北京大学：《数学物理方法》精品课程电子教案（A类）第二部分 数学物理方程_第11讲 柱函数（三）

遗留的问题 求解常微分方程 1 d dR n7 m P=0 rdr d h 令k2=-(nx/h)2,可以预料,作变 换z=kr,能就化为 Bessel方程 现在分两步走:先令=(/h),然后再作

Modified Bessel Functions Bessel Functions with Half Odd Integer Order Spherical Bessel Functions Modified Bessel Equations Modified Bessel Functions Asymptotic Behaviors of Modified Bessel Ftns ¢3¯K ¦)~‡©§ 1 r d dr  r dR dr  +  − nπ h 2 − m2 r 2  R = 0 -k 2 = −(nπ/h) 2 §Œ±ý§ŠC †z = kr§UÒzBessel§ y3©üÚrµk-x = (nπ/h)r§,￾2Š C†z = ix C. S. Wu 1›ù μê(n)

遗留的问题 求解常微分方程 1 d dR n7 m P=0 rdr d h 令k2=-(mπ/h)2,可以预料,作变 换z=kr,能就化为 Bessel方程 现在分两步走:先令x=(nT/h)r,然后再作 变换z=ir

Modified Bessel Functions Bessel Functions with Half Odd Integer Order Spherical Bessel Functions Modified Bessel Equations Modified Bessel Functions Asymptotic Behaviors of Modified Bessel Ftns ¢3¯K ¦)~‡©§ 1 r d dr  r dR dr  +  − nπ h 2 − m2 r 2  R = 0 -k 2 = −(nπ/h) 2 §Œ±ý§ŠC †z = kr§UÒzBessel§ y3©üÚrµk-x = (nπ/h)r§,￾2Š C†z = ix C. S. Wu 1›ù μê(n)

遗留的问题 求解常微分方程 1 d dR n丌\2m2 R=0 rdr d h ●作变换x=(n/h)r,y(x)=R(r),方程变为 1 d/ d 72 +(-1 d 再作变换2=1x.(2)=0(x),化为Bee方程 1 d/ de

Modified Bessel Functions Bessel Functions with Half Odd Integer Order Spherical Bessel Functions Modified Bessel Equations Modified Bessel Functions Asymptotic Behaviors of Modified Bessel Ftns ¢3¯K ¦)~‡©§ 1 r d dr  r dR dr  +  − nπ h 2 − m2 r 2  R = 0 ŠC†x = (nπ/h)r, y(x) = R(r)§§C 1 x d dx  x dy dx  +  −1 − m2 x 2  y(x) = 0 2ŠC†z = ix, w(z) = y(x)§zBessel§ 1 z d dz  z dw dz  +  1 − m2 z 2  w(z) = 0 C. S. Wu 1›ù μê(n)

遗留的问题 求解常微分方程 1 d dR n丌\2m2 R=0 rdr d h ●作变换x=(mπ/h)r,y(x)=B(r),方程变为 1 d 出)+(1-m))=0 再作变换z=ix,w(z)=y(x),化为 Bessel方程 1 d/ dw (2)= 0

Modified Bessel Functions Bessel Functions with Half Odd Integer Order Spherical Bessel Functions Modified Bessel Equations Modified Bessel Functions Asymptotic Behaviors of Modified Bessel Ftns ¢3¯K ¦)~‡©§ 1 r d dr  r dR dr  +  − nπ h 2 − m2 r 2  R = 0 ŠC†x = (nπ/h)r, y(x) = R(r)§§C 1 x d dx  x dy dx  +  −1 − m2 x 2  y(x) = 0 2ŠC†z = ix, w(z) = y(x)§zBessel§ 1 z d dz  z dw dz  +  1 − m2 z 2  w(z) = 0 C. S. Wu 1›ù μê(n)

Bessel Functions with 1 d dw m d(2d)+(1-2)m()=0的解是 u(2)=CJm(2)+DNm(2) (x)=0的解是 y()=CUm(ir)+ DNm(ic)

Modified Bessel Functions Bessel Functions with Half Odd Integer Order Spherical Bessel Functions Modified Bessel Equations Modified Bessel Functions Asymptotic Behaviors of Modified Bessel Ftns 1 z d dz  z dw dz  +  1 − m2 z 2  w(z) = 0)´ w(z) = CJm(z) + DNm(z) 1 x d dx  x dy dx  +  −1 − m2 x 2  y(x) = 0)´ y(x) = CJm(ix) + DNm(ix) 1 r d dr  r dR dr  +  − nπ h 2 − m2 r 2  R = 0)´ R(r) = CJm  inπ h r  + DNm  inπ h r  C. S. Wu 1›ù μê(n)