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4
Elementary Functions
Hyperbolic Functions
4.35
Identities
4.37
Inverse Hyperbolic Functions
§4.36
Infinite Products and Partial Fractions
ⓘ
Keywords:
hyperbolic functions
,
infinite products
,
partial fractions
Notes:
Replace
z
by
i
z
in §
4.22
.
Permalink:
http://dlmf.nist.gov/4.36
See also:
Annotations for
Ch.4
4.36.1
sinh
z
=
z
∏
n
=
1
∞
(
1
+
z
2
n
2
π
2
)
,
ⓘ
Symbols:
π
: the ratio of the circumference of a circle to its diameter
,
sinh
z
: hyperbolic sine function
,
n
: integer
and
z
: complex variable
A&S Ref:
4.5.68
Permalink:
http://dlmf.nist.gov/4.36.E1
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§4.36
and
Ch.4
4.36.2
cosh
z
=
∏
n
=
1
∞
(
1
+
4
z
2
(
2
n
−
1
)
2
π
2
)
.
ⓘ
Symbols:
π
: the ratio of the circumference of a circle to its diameter
,
cosh
z
: hyperbolic cosine function
,
n
: integer
and
z
: complex variable
A&S Ref:
4.5.69
Permalink:
http://dlmf.nist.gov/4.36.E2
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§4.36
and
Ch.4
When
z
≠
n
π
i
,
n
∈
ℤ
,
4.36.3
coth
z
=
1
z
+
2
z
∑
n
=
1
∞
1
z
2
+
n
2
π
2
,
ⓘ
Symbols:
π
: the ratio of the circumference of a circle to its diameter
,
coth
z
: hyperbolic cotangent function
,
n
: integer
and
z
: complex variable
Permalink:
http://dlmf.nist.gov/4.36.E3
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§4.36
and
Ch.4
4.36.4
csch
2
z
=
∑
n
=
−
∞
∞
1
(
z
−
n
π
i
)
2
,
ⓘ
Symbols:
π
: the ratio of the circumference of a circle to its diameter
,
csch
z
: hyperbolic cosecant function
,
i
: imaginary unit
,
n
: integer
and
z
: complex variable
Permalink:
http://dlmf.nist.gov/4.36.E4
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§4.36
and
Ch.4
4.36.5
csch
z
=
1
z
+
2
z
∑
n
=
1
∞
(
−
1
)
n
z
2
+
n
2
π
2
.
ⓘ
Symbols:
π
: the ratio of the circumference of a circle to its diameter
,
csch
z
: hyperbolic cosecant function
,
n
: integer
and
z
: complex variable
Permalink:
http://dlmf.nist.gov/4.36.E5
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§4.36
and
Ch.4