柱坐标 Bessel函数

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1、柱坐标Bessel函数basisJ0(ωnr)J1(ωnr)J2(ωnr)N0(ωnr)N1(ωnr)N2(ωnr)b.c.J0(ωa)=0J0'(ωa)=0J1(ωa)=0springUST©-math-phyweihuang§3Bessel贝塞尔nJ0零点x0In

2、J1(x0n)

3、J0＇或J1零点012.40480.51913.831725.52010.34037.015638.65370.271510.1735411.79150.232513.3237514.93090.206516.4706618

4、.07110.187719.6159721.21160.173322.7601824.35250.161725.9037927.49350.152229.046810BesselJZero[0,10]//N=30.6346BesselJ[1,BesselJZero[0,10]]//N//Abs=0.1442N[BesselJZero[1,10]]=32.1897fzero(@(x)besselj(1,x),32)←(*Mathematica*)(或Maple符号计算Matlab工程/数值)UST©-mat

5、h-phyweihuang打靶:按问题给的b.c.找Jν→0的那些x=ωa找得可数(无穷多个离散的可排序的)非负实本征值J0(ωa)=0,boundarya=1J0'(ωa)=0↔J1(ωa)=0springUST©-math-phyweihuang一种完备正交函数系{J0(ω0Inr)}另一种完备正交函数系{J0(ω0IInr)}n>0对应I齐b.c.n≥0对应II齐b.c.类比三角函数系{cosωnr}springUST©-math-phyweihuang完备正交函数系{Jν(ωνI/II/IIInr

6、)}{J0(ω0Inr)}{J0(ω0IInr)}Ib.c.IIb.cspringUST©-math-phyweihuang又一种完备正交函数系{J1(ω0Inr)}I齐b.c.,n>0,柱径取为1类比三角函数系{sinωnr}springUST©-math-phyweihuang{J1(ω1Inr)}{J1(ω1IInr)}Ib.c.IIb.cspringUST©-math-phyweihuangspringUST©-math-phyweihuangspringUST©-math-phyweihuang

7、*(k2-μ<0情况)虚宗量Bessel方程的通解springUST©-math-phyweihuang*A教材例3.4.7springUST©-math-phyweihuang*(球问题)球Bessel方程的通解和固有值问题的解springUST©-math-phyweihuang