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

596 Chapter 19 Ordinary differential equations I
Recallalsothatthereisnothingspecialaboutlabellingthedummyvariablen.Anyother
letter could have been used and the statement of the expansion would be equivalent.
Now replacexfirstly by xt
2 and then by − x in Equation 19.31 to produce the following
expansions where we have usedr andsasthe dummy
2t
variables:
and
e xt
2 =
e − x 2t =
∞∑
r=0
∞∑
s=0
1
r!
( xt
) r
2
1
(
− x ) s
s! 2t
Multiplying these two expansions together wefind
e xt
2 e
− x 2t = e x 2 (t−1 t ) =
∞∑
r=0
1
r!
( xt
) r ∑ ∞
1
(
− x ) s
2 s! 2t
s=0
or, more explicitly,
[
]
e x 2 (t−1 t ) = 1 + xt
1!2 + x2 t 2
2!2 2 + x3 t 3
3!2 3 + x4 t 4
4!2 4 +... ×
[
]
1 − x
1!2t + x2
2!2 2 t 2 − x3
3!2 3 t 3 + x4
4!2 4 t 4 −...
(19.32)
Now imagine multiplying out the brackets on the right and observe that the resulting
expression can be considered as the sum or difference of terms involving powers oft,
that is, t, t 2 ,...t n ... and t −1 ,t −2 ,...t −n ,.... In fact we can express Equation 19.32
concisely as
e x 2 (t−1 t ) =
∞∑
n=−∞
y n
(x)t n (19.33)
where eachy n
(x) is a function ofxthat we can regard as the coefficient of the termt n
in the expansion. For example, let us focus first on the those terms in Equation 19.32
whichwhenmultipliedoutresultinmultiplesoft 0 ,aswillhappenbyformingproducts
such as
( )( )
1 1 ,
( xt
) (
− x )
,
1!2 1!2t
( )( )
x 2 t 2 x 2
2!2 2 2!2 2 t 2 ,
(
x 3 t 3
3!2 3 )(
− x3
3!2 3 t 3 )
Then together, these termswill formy 0
(x)t 0 inEquation 19.33, that is
y 0
(x)=1−
or more concisely,
y 0
(x) =
∞∑
n=0
x2
(1!) 2 2 + x 4
2 (2!) 2 (2 2 ) − x 6
2 (3!) 2 (2 3 ) +... 2
and soon.
(−1) n x 2n
(n!) 2 2 2n (19.34)
Comparing Equation 19.34 with Equation 19.27 we see thaty 0
(x) is simply the Bessel
function oforder zero,J 0
(x).