(a) The generating function for Bessel functions of integral order p = n is Φ ( x , h ) = e ( 1 / 2 ) x h − h − 1 = ∑ n = − ∞ ∞ h n J n ( x ) . By expanding the exponential in powers of x h − h − 1 show that the n = 0 term is J 0 ( x ) as claimed. (b) Show that x 2 ∂ 2 Φ ∂ x 2 + x ∂ Φ ∂ x + x 2 Φ − h ∂ ∂ h 2 Φ = 0 . Use this result and Φ ( x , h ) = ∑ n = − ∞ ∞ h n J n ( x ) to show that the functions J n ( x ) satisfy Bessel’s equation. By considering the terms in h n in the expansion of e ( 1 / 2 ) x h − h − 1 in part (a), show that the coefficient of h n is a series startingwith the term ( 1 / n ! ) ( x / 2 ) n . ( You have then proved that the functions called J n ( x ) in the expansion of Φ ( x , h ) are indeed the Bessel functions of integral order previously defined by ( 12.9 ) and ( 13.1 ) with p = n . )
(a) The generating function for Bessel functions of integral order p = n is Φ ( x , h ) = e ( 1 / 2 ) x h − h − 1 = ∑ n = − ∞ ∞ h n J n ( x ) . By expanding the exponential in powers of x h − h − 1 show that the n = 0 term is J 0 ( x ) as claimed. (b) Show that x 2 ∂ 2 Φ ∂ x 2 + x ∂ Φ ∂ x + x 2 Φ − h ∂ ∂ h 2 Φ = 0 . Use this result and Φ ( x , h ) = ∑ n = − ∞ ∞ h n J n ( x ) to show that the functions J n ( x ) satisfy Bessel’s equation. By considering the terms in h n in the expansion of e ( 1 / 2 ) x h − h − 1 in part (a), show that the coefficient of h n is a series startingwith the term ( 1 / n ! ) ( x / 2 ) n . ( You have then proved that the functions called J n ( x ) in the expansion of Φ ( x , h ) are indeed the Bessel functions of integral order previously defined by ( 12.9 ) and ( 13.1 ) with p = n . )
(a) The generating function for Bessel functions of integral order
p
=
n
is
Φ
(
x
,
h
)
=
e
(
1
/
2
)
x
h
−
h
−
1
=
∑
n
=
−
∞
∞
h
n
J
n
(
x
)
.
By expanding the exponential in powers of
x
h
−
h
−
1
show that the
n
=
0
term is
J
0
(
x
)
as claimed.
(b) Show that
x
2
∂
2
Φ
∂
x
2
+
x
∂
Φ
∂
x
+
x
2
Φ
−
h
∂
∂
h
2
Φ
=
0
.
Use this result and
Φ
(
x
,
h
)
=
∑
n
=
−
∞
∞
h
n
J
n
(
x
)
to show that the functions
J
n
(
x
)
satisfy Bessel’s equation. By considering the terms in
h
n
in the expansion of
e
(
1
/
2
)
x
h
−
h
−
1
in part (a), show that the coefficient of
h
n
is a series startingwith the term
(
1
/
n
!
)
(
x
/
2
)
n
.
( You have then proved that the functions called
J
n
(
x
)
in the expansion of
Φ
(
x
,
h
)
are indeed the Bessel functions of integral order previously defined by
(
12.9
)
and
(
13.1
)
with
p
=
n
.
)
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