已有结果 Bessel方程(约定Rev≥0) 1d「da(2) zdz dz (z)=0 当=整数m时,J()和」n(z)线性相关 J-n(z)=(-)J(2) ●此时, Bessel方程的第一解仍是Jn(z),第二解 则可取为 Nn(2)= lim Nv(a)=lim Cos丌J(2)-J-(2) SIn v7 尜
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()
已有结果 Bese程(约定Re≥0 1d「da(=2) zd d (2)=0 应用 L'Hospital法则,可得 N()=1()一元一 k!(k+n) (n+k+1)+(k+1) 并且约完,当=0时,需去掉表达式中第二项的有限和
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 A^L’Hospital{K§ Nn(z) =2 π Jn(z) ln z 2 − 1 π Xn−1 k=0 (n − k − 1)! k! z 2 2k−n − 1 π X∞ k=0 (−) k k!(k+n)! ψ(n+k+1)+ψ(k+1) z 2 2k+n ¿ ½§n = 0§IKLª¥1kÚ C. S. Wu 1Êù μê()
Bessel函数的图形 意 Bessel函数图形的特点
Bessel ftns & Neumann ftns Properties of Bessel ftns with Integer Order Fundamental Solutions to Bessel Equation Recurrence Relations Asymptotic Expansion Bessel¼êã/ 5¿Bessel¼êã/A: oNCzª³ x = 0NC1 ":©Ù C. S. Wu 1Êù μê()
Bessel函数的图形 注意Be函数图形的特点 总体变化趋势 0附近的行为 占
Bessel ftns & Neumann ftns Properties of Bessel ftns with Integer Order Fundamental Solutions to Bessel Equation Recurrence Relations Asymptotic Expansion Bessel¼êã/ 5¿Bessel¼êã/A: oNCzª³ x = 0NC1 ":©Ù C. S. Wu 1Êù μê()
Bessel函数的图形 注意Be函数图形的特点 总体变化趋势 ●x=0附近的行为 零点的分布
Bessel ftns & Neumann ftns Properties of Bessel ftns with Integer Order Fundamental Solutions to Bessel Equation Recurrence Relations Asymptotic Expansion Bessel¼êã/ 5¿Bessel¼êã/A: oNCzª³ x = 0NC1 ":©Ù C. S. Wu 1Êù μê()