082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

19.8 Bessel’s equation and Bessel functions 589
J o (x)
1
0
–0.4
10 20 30
x
Figure19.16
TheBesselfunction ofthe firstkind oforder
zero,J 0 (x).
function called a Bessel function of the first kind of order zero that is conventionallydenotedJ
0
(x). Itispossible toshow thatitcan beexpressed concisely usingsigma
notation as
J 0
(x) =
∞∑
m=0
(−1) m x 2m
2 2m (m!) 2 . (19.27)
You shouldverifythisbywritingoutthefirstfewtermsofthisinfiniteseries.Note that
this method of solving the differential equation produces a power series solution and is
usedtointroduceanewfunction,J 0
(x).ItisnotpracticaltotrytodrawagraphofJ 0
(x)
by hand but this can be obtained readily using a technical computing language such as
MATLAB ® .Figure19.16showsagraphobtainedinthisway.ObservethatJ 0
hassome
similar characteristics to a cosine wave: its value whenx = 0,J 0
(0), equals 1, and the
functionoscillates,albeitwithdecreasingamplitude.Thevaluesofxwherethefunction
crosses the horizonal axis are known as zeros of the Bessel function and these values
arise in important applications as we shall see in Engineering application 19.12: The
modes of a cylindrical microwave cavity.
WhenaspecificvalueofaBesselfunctionisrequireditisusualpracticetolookthis
up rather than work directly with the power series definition. Both printed tables and
computer packages are available forthispurpose.
Recall from Section 19.5 that the general solution of a second order linear equation
requires two independent solutions. The function J 0
(x) is one such solution. Another
independent solution is called a Bessel function of the second kind of order
zero and is denoted byY 0
(x). Derivation of this function is beyond the scope of this
book but, like J 0
(x), values ofY 0
(x) can be obtained from tables and by using computer
software. The general solution of Bessel’s equation of order zero can then be
written
y =AJ 0
(x) +BY 0
(x)
whereAandBare arbitraryconstants.
19.8.2 Bessel’sequationoforderone
Here we adopt the previous approach to obtain a solution of Bessel’s equation of order
one, and thereby introduce the Bessel function of the first kind of order one.