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10
Bessel Functions
Modified Bessel Functions
10.27
Connection Formulas
10.29
Recurrence Relations and Derivatives
§10.28
Wronskians and Cross-Products
ⓘ
Keywords:
Wronskians
,
cross-products
,
modified Bessel functions
Notes:
For the Wronskians use (
1.13.5
) and the limiting forms in §
10.30
. For the cross-products apply (
10.29.2
).
Permalink:
http://dlmf.nist.gov/10.28
See also:
Annotations for
Ch.10
10.28.1
𝒲
{
I
ν
(
z
)
,
I
−
ν
(
z
)
}
=
I
ν
(
z
)
I
−
ν
−
1
(
z
)
−
I
ν
+
1
(
z
)
I
−
ν
(
z
)
=
−
2
sin
(
ν
π
)
/
(
π
z
)
,
ⓘ
Symbols:
𝒲
: Wronskian
,
π
: the ratio of the circumference of a circle to its diameter
,
I
ν
(
z
)
: modified Bessel function of the first kind
,
sin
z
: sine function
,
z
: complex variable
and
ν
: complex parameter
A&S Ref:
9.6.14
Permalink:
http://dlmf.nist.gov/10.28.E1
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§10.28
and
Ch.10
10.28.2
𝒲
{
K
ν
(
z
)
,
I
ν
(
z
)
}
=
I
ν
(
z
)
K
ν
+
1
(
z
)
+
I
ν
+
1
(
z
)
K
ν
(
z
)
=
1
/
z
.
ⓘ
Symbols:
𝒲
: Wronskian
,
I
ν
(
z
)
: modified Bessel function of the first kind
,
K
ν
(
z
)
: modified Bessel function of the second kind
,
z
: complex variable
and
ν
: complex parameter
A&S Ref:
9.6.15
Permalink:
http://dlmf.nist.gov/10.28.E2
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§10.28
and
Ch.10
