Bessel ( 06/11/21) Bessel 1 ( ) 1.1 0, 1,..., n n J 0 (x), J 1 (x),..., J n (x) I 0 (x), I 1 (x),..., I n (x) Miller (Miller algorithm) Bess

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1 Bessel ( 6/11/1) Bessel 1 ( ) 1.1, 1,..., n n J (x), J 1 (x),..., J n (x) I (x), I 1 (x),..., I n (x) Miller (Miller algorithm) Bessel (6 ) ( ) [1] n n d j J n (x), d j I n (x) Deuflhard j= j=.1 C Numerical Recipes Bessel, Bessel, 1 ( jp/~ooura/bessel-j.html) C Fortran. UNIX UNIX double j(double), double j1(double), double jn(int, double), double y(double), double y1(double), double yn(int, double) FreeBSD Linux, Cygwin C, C++ Fortran Bessel 1

2 .3 GSL GNU Scientific Library 1 (GSL) Bessel C, C++, Fortran /* * * gcc -I/usr/local/include -L/usr/local/lib -lgsl -lgslcblas testbessel.c -o testbessel */ #include <stdio.h> #include <gsl/gsl_specfunc.h> /* * gsl_sf_bessel_j() * gsl_sf_bessel_j1() * gsl_sf_bessel_jn() * gsl_sf_bessel_i() * gsl_sf_bessel_i1() * gsl_sf_bessel_in() */ int main() { int i,n,nu,maxnu = 3; double x,dx,xmin,xmax; n = 1; xmin =.; xmax = 1.; dx = (xmax - xmin) / n; for (i = ; i <= n; i++) { x = xmin + i * dx; printf("%f ", x); for (nu = ; nu <= MaxNu; nu++) printf("%f ", gsl_sf_bessel_in(nu, x)); printf("\n"); } return ; }.4 Java Colt Project ( colt.jar CLASSPATH Cygwin Cygwin tcsh setenv CLASSPATH c:\jsdk1.4._6\lib\colt.jar;.; 1

3 cern.jet.math Bessel TestBessel.java // TestBessel.java import cern.jet.math.*; public class TestBessel { public static void main(string []args) { System.out.println(""+Bessel.j(1.)); } }, 1 Colt jar jar xvf colt.jar cern/jet/math/bessel.class jar xvf colt.jar cern/jet/math/constants.class cern/jet/math/{bessel,constants}.class CLASSPATH.5 Mathematica Mathematica BesselJ[n,x] n Bessel J n g1=plot[besselj[,x],{x,,1}] g=plot[evaluate[table[besselj[n, x], {n,, 1}]],{x,,1}] B.7 ( ) UNIX jar tar 3 Mathematica Export["Bessel.eps", g1] OK. 3

4 FindRoot[BesselJ[,x]==,{x,},WorkingPrecision->5] Table[FindRoot[BesselJ[,x]==,{x,m*Pi+},WorkingPrecision->5],{m,,}] AccuracyGoal-> ( Automatic ) n MaxIterations-> ( 1 ) Version 6 Mathematica Bessel BesselJZero[n,k] (J n k ), BesselJZero[n,k,x ] (J n x k ), BesselYZero[n,k] (Y n k ), BesselYZero[n,k,x ] (Y n x k ) bjz[n_,m_]:=bjz[n,m]=n[besseljzero[n,m],3] bjz[n][m] ( ) Version 5. NumericalMath BesselZeros ( wolfram.com/mathematica/add-onslinks/standardpackages/numericalmath/besselzeros.html) BesselJZeros[], BesselJPrimeZeros[] BesselJZero[] (N[BesselJZero[1,],3] ) J n(z) (n =, 1,..., 1) <<NumericalMath BesselZeros For[n=,n<=1,n++,bjpz[n]=BesselJPrimeZeros[n,,WorkingPrecision->3]] bjpz[n][[m]] J n m 3.6 Maple Maple BesselJ(n,x) n Bessel J n plot(besselj(,x),x=..1) plot([seq(besselj(n,x),n=..5)],x=..1) plot({seq(besselj(n,x),n=..5)},x=..1) 4 4 Mathematica Export["Bessel.eps", %] OK. 4

5 J n m ν n,m BesselJZeros(n,m) evalf[5](besselzeros(,1)) evalf[5](besselzeros(,1..)) (Maple ).7 gnuplot gnuplot besj(), besj1() plot besj(x),besj1(x) J, J 1 n J n (x).8 Bessel 3 Bessel Bessel.pdf NUMPAC NetNUMPAC ( 5

6 4 17/1 Netlib AMOS 5 README ======== readme for AMOS ======= A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order This algorithm is a package of subroutines for computing Bessel functions and Airy functions. The routines are updated versions of those routines found in TOMS algorithm 644. A Bessel Bessel Watson [] [3] ( ) [] Amazon A.1 Bessel A.1.1 Bessel ( ) ν (1) () ( z ν ( 1) J ν (z) := ) k ( z ) k, k!γ(ν + k + 1) k= cos µπj µ (x) J µ (x) Y ν (z) = lim. µ ν sin µπ J ν (z) ν 1, Y ν (z) ν () A.1.3 Γ ( ) B.5 m Γ(m + 1) = m! 5 6

7 A.1. ( ) Bessel ( ) (Friedrich Wilhelm Bessel, , Minden Königsberg, ) Kepler Bessel A.1 [4] Fourier ( ) ( [3]) ( ) 6 ( ) ( ) ( ) Bessel A.1.3 ( ) J ν (z) ( ) 7 (1) z ( ) C ( ) (z/) ν ν (z/) ν = exp (ν log(z/)) w = z w = log z log C 8 Ω := C \ {x R; x } 6 (1) () 7 () 8 ( ) log ( ) 7

8 ν R x > J ν (x) R ν J ν () : J ν () = (ν > ), J () = 1. Bessel () ν C J ν (z) ( ) C \ {x; x } ν J ν (z) C A. Bessel ν (3) y + 1 ) x y + (1 ν y = (ν ) Bessel Bessel J ν (x), Y ν (x) ( ) y = AJ ν (x) + BY ν (x) x A..1 (3) ( ) ( ) 1 () y + p(x)y + q(x)y = p(x), q(x) Bessel [a, b] ( < a < b < ) {φ 1, φ } y C 1, C (4) y = C 1 φ 1 + C φ φ 1, φ Ω := C \ {x; x } ( ) J ν (z), Y ν (z) 8

9 φ 1 (z), φ (z) ( ) y Ω y [a, b] C 1, C (4) ( ) Ω (4) A.. Frobenius Bessel Frobenius Frobenius y = x λ k=... ( ) J ν (x), J ν (x) A.1 ν C J ν (x), J ν (x) Bessel (3) b k x k A..3 ν C \ Z J ν (x), J ν (x) 1 ( (3) ) x A. ν C \ Z sin νπ W (J ν (x), J ν (x)) = πx. W (f, g) f, g : W (f, g) := ( ) f g f g. A. ν C \ Z J ν (x), J ν (x) Bessel ν Z ν = n Z J ν (x), J ν (x) 1 J n (z) = ( 1) n J n (z) () Y ν (x) 9

10 ν C \ Z Y ν (x) := cos νπj ν(x) J ν (x) sin νπ J ν (x), J ν (x) (3) A. (5) W (J ν (x), Y ν (x)) = πx (ν C \ Z) J ν (x), Y ν (x) (3) n Z (3) (5) 9 lim Y ν(x) ν n Y n (x) := lim ν n Y ν (x) W (J n (x), Y n (x)) = lim ν n W (J ν (x), Y ν (x)) = πx J n (x), Y n (x) (ν = n ) (3) A.3 (Bessel ) ν C J ν (x), Y ν (x) Bessel (3) A.3 Bessel A.4 Bessel A.5 Fourier-Bessel A.4 (Bessel ) ν 1/ 1 rj ν (µ ν,n r)j ν (µ ν,m r)dr = J ν+1(µ ν,n ) δ nm (n, m N) 9 1

11 A.5 (Fourier-Bessel ) f : (, c) C c xf(x) dx f I (, c) x, n N {} f(x) = A j J n (λ j x) j=1 A j := c J n+1 (λ j c) c λ j J n : xf(x)j n (λ j x) dx (j N) {x > ; J n (x) = } = {λ j } j N, < λ 1 < λ < < λ j < λ j+1 <. A.6 ( ) f xf(x) (, c) Lebesgue f(x + ) + f(x ) = A j J n (λ j x) Fourier A.7 {(x, y); x + y < 1} j=1 f(r, θ) = 1 A nj = B nj = A j J (µ j r) + j=1 πj n+1 (µ nj ) πj n+1 (µ nj ) n=1 j=1 1 ( π 1 ( π J n (µ nj r)(a nj cos nθ + B nj sin nθ) ) f(r, θ)j n (µ nj r) cos nθ dθ dr, ) f(r, θ)j n (µ nj r) sin nθ dθ dr. r (, 1) θ f(r, θ) Fourier f(r, θ) = a (r) + (a n (r) cos nθ + b n (r) sin nθ), n=1 a n (r) := 1 π π f(r, θ) cos nθ dθ, b n (r) := 1 π π f(r, θ) sin nθ dθ a n (r), b n (r) (, 1) Fourier-Bessel a n (r) = A nj J n (µ nj r), A nj := j=1 J n+1 (µ nj ) 1 ra n (r)j n (µ nj r) dr, b n (r) = B nj J n (µ nj r), B nj := j=1 J n+1 (µ nj ) 1 rb n (r)j n (µ nj r) dr. 11

12 A.8 {(x, y); x + y < R } f(r, θ) = 1 A nj = B nj = A j J (µ j r) + j=1 n=1 j=1 R ( π πr J n+1 (µ nj R) πr J n+1 (µ nj R) R ( π J n (µ nj r)(a nj cos nθ + B nj sin nθ) ) f(r, θ)j n (µ nj r) cos nθ dθ dr, ) f(r, θ)j n (µ nj r) sin nθ dθ dr. A.6 1: Ω = {(x, y) R ; x + y < 1} (6) (7) (8) u = λu (in Ω), u = (on Ω), u 3 u λ λ u λ x = r cos θ, y = r sin θ, U(r, θ) = u(x, y) (9) (1) (11) U rr + 1 r U r + 1 r U θθ = λu U(1, θ) = (θ [, π]), U. ( < r < 1, θ [, π]), U(r, θ) = R(r)Θ(θ) (9) R(r)Θ(θ) R R + 1 R r R + 1 Θ r Θ = λ. r R + rr + λr R = Θ R Θ. µ Θ π Θ = µθ. ( n N {}) ( A, B C) s.t. µ = n, Θ(θ) = A cos nθ + B sin nθ. µ = n R(r) r R + rr + (λr n )R =. 1

13 λ > R + 1 ) r R + (λ n R =. z = λr, r R(r) = f(z) λf ( λr ) + λ r f ( λr ) + (λ n f (z) + 1 ) z f (z) + (1 n f(z) =. z r ) ( ) f λr =. Bessel f(z) z = C C s.t. f(z) = CJ n (z). ( ) R(r) = f(z) = f λr = CJ n ( λr). R(1) = CJ n ( λ) =. C λ J n λ = µ nm. m N s.t. λ = µnm. A.9 (λ > ) u = λu u λ u dx = u u dx = u dx > ( ) Ω Ω Ω λ > λ > λ > ( ) λ := µ nm u nm (r, θ) := J n (µ nm r)(a cos nθ + B sin nθ) λ ( ) Ω u t = κ u (in Ω (, )), u(x, t) = (on Ω (, )), u(x, ) = f(x) (x Ω) 13

14 u(r, θ, t) = 1 A m J (µ m r) + m=1 A nj = B nj = πj n+1 (µ nj ) πj n+1 (µ nj ) n=1 m=1 1 ( π 1 ( π exp ( κµ nmt ) J n (µ nm r)(a nm cos nθ + B nm sin nθ), ) f(r cos θ, r sin θ)j n (µ nj r) cos nθ dθ dr, ) f(r cos θ, r sin θ)j n (µ nj r) sin nθ dθ dr. R Neumann 1 c u tt = u (in Ω (, )), u(x, t) = (on Ω (, )), u(x, ) = ϕ(x) (x Ω), u t (x, ) = ψ(x) (x Ω) u(r, θ, t) = A m J (µ m r) cos(µ m t) + m=1 m=1 C m J (µ m r) sin(µ mt) µ m + n=1 m=1 n=1 m=1 J n (µ nm r) cos(µ mn t) (A nm cos nθ + B nm sin nθ) J n (µ nm r) sin(µ nmt) µ nm (C nm cos nθ + D nm sin nθ). B B.1 Kepler E, M, e [, 1) E e sin E = M Kepler Bessel E = M + n=1 n J n(ne) sin nm (M e E Newton ) 14

15 ( ) (x/a) + (y/b) = 1 (c, ) (c = a b ) (a, ) π = (1, ) S = (e, ) (e := c/a) M (cos M, sin M) P = (x, y) P P = (x, 1 x sign y) E = πop E πsp = ν tan ν = 1 + e 1 e tan E B.1 Kepler ( (by ) ) B. Bessel y + 1 ) x y (1 + ν y = Bessel x = ix Bessel J ν (ix), Y ν (ix) ( x ) ν 1 ( x ) m I ν (x) :=, m!γ (ν + m + 1) m= K ν (x) := π lim I α (x) I α (x) α ν sin απ I ν (x), K ν (x) x = I ν (x) x I ν (x) = exp ( 1 ) νπi J ν (ix) y + 1 ) x y (k + ν y = ( Bessel ) x y = AI ν (kx) + BK ν (kx). B.3 n Bessel < ν n,1 < ν n, < < ν n,m < ν n,m+1 < 15

16 (1) J n J n+1 ν n,1 < ν n+1,1 < ν n, < ν n+1, < < ν n,m < ν n+1,m < ν n,m+1 < ν n+1,m+1 < () lim m ν n,m =. (3) lim m (ν n,m+1 ν n,m ) = π. (4) lim n (ν n+1,1 ν n,1 ) = 1. B.4 (generating function) [ ( 1 exp z t 1 )] = J n (z)t n. t n Z C J n (z) = 1 [ ( 1 exp πi z t 1 )] t (n+1) dz. t C J n (z) = 1 π e i(z sin θ nθ) dθ = 1 π cos(z sin θ nθ) dθ. π π J 4 (5) ( [5]) B.5 (Euler ) Re z > z C Γ(z) = e x x z 1 dx (Re z > )

17 z C n N Γ(z + 1) = zγ(z) Γ(n) = (n 1)!. B.6 < x 1 J n (x) 1 n n! xn (n N {}), Y (x) π log x, Y n(x) n (n 1)! π x n (n N). x n J n (x) (x πx cos (n + 1) π ), Y n (x) (x 4 πx sin (n + 1) π ). 4 B.7 Mathematica (* n 1 Bessel x *) rootnear[n_, x_]:= FindRoot[BesselJ[n, x] ==, {x, x}, WorkingPrecision -> 1, AccuracyGoal -> 9] (* n 1 Bessel x *) rootnear[n_,x_]:= x /. rootnear[n,x][[1]] (* n 1 Bessel m *) Jroot[,1]=rootnear[,] Jroot[n_,1]:=Jroot[n,1]=rootnear[n,Jroot[n-1,1]+Pi/] Jroot[n_,m_]:=Jroot[n,m]=rootnear[n,Jroot[n,m-1]+Pi] ( Mathematica ) m > 1 ν n,m J n (x) = ν n,m 1 + π lim (ν n,m+1 ν n,m ) = π m (n m ν n,m+1 ν n,m π n ) m = 1 ν n,1 lim (ν n,1 ν n 1,1 ) = 1 n ν n 1,1 + 1 ν n 1,1 < ν n,1 < ν n 1, ν n 1,1 ν n 1,1 ν n 1,1 + π/ 17

18 ν n+1,1 ν n,1 ListPlot[Table[Jroot[n,1]-Jroot[n-1,1],{n,1}]] ν n 1,1 + 1 FindRoot[] B.7.1 J B.7. J

19 B.7.3 J B.7.4 n B µ n,m n m µ 61 = 9.93, µ 3 = 9.76, µ 13 =

20 C 1 [6] C.1 R I p(x), q(x) y + p(x)y + q(x)y = f, g W (x; f, g) := det ( f(x) f (x) g(x) g (x) ) f, g x, x I W (x; f, g) = W (x ; f, g) exp ( x ) p(t) dt x C.1 (i), (ii), (iii) (i) f, g I 1 (ii) x I W (x; f, g). (iii) x I W (x; f, g). C. (1) a(x)y + b(x)y + c(x)y = x x < r ( ) a(x ) x a(x ) = x x p(x) := b(x)/a(x), q(x) := c(x)/a(x) y + p(x)y + q(x)y = r > p(x), q(x) x x < r 1

21 C. p(x), q(x) x x < r y + p(x)y + q(x)y = (1), () (1) x x < r () C.3 () x x < r (1) a(x)y + b(x)y + c(x)y = x (i.e. a(x ) = ) p(x) := b(x)/a(x), q(x) := c(x)/a(x) x p(x), q(x) (13) y + p(x)y + q(x)y = p(x), q(x) x 1 x (1) x a(x) x C.3 (Euler ) a, b x y + axy + by = Euler x = e t C.4 ( ) [1],,,,,,, (6), UTF75F [] Watson, G. N.: A Treatise on the Theory of Bessel Functions, nd edition, Cambridge University Press (1944, 1995). [3], (1963),. [4] 1, (199). [5], notebook/numerical-integration.pdf (). [6], (1971 ( 4)). 1

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x [ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),