Bessel Functions

Steve Dunbar
Math 221H, Fall 1993

Covers Nagle & Saff, Section 8.8

O wonder!
  How many goodly creatures are there here!
  How beauteous mankind is! O brave new world
  That has such people in't!
   	
   William Shakespeare, The Tempest/5.1

The Bessel Equation and Functions

The Bessel equation of order  ν (``nu'') is  
	x2 y'' + x y' + (x2 - ν2) y = 0.
Notice that 0 is a regular singular point and so may not be able to be solved with the straightforward power series technique that we have developed so far.

The solution of the Bessle equation, particularly of integer order and half-integer order is important in physica and engineering.

The Bessel Function of Order 0, J0(x):

DSolve[ x^2 y''[x] + x y'[x] + x^2 y[x] == 0, y[x], x]

Plot[BesselJ[0,x], {x,0,10}]

Plot[BesselJ[0,x], {x,0,100}, PlotRange -> All]

Series[BesselJ[0,x],{x,0,20}]

Sum[ ( (-1)^n/(n!)^2) (x/2)^(2n), {n,0,10}]

Zeroes of the Bessel Function J0(x)

besselzeroes =
Table[ x /. FindRoot[BesselJ[0,x] == 0, {x, guess}], 
    {guess,1,20,2}]

besselzeroes[[3]] - besselzeroes[[2]]

besselzeroes[[5]] - besselzeroes[[3]]

besselzeroes[[6]] - besselzeroes[[5]]

besselzeroes[[8]] - besselzeroes[[6]]

Exercise:  What do you guess is the limit of the distance between successive zeroes of the Bessel function  J0(x)?  Can you make an intuitive justification of this by rewriting the equation in normal form?

The Complementary Bessel Function of Order 0, Y0(x)

Plot[BesselY[0,x], {x,0,10}]

yseries = Normal[Series[BesselY[0,x], {x,0,4}]]

Plot[{BesselJ[0,x], BesselY[0,x]}, {x, 0, 10}]

The Bessel Function of Order 1, J1(x)

DSolve[ x^2 y''[x] + x y'[x] + (x^2-1) y[x] == 0, y[x], x]

Plot[BesselJ[1,x], {x,0,10}]

DJ0 = D[BesselJ[0,x],x]

Plot[{DJ0, BesselJ[1,x]}, {x,0,10}]

Bessel Functions of Order 1/2

jhalf  = 
 DSolve[ x^2 y''[x] + x y'[x] + (x^2- 1/4) y[x] == 0,
  y[x], x]

BesselJ[1/2,x]

BesselJ[-1/2,x]