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10
Bessel Functions
Spherical Bessel Functions
10.55
Continued Fractions
10.57
Uniform Asymptotic Expansions for Large Order
§10.56
Generating Functions
ⓘ
Keywords:
generating functions
,
spherical Bessel functions
A&S Ref:
10.1.39, 10.1.40
(modified)
10.2.30, 10.2.31
Notes:
To verify (
10.56.1
) and (
10.56.2
) show that each side of both equations satisfies the differential equation
(
2
t
−
z
)
(
d
2
w
/
d
t
2
)
+
(
d
w
/
d
t
)
=
z
w
via the first of (
10.51.1
) and (
10.49.3
), (
10.49.5
). Then check the initial conditions at
t
=
0
. (
10.56.3
) and (
10.56.4
) follow from (
10.56.1
) and (
10.56.2
) via (
10.47.12
); then (
10.56.5
) follows from (
10.47.11
).
Permalink:
http://dlmf.nist.gov/10.56
See also:
Annotations for
Ch.10
When
2
|
t
|
<
|
z
|
,
10.56.1
cos
z
2
−
2
z
t
z
=
cos
z
z
+
∑
n
=
1
∞
t
n
n
!
𝗃
n
−
1
(
z
)
,
ⓘ
Symbols:
cos
z
: cosine function
,
!
: factorial (as in
n
!
)
,
𝗃
n
(
z
)
: spherical Bessel function of the first kind
,
n
: integer
and
z
: complex variable
Referenced by:
§10.56
,
Ch.10
Permalink:
http://dlmf.nist.gov/10.56.E1
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§10.56
and
Ch.10
10.56.2
sin
z
2
−
2
z
t
z
=
sin
z
z
+
∑
n
=
1
∞
t
n
n
!
𝗒
n
−
1
(
z
)
.
ⓘ
Symbols:
!
: factorial (as in
n
!
)
,
sin
z
: sine function
,
𝗒
n
(
z
)
: spherical Bessel function of the second kind
,
n
: integer
and
z
: complex variable
Referenced by:
§10.56
Permalink:
http://dlmf.nist.gov/10.56.E2
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§10.56
and
Ch.10
10.56.3
cosh
z
2
+
2
i
z
t
z
=
cosh
z
z
+
∑
n
=
1
∞
(
i
t
)
n
n
!
𝗂
n
−
1
(
1
)
(
z
)
,
ⓘ
Symbols:
!
: factorial (as in
n
!
)
,
cosh
z
: hyperbolic cosine function
,
i
: imaginary unit
,
𝗂
n
(
1
)
(
z
)
: modified spherical Bessel function
,
n
: integer
and
z
: complex variable
Referenced by:
§10.56
Permalink:
http://dlmf.nist.gov/10.56.E3
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§10.56
and
Ch.10
10.56.4
sinh
z
2
+
2
i
z
t
z
=
sinh
z
z
+
∑
n
=
1
∞
(
i
t
)
n
n
!
𝗂
n
−
1
(
2
)
(
z
)
,
ⓘ
Symbols:
!
: factorial (as in
n
!
)
,
sinh
z
: hyperbolic sine function
,
i
: imaginary unit
,
𝗂
n
(
2
)
(
z
)
: modified spherical Bessel function
,
n
: integer
and
z
: complex variable
Referenced by:
§10.56
Permalink:
http://dlmf.nist.gov/10.56.E4
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§10.56
and
Ch.10
10.56.5
exp
(
−
z
2
+
2
i
z
t
)
z
=
e
−
z
z
+
2
π
∑
n
=
1
∞
(
−
i
t
)
n
n
!
𝗄
n
−
1
(
z
)
.
ⓘ
Symbols:
π
: the ratio of the circumference of a circle to its diameter
,
exp
z
: exponential function
,
e
: base of natural logarithm
,
!
: factorial (as in
n
!
)
,
i
: imaginary unit
,
𝗄
n
(
z
)
: modified spherical Bessel function
,
n
: integer
and
z
: complex variable
Referenced by:
§10.56
,
Ch.10
Permalink:
http://dlmf.nist.gov/10.56.E5
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§10.56
and
Ch.10