ν zj νz z W J νz, J ν z C C z W J ν z, J ν z C z

11 9 W J ν z, J ν z C z z ν J ν z Γν + + Oz J νz z ν ν Γν + + Oz z ν Γν + Oz z ν + Oz W J ν z, J ν z J ν zj νz J ν zj νz z Γν + Γ ν Γ ν + Γν + Oz sin πν + z sin πν π π sin πν πz ν Z W J ν z, J ν z 0 J ν z J ν z ν Z.4 J ν z J ν z ν Z J ν z J ν z Y ν z Y ν z J ν z J ν z Y ν z W J ν z, J ν z sin πν Y ν z Y ν z J ν z J ν z C, C C Y ν z C J ν z + C J ν z W J ν z, Y ν z J ν zy νz Y ν zj νz J ν zc J νz + C J νz C J ν z + C J ν zj νz C J ν zj νz J ν zj νz + C J ν zj νz J ν zj νz C W J ν z, J ν z W J ν z, Y ν z J ν zy νz Y ν zj νz J ν zc J νz + C J νz C J ν z + C J ν zj νz C J ν zj νz J ν zj νz + C J ν zj νz J ν zj νz C W J ν z, J ν z C C sin πν J ν z J ν z sin πν ν Z sin πν 0 ν Z C C Y ν z J ν z W J ν z, Y ν z C W J ν z, J ν z πz C sin πν C ν Z C W J ν z, Y ν z C W J ν z, J ν z C C sin πν

12 0 Y ν z C J ν z J ν z sin πν n Zn 0 ν n 0.4 lim ν n C J ν z J ν z C J n z n J n z C n J n z ν n C n C C cos πν ν n lim Y cos πνj ν z J ν z νz lim ν n ν n sin πν lim cos πν ν n π cos πν n n π ν J νz νn ν J νz νn π ν J νz n νn ν J νz νn ν J νz π sin πνj ν z ν J νz J ν z J ν z ν J ν z J ν z ν J νz J ν z log z + m0 m0 m z ν Γν + + m J ν z log z m ψν + + m z m+ν Γν + + m ν J νz J ν z log z + m0 m m+ν ψν + + m z m+ν Γν + + m ψν + + m ν n lim ν n lim ν n ν J νz J n z log z m0 m ν J νz n J n z log z + m0 n J n z log z n + + mn m0 ψn + + m z m+n Γn + + m m m m ψ n + + m z m n Γ n + + m n J n z log z n + + n m0 m0 m m ψ + m m + n! Γ + m ψ n + + m z m n Γ n + + m ψ n + + m z m n Γ n + + m ψ n + + m z m n Γ n + + m z m+n J ν z ψ n + + m m < n

13 k Zx 0 lim ν n ψ k ψk + π cot πk ψ k + π cot πk ψk Γ k Γ k ψk + π cot πk Γk sin πk π ψkγk sin πk + Γk cos πk π k! k ν J νz n J n z log z n + + n m0 m0 m m ψ + m m + n! Γ + m n J n z log z n + n + n m0 m0 m ψ + m m + n! Γ + m n m! n m z m n z m+n n m! Y ν z ν n lim ν n Y νz π J nz log z π m m0 + π J nz log z π n m0 z ψn + + m n + n m! m+n z m n z m+n z m n m ψ + m z m+n π m + n! m0 π J nz log z m z m+n π n + ψn + + m + ψ + m π n m0 m0 n m! z m n.5 ν C z C J ν z Y ν z cos πνj νz J ν z sin πν Y ν z Bessel functions of the secon kin Neumann function N ν z ν Z n Z ν n Y ν z n 0 lim ν n Y νz π J nz log z π π n m0 m0 n m! m z m+n n + ψn + + m + ψ + m z m n

14 .6 ν C z C J ν z Y ν z.7 ν C z C Y ν z.3 z zν Y ν z z ν Y ν z, x z ν Y ν z z ν Y ν+ z. ν C z C Y ν z ν C ν J νz lim ν ν ν J ν z.8 n Z z C Y n z Y n z n Y n z. lim Y cos πνj ν z J ν z νz lim ν n ν n sin πν lim ν n n π n π n π cos πν π cos πν ν J νz π sin πνj ν z n ν J νz ν n ν J νz ν n n ν J νz + νn ν J νz νn ν J νz n νn ν J νz νn n lim ν n Y ν z ν J νz

15 3.6.6 ν C z C J ν z Y ν z H ν z J ν z + iy ν z, H ν z J ν z iy ν z H ν z, H ν z Hankel function.9 ν C z C H ν z, H ν z z zν H ν z z ν H ν z, z zν H ν z z ν H ν z, x z ν H ν z z ν H ν+ z x z ν H ν z z ν H ν+ z.3.7. ν C z C J ν z H ν z, H ν z H ν z J νz e iπν J ν z i sin νπ, H ν z eiπν J ν z J ν z i sin νπ H ν z J ν z + i cos νπj νz J ν z sin νπ i i sin νπj νz + cos νπj ν z J ν z sin νπ i e iνπ J ν z J ν z sin νπ J νz e iπν J ν z i sin νπ H ν z J ν z i cos νπj νz J ν z sin νπ i i sin νπj νz cos νπj ν z + J ν z sin νπ i eiνπ J ν z + J ν z sin νπ eiνπ J ν z J ν z i sin νπ

16 4.4.7 z C w fz ν C z w z + z w z z + ν w 0 z iz iz w fiz + iz iz iz iz + ν fiz 0 z w fiz + z z z z + ν fiz 0 z w fiz + z z z + z ν fiz 0 J ν iz i ν m0 m z m+ν Γν + + m z R J ν iz I ν z i ν J ν iz m0 m z m+ν Γν + + m z z ν arg z arg iz π, π] iz ν e ν log iz νlog z +arg iz e i ν z ν e ν log i e ν log z νlog z +arg i+arg z e arg iz arg i + arg z πν i I ν z e Jν e i π z π < arg z π I ν z e i 3πν Jν e i 3π z π < arg z π I ν z I ν z I νz i ν+ J νiz W I ν z, I ν z I ν zi νz I ν zi νz ij ν izj νiz J ν izj νiz iw J ν iz, J ν iz sin πν πz

17 5 ν Z I ν z I ν z ν Z I ν z i ν J ν iz i ν J ν iz i ν J ν iz I ν z I ν z K ν z C, C C K ν z C I ν z + C I ν z W I ν z, K ν z C W I ν z, I ν z, W I ν z, K ν z C W I ν z, I ν z C C π sin πν K ν z π I νz I ν z sin πν. K ν z iπ ei π ν J ν iz e i π ν J ν iz i sin πν K ν z π iν+ H ν iz, K ν z π i ν+ H ν iz K ν z n Z ν n lim K π νz lim ν n ν n iν+ H ν iz π lim ν n in+ J ν iz + iy ν iz π in+ J n iz + i n+ J n iz log iz in+ in+ n m n n m! iz m0 i π log i log z i n J n iz n+ + n m n m! z m n m0 m0 n m n m! n+ m0 m0 n m n m! n+ m0 z m0 m0 m n log z n I n z z m+n n + ψn + + m + ψ + m z m n log z n m m+n iz ψn + + m + ψ + m n + z m+n n + ψn + + m + ψ + m m0 n + ψn + + m + ψ + m z m z m+n n + m+n n m n m! z m n m0 + n+ log z z m+n n + ψn + + m ψ + m m0

18 6.8 ν C z C I ν z m0 m z m+ν Γν + + m I ν z moifie Bessel function of the first kin J ν z arg z arg iz π, π] πν i I ν z e Jν e i π z π < arg z π I ν z e i 3πν Jν e i 3π z π < arg z π z R I ν z i ν J ν iz K ν z π I νz I ν z sin πν π iν+ H ν iz π i ν+ H ν iz K ν z moifie Bessel function of the secon kin ν Z n Z ν n K ν z n 0.0 lim K νz n m n m! z m n ν n m0 + n+ log z z m+n n + ψn + + m ψ + m m0 ν C z C I ν z K ν z. ν C z C I ν z K ν z z zν I ν z z ν I ν z, z zν K ν z z ν K ν z, x z ν I ν z z ν I ν+ z x z ν K ν z z ν K ν+ z I ν z.3 K ν z z zν K ν z z ν π I ν z I ν z z ν π sin πν I ν z I ν z z ν K ν z sin πν x z ν K ν z z ν π I ν+z I ν+ z z ν π sin πν I ν+z I ν+ z z ν K ν+ z sin πν +

19 7.5 ψ R 3 k R + k ψ 0 ψ { x r cos θ y r sin θ ψr, θ, z RrΘθ r R Θ + Θ r r r r θ R + k RΘ 0 r r R Θ r θ r + R Θ + r k 0 θ r R kr x r r r R r + r k R r r r R r Θ θ Θ λ constant r R + r k λ R 0 r + r R r + r k λ R 0 r x r x k x x R x + xr x + x λ R 0 Rr J λ kr ψ x r sin θ cos ϕ y r sin θ sin ϕ z r cos θ ψr, θ, ϕ RrΘθΦϕ Rr r r R ΘΦ + r r r sin θ sin θ Θ RΦ + θ θ r sin θ r r r R r R r sin θ + sin θ Φ ϕ RΘ + k RΘΦ 0 Φ ϕ Θ + k r sin θ Φ ϕ Φ

20 8 ϕ Φ ϕ Φ µ constant r sin θ r r r θ R r r r kr x r R Φ r ϕ + sin θ R r r Θ + k r sin θ µ r R r R r r R r + k r µ R sin θ sin θ r R r r r r + k r µ sin θ sin θ Φ ϕ Θ r R r + k r λ R R + r r + k r λr 0 x R x + xr x + x λr 0 λ constant Rx x Rx Φ ϕ Θ x 3 4 x 5 R x 3 R x + x x 3 4 x R x R x + R x R x + x x 3 R + x + x R + x λx R 0 x x R + R x x R R + x x x R R + x x x + + x λ R 0 x 4 R + x λ R 0 x 4λ R 0 4 R λ νν + R 4λ 4 νν + 4 ν Rr x Jν kr

21 9.9 z C w fz ν C z w w + z z z + z νν + w 0 π j ν z z J ν z spherical Bessel function of the first kin spherical Bessel function y ν z π z Y ν z spherical Bessel function of the secon kin spherical Neumann function n ν z π π h ν j ν z + iy ν z z H z, h ν ν j ν z iy ν z z H z ν spherical Hankel function

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou (Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fourier) (Fourier Bessel).. V ρ(x, y, z) V = 4πGρ G :.

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