DLMF
Index
Notations
Search
Help?
Citing
Customize
Annotate
UnAnnotate
About the Project
NIST
10
Bessel Functions
Modified Bessel Functions
10.28
Wronskians and Cross-Products
10.30
Limiting Forms
§10.29
Recurrence Relations and Derivatives
ⓘ
Permalink:
http://dlmf.nist.gov/10.29
See also:
Annotations for
Ch.10
Contents
§10.29(i)
Recurrence Relations
§10.29(ii)
Derivatives
§10.29(i)
Recurrence Relations
ⓘ
Keywords:
modified Bessel functions
,
recurrence relations
Notes:
See
Watson (
1944
, p. 79)
.
Referenced by:
§10.43(ii)
Permalink:
http://dlmf.nist.gov/10.29.i
See also:
Annotations for
§10.29
and
Ch.10
With
𝒵
ν
(
z
)
defined as in §
10.25(ii)
,
10.29.1
𝒵
ν
-
1
(
z
)
-
𝒵
ν
+
1
(
z
)
=
(
2
ν
/
z
)
𝒵
ν
(
z
)
,
𝒵
ν
-
1
(
z
)
+
𝒵
ν
+
1
(
z
)
=
2
𝒵
ν
′
(
z
)
.
ⓘ
Symbols:
𝒵
ν
(
z
)
: modified cylinder function
,
z
: complex variable
and
ν
: complex parameter
A&S Ref:
9.6.26
Referenced by:
§10.29(ii)
,
§10.51(ii)
,
§7.6(ii)
Permalink:
http://dlmf.nist.gov/10.29.E1
Encodings:
TeX
,
TeX
,
pMML
,
pMML
,
png
,
png
See also:
Annotations for
§10.29(i)
,
§10.29
and
Ch.10
10.29.2
𝒵
ν
′
(
z
)
=
𝒵
ν
-
1
(
z
)
-
(
ν
/
z
)
𝒵
ν
(
z
)
,
𝒵
ν
′
(
z
)
=
𝒵
ν
+
1
(
z
)
+
(
ν
/
z
)
𝒵
ν
(
z
)
.
ⓘ
Symbols:
𝒵
ν
(
z
)
: modified cylinder function
,
z
: complex variable
and
ν
: complex parameter
A&S Ref:
9.6.26
Referenced by:
§10.28
,
§10.43(i)
,
§10.51(ii)
Permalink:
http://dlmf.nist.gov/10.29.E2
Encodings:
TeX
,
TeX
,
pMML
,
pMML
,
png
,
png
See also:
Annotations for
§10.29(i)
,
§10.29
and
Ch.10
10.29.3
I
0
′
(
z
)
=
I
1
(
z
)
,
K
0
′
(
z
)
=
-
K
1
(
z
)
.
ⓘ
Symbols:
I
ν
(
z
)
: modified Bessel function of the first kind
,
K
ν
(
z
)
: modified Bessel function of the second kind
and
z
: complex variable
A&S Ref:
9.6.27
Permalink:
http://dlmf.nist.gov/10.29.E3
Encodings:
TeX
,
TeX
,
pMML
,
pMML
,
png
,
png
See also:
Annotations for
§10.29(i)
,
§10.29
and
Ch.10
§10.29(ii)
Derivatives
ⓘ
Keywords:
derivatives
,
explicit forms
,
modified Bessel functions
Notes:
See
Watson (
1944
, p. 79)
. For (
10.29.5
) use induction combined with the second of (
10.29.1
).
Permalink:
http://dlmf.nist.gov/10.29.ii
See also:
Annotations for
§10.29
and
Ch.10
For
k
=
0
,
1
,
2
,
…
,
10.29.4
(
1
z
d
d
z
)
k
(
z
ν
𝒵
ν
(
z
)
)
=
z
ν
-
k
𝒵
ν
-
k
(
z
)
,
(
1
z
d
d
z
)
k
(
z
-
ν
𝒵
ν
(
z
)
)
=
z
-
ν
-
k
𝒵
ν
+
k
(
z
)
.
ⓘ
Symbols:
d
f
d
x
: derivative of
f
with respect to
x
,
𝒵
ν
(
z
)
: modified cylinder function
,
k
: nonnegative integer
,
z
: complex variable
and
ν
: complex parameter
A&S Ref:
9.6.28
Permalink:
http://dlmf.nist.gov/10.29.E4
Encodings:
TeX
,
TeX
,
pMML
,
pMML
,
png
,
png
See also:
Annotations for
§10.29(ii)
,
§10.29
and
Ch.10
10.29.5
𝒵
ν
(
k
)
(
z
)
=
1
2
k
(
𝒵
ν
-
k
(
z
)
+
(
k
1
)
𝒵
ν
-
k
+
2
(
z
)
+
(
k
2
)
𝒵
ν
-
k
+
4
(
z
)
+
⋯
+
𝒵
ν
+
k
(
z
)
)
.
ⓘ
Symbols:
(
m
n
)
: binomial coefficient
,
𝒵
ν
(
z
)
: modified cylinder function
,
k
: nonnegative integer
,
z
: complex variable
and
ν
: complex parameter
A&S Ref:
9.6.29
Referenced by:
§10.29(ii)
Permalink:
http://dlmf.nist.gov/10.29.E5
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§10.29(ii)
,
§10.29
and
Ch.10