已有结果 Bessel方程(约定Rev≥0) 1d「da(2) ed d u(z)=0 Bessel方程有两个奇点:z=0(正则奇点)和 2=0(非正则奇点) 在正则奇点2=0处,指标P 程的两个(线性无关
Bessel ftns & Neumann ftns Properties of Bessel ftns with Integer Order Fundamental Solutions to Bessel Equation Recurrence Relations Asymptotic Expansion ®k(J Bessel§(½Re ν ≥ 0) 1 z d dz z dw(z) dz + 1 − ν 2 z 2 w(z) = 0 Bessel§küÛ:µz = 0(KÛ:)Ú z = ∞(KÛ:) 3KÛ:z = 0?§Iρ = ±ν ν 6= ê§Bessel§ü(5Ã') K)´ J±ν(z) = X ∞ k=0 (−) k k!Γ (k ± ν + 1) z 2 2k±ν C. S. Wu 1Êù μê()
已有结果 Bessel方程(约定Rev≥0) 1d「da(2) ed d u(z)=0 Bessel方程有两个奇点:z=0(正则奇点)和 z=0(非正则奇点) 在正则奇点z=0处,指标p=± 当1≠整数时,Beel方程的两个(线性无关 正则解是 k「(k土1+1
Bessel ftns & Neumann ftns Properties of Bessel ftns with Integer Order Fundamental Solutions to Bessel Equation Recurrence Relations Asymptotic Expansion ®k(J Bessel§(½Re ν ≥ 0) 1 z d dz z dw(z) dz + 1 − ν 2 z 2 w(z) = 0 Bessel§küÛ:µz = 0(KÛ:)Ú z = ∞(KÛ:) 3KÛ:z = 0?§Iρ = ±ν ν 6= ê§Bessel§ü(5Ã') K)´ J±ν(z) = X ∞ k=0 (−) k k!Γ (k ± ν + 1) z 2 2k±ν C. S. Wu 1Êù μê()
已有结果 Bessel方程(约定Rev≥0) 1d「da(2) ed d Bessel方程有两个奇点:z=0(正则奇点)和 z=0(非正则奇点) 在正则奇点z=0处,指标p=±v 当≠整数时,Besl方程的两个(线性无关 正则解是 J(2)=∑ (-)(2)2 k「(k±v+1)(2
Bessel ftns & Neumann ftns Properties of Bessel ftns with Integer Order Fundamental Solutions to Bessel Equation Recurrence Relations Asymptotic Expansion ®k(J Bessel§(½Re ν ≥ 0) 1 z d dz z dw(z) dz + 1 − ν 2 z 2 w(z) = 0 Bessel§küÛ:µz = 0(KÛ:)Ú z = ∞(KÛ:) 3KÛ:z = 0?§Iρ = ±ν ν 6= ê§Bessel§ü(5Ã') K)´ J±ν(z) = X ∞ k=0 (−) k k!Γ (k ± ν + 1) z 2 2k±ν C. S. Wu 1Êù μê()
已有结果 Bec方程(约定Rev≥0) 2[+21+1-)=0 也可取 Bessel方程的第一解为 2k+v J(2)=∑ k=0 k!「(k+v+1)(2 而第二解为」(2)的线性组合 N(2) cosJ(2)-J-(2) sn丌
Bessel ftns & Neumann ftns Properties of Bessel ftns with Integer Order Fundamental Solutions to Bessel Equation Recurrence Relations Asymptotic Expansion ®k(J Bessel§(½Re ν ≥ 0) 1 z d dz z dw(z) dz + 1 − ν 2 z 2 w(z) = 0 Bessel§1) Jν(z) = X ∞ k=0 (−) k k!Γ (k + ν + 1) z 2 2k+ν 1)J±ν(z)5|Ü Nν(z) = cos νπJν(z) − J−ν(z) sin νπ C. S. Wu 1Êù μê()
已有结果 Bessel方程(约定Rev≥0) 1d「da(2) ed dz (z)=0 当=整数n时,Jn(2)和」n(z)线性相关 」-n(z)=(-)2Jn(z) 此时, Bessel方程的第一解仍是」(=),第二解 则可取为 Nn(=)=imN(=)= COsVTJ,(-J
Bessel ftns & Neumann ftns Properties of Bessel ftns with Integer Order Fundamental Solutions to Bessel Equation Recurrence Relations Asymptotic Expansion ®k(J Bessel§(½Re ν ≥ 0) 1 z d dz z dw(z) dz + 1 − ν 2 z 2 w(z) = 0 ν = ên§Jn(z)ÚJ−n(z)5' J−n(z) = (−) n Jn(z) d§Bessel§1)E´Jn(z)§1) K Nn(z) = lim ν→n Nν(z) = lim ν→n cos νπJν(z)−J−ν(z) sin νπ C. S. Wu 1Êù μê()