J. Bernoulli 694 Riccti dy dx + ψy + φy + χ = (ψ, φ, χ x ) Leibniz Riccti 73 Leibniz Bessel ( )Bessel. 738 J. Bernoulli. 764 Novi Comm. Acd. Petrop. L

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1 Bessel ) Lommel Bessel ( + ) Γ. Γ Γ ( ) Bessel I: 4 3. Bessel J n (z) (n Z) Bessel II: & Fourier-Bessel 3 5. Bessel J ν (z) Fourier-Bessel Reference37 ( + ) Bessel

2 J. Bernoulli 694 Riccti dy dx + ψy + φy + χ = (ψ, φ, χ x ) Leibniz Riccti 73 Leibniz Bessel ( )Bessel. 738 J. Bernoulli. 764 Novi Comm. Acd. Petrop. L. Euler Hist. de l Acd. R. des Sci. de Berlin Lngnge Kepler 4. 8 Mém. de l Acd. des Sci. Fourier (L Théorie nlytique de l chleur) Jour. de École Polytechnique Poisson Bessel Bessel 84 Berliner Abh. Bessel Untersuchung des Theils der plnetrischen Störungen, welcher us der Bewegung der Sonne entsteht. Kepler Bessel systemtic 8 Hnkel, Lommel, Neumnn, Schläfli [W] Bessel 8 Bessel Γ x! Γ [AAR]

3 . Γ C meromorphic function x Z f(x) = x! f(x) x Z (x + n)! x! = (x + )(x + ) (x + n) n!(n + )(n + ) (n + x) = (x + )(x + ) (x + n) n!n x (n + )(n + ) (n + x) = (x + )(x + ) (x + n) n x (n + )(n + ) (n + x) lim n n x = lim x n k= n!n x f(x) = lim n (x + )(x + ) (x + n) ( + k ) = n () x C\Z < k Z > f(k) = k! ( n!n x n (x + )(x + ) (x + n) = n + ) x n j= ( + x ) ( + ) x j j ( + x ) ( + ) x ( ) x(x ) = + + O j j j j 3. ) x C \ Z <. C meromorphic function Γ(z) ( ) n!n x Γ(x) := lim n x(x + )(x + ) (x + n ) n!n x = lim n x(x + )(x + ) (x + n) () Euler Γ(x + ) = xγ(x), (3) Γ(n + ) = n! (n Z ). (4) 3

4 . (3), (4) () Γ(x) x Z Γ(x) Weierstrß. (Weierstrß ) x C {( Γ(x) = xeγx + x ) } e x n n γ = lim n Euler n= ( n k= ) k log n (5) Γ(x) = lim x(x + ) (x + n) n n!n x = lim n x ( + x = lim n xe x(+ + =xe γx n= ) ( + x n log n) n ) ( + x ) x log n e n k= {( + x ) } e x n n {( + x ) } e x k k x C (.3 ) Euler reflection formul. x Z Γ(x)Γ( x) = π sin πx. (6) Weierstrß (5) Γ(x)Γ( x) = ( x)γ(x)γ( x) =xe γx e γx =x =x n= n= {( + x n ) ( x n= n {( + x ) e x n n ) } {( e x n x n sin πx = π 4 } n= ) e x n {( x ) } e x n n }

5 3, (.4 ) dupliction formul.3 x C \ Z Γ(x)Γ(x + ) = π x Γ(x). (7) Weierstrß (5) log Γ(x) = log x + γx + k> { ( log + x ) x } k k C \ Z (.5 ) d dx log Γ(x) = x + γ + { x + k } k k> C \ Z d dx log Γ(x) = k (x + k) F (x) := Γ(x)Γ(x + ) x C \ Z d dx log F (x) = k =4 k = d (x + k) + (x + + k k) (x + k) log Γ(x) dx A, B F (x) = AB x Γ(x) A, B (3) F (x+ ) = Γ(x+ )Γ(x+) = xf (x) F (x + ) = ABx+ Γ(x + ) = B xf (x) B = F (x) = A x Γ(x) F ( ) = Γ( )Γ() = A Γ() A = Γ( ) reflection formul (6) Γ( ) = π 7) Guß.4 (Guß) n Z > x C \ Z n ( Γ(x)Γ x + ) ( Γ x + n ) = (π) n n nx Γ(nx). (8) n n.6 (Hint: Stirling 5

6 . Γ Stirling t = te xt Tylor e t. (Bernoulli). n Z B n (x) Bernoulli te xt e t = n= B n (x) tn n! (bout t = ).. Bernoulli B n := B n () Bernoulli.5. B = B k+ = (k Z > ) B =, 6 B 4 =, B 3 6 = 4. B (x) = x, B (x) = x x + 6. k Z > n = ( )k+ k B k k π k (k)! n> (.7 ) Hint x cot x = x + x cot x x cot x = + n= x e x x x n π.6 (Euler-Mclurin) f C s - (s Z > ) n k=m+ f(k) = n m f(x)dx + + ( )s s! s l= n m ( ) l B l l! {f (l ) (n) f (l ) (m)} B s (x [x])f (s) (x)dx. x R [x] x 6 (9)

7 .7 n Z > x R -periodic B n (x). B (x) := x [x],. n > B n(x) = B n (x) B n (x)dx = B n (x) = n! B n(x [x]) x..7 B (x) := G(x, t) := n= B n (x)t n < x < x G x = n= B n(x)t n = n= B n (x)t n = t n= B n (x)t n = tg G(x, t) = A(t)e xt x B n (x)dx = G(x, t) = = A(t) et t text e t = j j+ n= B n (x) tn n! < x < B n.6 s s = m j < n j Z j+ [( d (f(j) + f(j + )) = x j ) ] f(x) dx dx = j f(x)dx + j+ j ( x j ) f (x)dx j = m, m +,, n n k=m+ f(k) + n (f(m) + f(n)) = f(x)dx + m 7 n m ( x [x] ) f (x)dx

8 B = B (x) = x n n n f(k) + B (f(n) f(m)) = f(x)dx + B (x [x])f (x)dx k=m+ m s =.7 B n (x) n n s f(k) = f(x)dx + ( ) l Bl (){f (l ) (n) f (l ) (m)} k=m+ m + ( ) s n m l= B s (x)f (s) (x)dx..7.8 z C \ (, ] ( log Γ(z) = z ) m B j log z z + j(j ) z j () + Γ(z) = lim n log Γ(z) = lim n log π m [ n l= j= l z + l (z ) log(n + ) m B m (x [x]) dx. (x + z) m ( l + l n l= ) x ] log z + l l log brnch, C \ (, ] f(x) := log x + z = log(x + z ) log x x.6 n l= log z + l l = log z + = log z + + m l= n l= n log z + l l {log(x + z ) log x}dx ( ) l B l l! {f (l ) (n) f (l ) ()} + ( )m (m)! 8 n B m (x [x])f (m) (x)dx ()

9 .5 k Z > f (k) (x) = ( ) k (k )![(x + z ) k x k ] n l= log z + l l = log z + + m j= n {log(x + z ) log x}dx B j j(j ) + [log(n + z ) log n log z] + n [ B m (x [x]) m [ (n + z ) j n ] j z + j (x + z ) m x m n (n + z ) log(n + z ) n log n + log n + z n m [ B j + j(j ) (n + z ) ] j n j j= ] dx. (z ) log(n + ) n () ( log Γ(x) = z ) m ( ) B j log z z + + j(j ) z j j= [ B m (x [x]) m (x + z ) ] () dx. m x m [ ( lim log Γ(z) z ) ] Γ(z) log z + z = lim log z z z z e z () m :=.5.7 [ ( lim log Γ(z) z ) ] log z + z = z + B (x [x]) x dx = + B (x [x]) x dx (7) lim z Γ(z) Γ(z) = lim z z e z z (z) z e =(π) =(π) lim z { lim z = (π) z Γ(z) z z e z Γ(z) z z e z z z lim Γ(z + ) (z + )z e z } Γ(z)Γ(z + ) (z) z e z e ( + z ) z 9

10 [ ( lim log Γ(z) z ) ] log z + z = log π z () z m j= B j j(j ) + m B m (x [x]) dx = log π x m ().8 Rex (.8 ( ) B m (x [x]) dx = O (x + z) m x m.9 (Stirling-de Moivre) m Z > Γ(z) =(π) z z e z+ P m j= (π) (z ) z e (z )+ P m j= B j j(j ) z (j ) +O(z (m+) ) B j j(j ) z (j ) +O(z (m+)). rg z = π δ δ > Γ(z) (π) (z ) z e (z ) Stirling z.3 ( ) s R > Γ Γ(s) Γ(s) := e x x s dx x () Res > s C well-defined Γ C \ Z Euler () s R > n Z > ( x ) n e x x n n

11 n Γ(s) = lim n ( x n) n x s dx (3) (3) z = nx n ( x ) n x s dx =n s ( z) n z s dz n =n s B(n +, s) s Γ(n + )Γ(s) =n Γ(n + + s) n s n! = s(s + )(s + ) (s + n) (3) (3) Euler () B(p.q) Euler Bet Rep > Req > B(p, q) = ( x) p x q dx (4) B(p, q) = Γ(p)Γ(q) Γ(p + q).9 (5) p, q R > (5). e x y x p y q (x, y) R > ( ). Fubini Γ(p)Γ(q) = = e x x p dx e y y q dy e x y x p y q dxdy x = X, y = Y 3. X = R cos θ, Y = R sin θ x = sin θ, y = R () e x x s x = Res > Γ s C \ Z () s C \ Z e x x s Γ

12 . s C \ Z Γ(s) = e z z s dz e π s C z (6) () C Hnkel rg z = rg z = π : C Res > () e z z s C \ {} Cuchy C ε C rg z = C rg z = π C 3 C :=C C C 3, C =C 3 := (ε, ], C := {z C z = ε}, ε R > e z z s dz = e z z s dz C C e z z s dz = C e z z s dz = C 3 ε ε e z z s dz, e z (e π z) s dz = e π s z = εe θ e z z s dz = π e θ(εe εe θ ) s dθ C ε e z z s dz

13 ε θ π e εe θ(εe θ ) s ε Res e ε+π Ims limit (. ) C e z z s dz e z z s dz (e π s ) e z z s dz C C 3 e z z s dz = lim e z z s dz = (e π s ) C ε C Γ(s) e z z s dz. s C \ Z Γ(s) Γ(s) = π e z z s dz. (7) C C Hnkel rg z = π rg z = π : C Hnkel ( 6) s Γ(s) = eπ e π s C = sin πs reflection formul ( 6) Γ(s) = Γ(s)Γ( s) π C e z (e π z) s dz z C e z (e π z) s dz z. e z (e π z) s dz z w = e π z Γ( s) = π e w w s dw γ s s 3

14 . w = e π z w = e π z. Bet B(p, q) p, q Z B(p, q) = z p ( z) q dz. ( e π p )( e π q ) c c A A rg z = rg( z) = brnch Pochhmmer 3 Bessel I: n Z n Bessel J n (z) [W] 3. Bessel J n (z) (n Z) z C t C e z(t t ) t = Lurent e zt t e z t t 3. n Z J n (z) e z(t t ) = n Z J n (z)t n (8) J n (z) n Bessel t t (8) e z(t t ) = J n (z)( t ) n n Z = J n (z)( t) n n Z n Z J n (z) = ( ) n J n (z) (9) 4

15 J n (z) J n (z) = π e z(t t ) t n dt. () ζ 3. ζ C () t = e θ J n (z) = π e (z sin θ nθ) dθ = π cos(z sin θ nθ)dθ. () π π π 3. cos J n (z) = π π {cos(z sin θ) cos nθ + sin(z sin θ) sin nθ}dθ θ π θ π cos(z sin θ) cos nθdθ π J n (z) = π sin(z sin θ) sin nθdθ π n : even, n : odd, [, π] [ π, π] J n (z) = π π π π cos(z sin θ) cos nθdθ sin(z sin θ) sin nθdθ n : even, n : odd, J n (z) e zt e zt () e z(t t ) = ( z) r t r r= r! m= ( z) m t m m! n Z t n ( z) n+m ( z) m, (n + m)! m! m= n Z J n (z) = ( ) ( m z) n+m, (3) m!(n + m)! m= 5

16 n Z < Γ J n (z) = m= ( ) m ( z) n+m m!γ(n + m + ), (4) n Z 3.3 (4) (9) n Z (3) J n (z) order z C n J n (z) z z m m!(n + m)! m= z n z m n! m!(n + ) m m= J n (z) z n ( ) 4 exp z n! n + z n ( ) exp n! 4 z (5) 3. {J n (z)} (8) Bessel J n (z) ide 3. n Z J n (z) + J n+ (z) = n z J n(z), J n (z) J n+ (z) = J n(z), (6) (8) t z ( 3.4 (6) zj n(z) + nj n (z) = zj n (z), zj n(z) nj n (z) = zj n+ (z), d dz {zn J n (z)} = z n J n (z), d dz {z n J n (z)} = z n J n+ (z). 6

17 n n d dz {zn J n (z)} = z n J n (z), d dz {z n J n (z)} = z n J n (z), J n (z) [ d z n d ] dz dz {zn J n (z)} = z n J n (z) 3. n Bessel J n (z) z d w dz + z dw dz + (z n )w =. (7) (7) J n (z) n n J n (z) (9) (8) e x(t t ) e y(t t ) = e (x+y)(t t ) r Z J r (x)t r m Z J m (y)t m = n Z J n (x + y)t n t n 3.3 n Z J n (x + y) = m Z J n m (x)j m (y). (8) 3.5 e z(t t ) e z( t+ t ) = J (z) + J r (z) =. r= 7

18 3.3 Poisson 83 Jour. Éc. Polytech. Bessl J n (z) n Z n Z ( J n (z) = z) n π Γ(n + )Γ( ) cos(z cos θ) sin n θdθ (9) Poisson cos(z cos θ) ( z) n Γ(n + )Γ( ) π r= ( ) r (r)! (z cos θ)r sin n θdθ ( 3.6 ) ( z) n Γ(n + )Γ( ) r= (9) π (3) Γ(r + ) ( Γ( ) = r ) cos r θ sin n θdθ = ( ) r π (r)! zr cos r θ sin n θdθ π cos r θ sin n θdθ =B(r +, n + ) = Γ(r + )Γ(n + ) Γ(r + n + ) ( ) n z ( ) r Γ(r + ) z r (r)! Γ( ) Γ(r + n + ) r= ( r 3 ) (r )(r 3) = = (r)! r r r! (9) (4) (9) ( J n (z) = z) n π Γ(n + )Γ( ) e z cos θ sin n θdθ θ π θ ( 3.7 ) t = cos θ ( J n (z) = z) n Γ(n + )Γ( ) e zt ( t ) n dt (3) 8

19 (9) () n Z () θ π θ J n (z) = ( ) n π π cos(z cos θ) cos nθdθ 3. π 3.8 cos nθ = π cos(z cos θ) sin n θdθ = = π cos(z cos θ) sin n θdθ cos(z sin θ) cos n θdθ n ( ) m 4n {4n } {4n (m ) } sin m θ (m)! m= 3.9 (9) J n (z) = ( ) n π n ( ) m 4n {4n } {4n (m ) } (m)! m= (3) (6) Γ(m + )Γ( ) π(m)! ( z) m = ( m ) ( ) m 3 (m)! ( z) m (m )(m 3) 3 = (m)!z m = m m!z m Lommel 868 Mth. Ann. J n (z) = ( ) n n m= Γ(m + )Γ( ) ( z) m J m (z) ( ) m 4n {4n } {4n (m ) } J m (z) m m! z m 4 Bessel II: ν C Bessel J ν (z) 9

20 4. (7) z d w dz + z dw dz + (z ν )w =. (3) w(z) = z ρ k= c k z k c (3) z d w dz = zρ (ρ + k)(ρ + k )c k z k, z dw dz = zρ k= (ρ + k)c k z k, k= (z ν )w = z ρ (c k ν c k )z k, c = c = z ρ+k (k =,,, ) k= {(ρ + k) ν }c k = c k (3) c ρ ν = ρ = ±ν ρ = ν ν C (3) k = c = c k+ = (k Z ) k k = l Z > (3) c l = = (ν + l) ν (ν + 4) ν (ν + ) ν c ( ) l l l!(ν + l) (ν + )(ν + ) c c l = c := ν Γ(ν + ) ( ) l ν+l l!γ(ν + l + ) ρ = ν ν ν (3) ( ) ν z ( ) ( ) k k ( ) ν k!γ(ν + k + ) z, z ( ) ( ) k k k!γ( ν + k + ) z k= k=

21 C ν C ν Bessel J ν (z) J ν (z) := C ( ) ν z ( ) ( ) k k k!γ(ν + k + ) z. k= n Z (4) 3 J ν (z) J ν (z) ν Z (9) ν C \ Z Wronskin J ν (z) J ν (z) W (J ν (z), J ν (z)) = J ν(z) J ν(z). = d z dz (3) d dz W (J J ν(z), J ν (z)) = ν(z) J ν(z) J ν(z) J ν(z) + J ν (z) J ν (z) J ν (z) J ν(z) W (J ν (z), J ν (z)) = z W (J ν(z), J ν (z)), dw dz = z W, ( ) Wronskin W (J ν (z), J ν (z)) = C (33) z C C 4. Bessel J ν (z) J ν(z) z = ( ) ν [ ] J ν (z) = z Γ(ν + ) + O(z ), J ν(z) = ( ) ν [ ] z Γ(ν) + O(z ).

22 Wronskin Gmm reflection formul 6) [ ] W (J ν (z), J ν (z)) =z Γ(ν + ) Γ( ν) Γ( ν + ) Γ(ν) + O(z ) sin νπ = + O(z) πz (33) sin νπ W (J ν (z), J ν (z)) = πz ν Z 4. ν C \ Z {J ν (z), J ν (z)} (3) ν Z 4. ν C Y ν (z) := cos νπj ν(z) J ν (z) sin νπ ν Neumnn Bessel ν C \ Z well-defined ν = n Z 9) (34) Y n (z) := lim ν n Y ν(z) (34) W (J ν (z), Y ν (z)) = πz 4. ν C {J ν (z), Y ν (z)} (3) ν = n Z Y ν (z) n Z [ { lim Y ν(z) = π sin νπj ν (z) + cos νπ J ν(z) J }] ν(z) ν n π cos νπ ν ν ν=n = [ ] [ ] Jν (z) + ( )n Jν (z) π ν π ν ν=n ν= n ψ(z) := d log Γ(z) dz ψ

23 3. Weierstrß (5) ψ(z) = z γ + n= ( n ) n + z k Z ψ(k + ) = k γ 4. J ν (z) = ( ) ν z ( ) ( ) k k k!γ(ν + k + ) z k= J ν (z) ν = J ν (z) log z ( z ) ν k= ( ) k ψ(ν + k + ) k!γ(ν + k + ) ( ) k z J ν (z) ν J ν (z) ν = J n (z) log z ν=n k= = J n (z) log z ν= n k= ( ) k ψ(n + k + ) k!(n + k)! ( ) k ψ( n + k + ) k!γ( n + k + ) ( ) n+k z, ( ) n+k z, n k= + k=n N Z > ψ( N) Γ( N) = d dt Γ(t) = ( ) N+ N! t= N (3) Γ(t) = t(t + ) (t + N) Γ(t + N + ) ( 3. ) (9) J ν (z) ν =( ) n J n (z) log z ν= n n (n k )! ( )n k! k= ( ) ( ) n ( ) k n+k ψ(k + ) k!(k + n)! z k= 3 ( ) k n z

24 4.3 n Z Y n (z) = π J n(z) log z ( ) k {ψ(k + ) + ψ(n + k + )} k!(n + k)! k= n ( ) n+k (n k)! k! z. k= ( ) n+k z 3.3 ν Z < Y ν (z) n Z Y n (z) = ( ) n Y n (z) (35) Bessel 3.4 ( ) ( ) J (z) = sin z, J (z) = cos z. πz πz 4. & Bessel 3. nïve 4.4 ν C J ν (z) + J ν+ (z) = ν z J ν(z), J ν (z) J ν+ (z) = J ν(z). d dz {zν J ν (z)} = z ν J ν (z), d dz {z ν J ν (z)} = z ν J ν+ (z). (36) ν C Y ν (z) + Y ν+ (z) = ν z Y ν(z), Y ν (z) Y ν+ (z) = Y ν(z) 4

25 4.4 J ν (z) = ν [ zj ] ν+(z) J ν (z) z νj ν (z) J ν (z) J ν (z) = z ν z J ν ν+(z) J ν (z) Bessel degree m + z J ν (z) J ν (z) = ν z 4ν(ν+) z 4(ν+m )(ν+m) z 4(ν+)(ν+) z (ν+m) J ν+m+(z) J ν+m (z). (37) 3.6 J ν (z) J ν (z) = ν z (ν+) z (ν+) z (ν+m) z J ν+m+ (z) J ν+m (z). (38) ν C t > z J ν (t + z) = m= J ν m (t)j m (z). r, R, R > r < R < z r, R t 3.7 [W] 4.4 ( t ) J ν m (t)j m (z) = {J z ν m(t)j m (z) J ν m (t)j m(z)} m= m= = {J ν m (t) J ν m+ (t)}j m (z) 5 m= m= J ν m (t){j m (z) J m+ (z)}

26 ( 3.8 ( t ) J ν m (t)j m (z) = z m= z < t t + z z = (9) (3) J ν (t) 4.3 Bessel J ν (z) (7) ( ) ( ) n ν+n J ν (z) = n!γ(ν + n + ) z n= = ( ) ν π z C ( ) ( n z) n e t t ν n dt n! n= ( ) ( n z) n ( ) z t n exp n! 4 t n= J ν (z) = ( ) ν π z e t t ν C n! n= = ( ) ν π z t ν exp C ( z {t z 4t ) n dt 4t } dt (39) Schläfli rg z < π t = zu { ( J ν (z) = π u ν exp C z u )} du (4) u Sonine ν Z () (3) ν C ν := z d dz + z d dz + z ν (3) Hint b z ν e zt T dt. (4) 6

27 T t { b } ν z ν e b zt T dt =z [ν(ν )z ν e zt T dt + b νz ν te zt T dt b ] z ν t e b zt T dt + z [νz ν e zt T dt + b ] z ν te b zt T dt + (z ν )z ν e zt T dt b =z ν+ ( t )e zt T dt + (ν + ) b z ν+ te zt T dt { b } ν z ν e zt T dt = [ z ν+ e ] b zt T (t ) + b [ z ν+ e zt (ν + )T t d ] dt {T (t )} dt (4) (3) d dt {T (t )} = (ν + )T t, [ e ] b zt T (t ) =. (4) T = (t ) ν Re(ν + ) > =, b = γ A A γ 3: γ, γ 7

28 A rg(t ± ) = 4.7 Poisson (3). ν + Z >. ν + Z > Rez > J ν (z) = Γ( ν) ( z) ν π Γ( ) e zt (t ) ν dt. γ J ν (z) = Γ( ν ( ν)eπ z) ν π Γ( ) e zt (t ) ν dt. γ Hnkel γ e zt Tylor e zt = z ν e zt (t ) ν dt = z ν γ ( z) m m= m! z m ( z) m z m t m (t ) ν dt m! γ m= t m (t ) ν m z ν e zt (t ) ν ( ) m z ν+m dt = t m (t ) ν dt γ (m)! γ m= ( 3.9 ) γ A u = t z ν e zt (t ) ν ( ) m z ν+m dt = γ (m)! m= u m γ (u ) ν du 8

29 u A rg(u ) = ( 3. justify ) evlute Re(ν + ) > (u ) ν du ={e π ν e π ν } u m ( u) ν du γ u m = cos νπ Γ(m + )Γ(ν + ) Γ(ν + m + ) ( 3. ) ν C ν C z ν e zt (t ) ν γ dt = cos νπγ(ν + ) m= ( ) m Γ(m + ) (m)!γ(ν + m + ) zν+m Γ(m + ) =(m )(m 3 ) Γ( (m )(m 3) ) = Γ( m ) = (m)! m m! Γ( ) z ν e zt (t ) ν dt = cos νπγ(ν + γ )Γ( )zν m= = ν+ cos νπγ(ν + )Γ( )J ν(z) ( ) ( ) m m m!γ(ν + m + ) z reflection formul (6) cos νπγ(ν+ ) = π Γ( ν) ν+ Z > (ν + Z > cos νπγ(ν + ) = ) γ t > 3π < rg t < π ( ) m (ν m ) (t ) ν = t ν ( ) ν ( t ) m m m= =( ) m m! (ν )(ν 3 ) (ν m + ) = m! ( ν + m) (3 ν)( ν) = Γ( ν + m) m!γ( ν) 9

30 (t ) ν Γ( ν + m) = m!γ( tν m ν) m= t > z ν e zt (t ) ν z ν Γ( ν + m) dt = γ m!γ( ν) e zt t ν m dt γ m= 3. justify zt = e π u e zt t ν m dt =e u (e π z u) ν m z e π du =( ) m+ e π ν z m ν e u ( u) ν m du e zt t ν m dt = ( ) m+ e π ν z m ν γ γ e u ( u) ν m du γ z essentil Hnkel (7) e u ( u) ν m π du = Γ(m ν + ) ( 3.3 ) γ z ν e zt (t ) ν π dt = γ Γ( ν)e π dupliction formul (7) ν m= Γ( ν + m) Γ(m ν + ) = ν m Γ( ) Γ( ν + m + ) ( ) m Γ( ν + m) m!γ(m ν + ) zm ν z ν e zt (t ) ν ν+ π Γ( ν dt = )e π γ Γ( ν) J ν (z) ν + Z > 3

31 [O] [W] ν C m Z (ν) m := ν(ν + ) (ν + m ), (ν, m) := ( ) m ( ν) m ( + ν) m m! = Γ(ν + m + ) m!γ(ν m + ), 869 Mth. Ann. Hnkel 4.8 z C rg z < π z p ( ) J ν (z) cos(z πz νπ 4 π) ( ) m (ν, m) (z) m m= p + sin(z νπ 4 π) m= ( ) m (ν, m ) + O(z p ) p Z (z) m >. 5 Fourier-Bessel Fourier-Bessel 5. Bessel J ν (z) J ν (z) Bessel J ν (z) (ν R ) relity 5. Bessel J ν (z) z = (simple zero) ν Leibniz rule ( ) n ( d d ν =z dz dz + nz ) n+ ( d + (n + )z dz ( ) n d + n(n ) dz n z C J ν (z) z=z = d dz J ν(z) = z=z ) n+ ( d + (z ν + n ) dz ( ) n d dz ) n 3

32 n Z d n dz J ν(z) n = z=z Tylor J ν (z) z ν J ν (z) ±j ν,, ±j ν,, Rej ν, Rej ν, Rej ν,n = imginy prt imginry prt A, B R > D C ±A ± B 4 J ν (z) ±j ν,n ( n m) π D z J ν+ (w) w(w z) J ν (w) dw z D \ {±j ν,n ( n m)} pole w = z, ±j ν,,, ±j ν,m 5. pole 5. w, w = z residue J ν+ (z) J ν (z) w = ±j ν,n residue (36) ± z j ν,n j ν,n π z J ν+ (w) D w(w z) J ν (w) dw = J ν+(z) m { + + } m + J ν (z) z j ν,n j ν,n n= n= { z + j ν,n j ν,n }. J ν+(z) J ν(z) 4.8 J ν+(z) J ν(z) Rez A, B z J ν+ (w) dw w(w z) J ν (w) π ( 5. ), J ν+(z) J ν (z) = n= D { + } + z j ν,n j ν,n 3 n= { }. z + j ν,n j ν,n

33 { z } J ν+ (t) exp J ν (t) dt = n= (36) {( z ) ( z exp j ν,n j ν,n )} n= {( + z ) ( exp z )} j ν,n j ν,n J ν+(z) J ν (z) = z ν J ν+ (z) z ν J ν (z) = d dz log{z ν J ν (z)} { z } ( ) ν J ν+ (t) exp J ν (t) dt = Γ(ν + ) z J ν (z) ( 5.3 ) 5. ν C \ Z < J ν (z) = ( z) ν Γ(ν + ) { z m= j ν,n }. ν = Euler 78 Bessel Bessel relity α, β C µ, ν C J µ (αz), J ν (βz) 3) ) d u du (α µ u =, (43) dz + z dz + z d v dz + dv (β z dz + ν z ) v =. (44) (z(v (43) u (44)) ), [ { d z v du }] ] dz dz udv + [(α β )z + ν µ uv =. (45) dz z µ = ν (45) [, b] (, b R s.t. < b) b zj ν (αz)j ν (βz)dz = α β [z {βj ν(αz)j ν(βz) αj ν(αz)j ν (βz)}] b. (46) α = β β = α + ε ε ( 5.4 ) b z{j ν (αz)} dz = { ) }] b [z {J ν(αz)} + ( ν {J α z ν (αz)}. (47) 33

34 ν R α β J ν (α) = J ν (β) = (46) zj ν (αz)j ν (βz)dz = (48) α = β 47) (36) z{j ν (αz)} dz = {J ν(α)} = {J ν+(α)}. (49) (48) Lommel 5.3 ν R Bessel J ν (z) α C J ν (z) α R α R m= ( ) m ( α)m m!γ(ν + m + ) summnd ν α J ν (z) (46) x t J ν (αt) dt = x = x α α tj ν (αt)j ν (αt)dt α α [ J ν (αx) d dx J ν(αx) J ν (αx) d dx J ν(αx) t J ν (αt) dt = Bessel J ν (z) j, j, (48) (49) 5.4 ν R zj ν (j m z)j ν (j n z)dz = { m n, {J ν+(j m )} m = n. ] 5. Fourier-Bessel Beesel 99 Proc. London Mth.Soc. Hobson [W] 34

35 5.4 Fourier f(x) Bessel {J ν (j m x)} m= f(x) = b m J ν (j m x) m= 5.4 b m b m = J ν+ (j m+ ) tf(t)j ν (j m t)dt. < x, < t T n (x, t) T n (t, x) := n m= J ν (j m x)j ν (j m t) J ν+ (j m ). Riemnn-Lebesgue 5.5, b R < < b < f(x) b t f(t)dt x [, ) (b, ] b tf(t)t n (t, x)dt = o() n >> [W] 5. Riemnn Lebesgue R [, b] Lebesgue f(x) b lim n f(x) sin nxdx =. 5.6 f(t) [, ] t f(t) [, ] Lebesgue ν R m := J ν+ (j m ) tf(t)j ν (j m t)dt 35

36 < < b <, b R f(x) [, b] x [, b] m J ν (j m x) m= {f(x + ) + f(x )} Fourier-Bessel 5.4 n m J ν (j m x) = m= tf(t)t n (t, x)dt {f(x + ) + f(x )} = lim n x ν f(x ) x + lim n x ν f(x + ) t ν+ T n (t, x)dt x t ν+ T n (t, x)dt lim n x lim n S n (x) := x t ν+ T n (t, x)dt = x ν, ( < x < ) t ν+ T n (t, x)dt = xν, ( < x < ) x + t ν+ {t ν f(t) x ν f(x )}T n (t, x)dt x t ν+ {t ν f(t) x ν f(x + )}T n (t, x)dt lim S n(x) = n t ν+ {t ν f(t) x ν f(x + )}T n (t, x)dt t ν f(t) x ν f(x + ) [x, b] [x, b] χ (t), χ (t) t ν f(t) x ν f(x + ) = χ (t) χ (t) 36

37 χ (x + ) = χ (x + ) = ε > δ > δ < b x x t x + δ t = x χ (t) < ε, χ (t) < ε t ν+ {t ν f(t) x ν f(x + )}T n (t, x)dt x+δ x+δ + x t ν+ {t ν f(t) x ν f(x + )}T n (t, x)dt t ν+ χ (t)t n (t, x)dt x+δ x t ν+ χ (t)t n (t, x)dt 5.5 n ε < ξ < δ ξ R x+δ x x+δ t ν+ χ (t)t n (t, x)dt = χ (x + δ) t ν+ T n (t, x)dt x+ξ n Uε ( U > ) 3 Uε S n (x) x dt n m= m J ν (j m x) {f(x ) + f(x + )} (4U + )ε [AAR] [I] [O] [W] Andrews G. E., Askey R. nd Roy R., Specil Functions, Encycl. Mth. nd its Appl. 7, Cmbridge, Frnk W. J. Olver, Asymptotics nd Specil Functions, AKP Clssics, A K Peters, 997. Wtson G.N., A Tretise on the theory of Bessel Functions, Cmbridge Mth. Librry, nd. ed., Cmbridge,

(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 :.