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

fromwhich
a 5
=− 1
24 a 3 = 1
192 a 1
using a 3
= − 1 8 a 1 .
19.8 Bessel’s equation and Bessel functions 591
Note that all non-zero coefficients are multiples ofa 1
. We can now write down the first
threenon-zero terms inthe power series solution:
(
y=a 1
x − 1 8 x3 + 1 )
192 x5 −...
(19.29)
Herea 1
isanarbitraryconstant.Wehavesucceededinobtainingapowerseriessolution
of Bessel’s Equation 19.28.
Although a 1
is an arbitrary constant, historically its value is chosen to equal 1 2 in
Equation19.29andtheresultingpowerseriesisusedtodefineafunctioncalledaBessel
functionofthefirstkindoforderonethatisconventionallydenotedJ 1
(x).Itispossible
toshow thatitcan beexpressed concisely using sigmanotation as
J 1
(x) =
∞∑
m=0
(−1) m x 2m+1
2 2m+1 m!(m +1)! . (19.30)
You should verify this by writing out the first few terms of this infinite series. Figure
19.17 shows a graph of J 1
(x) obtained using computer software. Observe that the
graph is similar to that of the sine function but has decreasing amplitude. Here,
J 1
(0) =0.
J 1 (x)
0.5
0
10 20 30
x
–0.3
Figure19.17
The Besselfunction ofthe firstkind of
order 1,J 1 (x).
A second independent solution is called a Bessel function of the second kind of
orderoneandisdenotedbyY 1
(x).Derivationofthisfunctionisbeyondthescopeofthis
bookbut,likeJ 1
(x),valuesofY 1
(x)canbeobtainedfromtablesandbyusingcomputer
software. The general solution ofBessel’s equation of order one can thenbe written
y =AJ 1
(x) +BY 1
(x)
whereAandBare arbitraryconstants.