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

19.8 Bessel’s equation and Bessel functions 595
Table19.2
Besselfunction zeros foramicrowave
cavity.
m X m1 X m2 X m3
0 2.4048 5.5201 8.6537
1 3.8317 7.0156 10.173
2 5.1356 8.4172 11.620
f 011
= 3×108
2π
√ (2.4048 ) 2
+
0.033
( ) 1 × π 2
= 3.70GHz
0.120
EXERCISES19.8.3
1 WehaveseenthatforpositiveintegerordertheBessel
functionsofthe firstkind are defined by
∞∑ (−1) m x 2m+n
J n (x) =
2 2m+n m!(m +n)! ,
m=0
for n=0,1,2,...
(a)Now suppose that weare interestedin Bessel
functionswithnegativeintegerorder.Replacenby
−nin thisdefinition to showthat
∞∑ (−1) m x 2m−n
J −n (x) =
2 2m−n m!(m −n)! ,
m=0
withn=1,2,...
(b)Now make the assumption that (m −n)!,which
appears in the denominator ofeach term,becomes
infinite whenm = 0,1,2, . . . ,n −1.Itis possible
to prove thisusingproperties ofthe Gamma
functionbutisbeyondthescopeofthisbook.With
thisassumption showthat
∞∑ (−1) m x 2m−n
J −n (x) =
2
m=n
2m−n m!(m −n)! ,
withn=1,2,...
(c)Byintroducing anew dummy variable,s,where
m =s +n, showthat
∞
J −n (x) = (−1) n ∑ (−1) s x 2s+n
2 2s+n m!s!
s=0
for n=0,1,2,...
and hence deduceJ −n (x) = (−1) n J n (x).
19.8.4 Besselfunctionsandthegeneratingfunction
A different set of modes known asTE modes may also be calculated using Bessel
functions. Rather than using zeros of the Bessel functions directly it is necessary to
usethederivativesoftheBesselfunctions,atopicbeyondthescopeofthisintroductorytreatment.
SeveralimportantpropertiesofBesselfunctionsusedinphysicsandengineeringapplications
can be deduced from knowledge of the so-called generating function. In this
sectionweintroducethisgeneratingfunctionanduseittoderivetheJacobi-Angeridentitieswhich
wethen employ inEngineering application 19.13.
Firstly, recall the power series expansion of e x introduced inSection 6.5.
∞
e x =1+x+ x2
2! + x3
3! + x4
4! +...= ∑ 1
n! xn . (19.31)
n=0