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

19.8 Bessel’s equation and Bessel functions 597
In a similar manner, suppose we focus attention on the terms in Equation 19.32 that
when multiplied outresultint 1 as will happen when weform products such as
( xt
)
(1) ,
1!2
( x 2 t 2 ) (
− x )
,
2!2 2 1!2t
( x 3 t 3
3!2 3 )( x
2
2!2 2 t 2 )
and so on.
Then, together these will formy 1
(x)t 1 inEquation 19.33, so that
y 1
(x) = x
1!2 − x3
2!2 3 + x5
3!2!2 5 ...
or more concisely,
y 1
(x) =
∞∑
m=0
(−1) m x 2m+1
(m +1)!m!2 2m+1 (19.35)
Comparing Equation 19.35 with 19.30 we see thaty 1
(x) is simply the Bessel function
of order one,J 1
(x).
Inthesameway,itispossibletoshowthatalltheothercoefficientsy n
(x)inEquation
19.33 are the Bessel functions J n
(x). The function e x 2 (t−1 t ) in Equation 19.33 is called
a generating function for the Bessel functions because the coefficient oft n in the expansion
isthe Bessel functionJ n
(x). Thus the expansion isused togenerate the Bessel
functions.We have the following important result:
Generatingfunctionforthe Bessel functions:
∞∑
e x 2 (t−1 t ) = J n
(x)t n (19.36)
n=−∞
Ifthefunctionontheleftisexpandedinpowersoft thenthecoefficientoft n isthe
Bessel function of ordern.
EXERCISES19.8.4
1 Usethe generating functionto show thatifnis an
even integer, thenthe BesselfunctionJ n (x)is an even
function, thatisJ n (−x) =J n (x).Ifnisan odd
integer, show thatJ n (x) isan odd function, thatis
J n (−x) = −J n (x).
19.8.5 TheJacobi-Angeridentities
Two results which relate to Bessel functions and which are important in communicationsengineering
are known as the Jacobi-Anger identities. We first statethem before
deriving them from the generatingfunction.