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NIST
4
Elementary Functions
Hyperbolic Functions
4.30
Elementary Properties
4.32
Inequalities
§4.31
Special Values and Limits
ⓘ
Keywords:
hyperbolic functions
,
limits
,
special values
Notes:
See
Hobson (
1928
, p. 326)
,
Levinson and Redheffer (
1970
, p. 61)
.
Permalink:
http://dlmf.nist.gov/4.31
See also:
Annotations for
Ch.4
Table 4.31.1:
Hyperbolic functions: values at multiples of
1
2
π
i
.
z
0
1
2
π
i
π
i
3
2
π
i
∞
sinh
z
0
i
0
-
i
∞
cosh
z
1
0
-
1
0
∞
tanh
z
0
∞
i
0
-
∞
i
1
csch
z
∞
-
i
∞
i
0
sech
z
1
∞
-
1
∞
0
coth
z
∞
0
∞
0
1
ⓘ
Symbols:
π
: the ratio of the circumference of a circle to its diameter
,
csch
z
: hyperbolic cosecant function
,
cosh
z
: hyperbolic cosine function
,
coth
z
: hyperbolic cotangent function
,
sech
z
: hyperbolic secant function
,
sinh
z
: hyperbolic sine function
,
tanh
z
: hyperbolic tangent function
,
i
: imaginary unit
and
z
: complex variable
A&S Ref:
4.5.61
Permalink:
http://dlmf.nist.gov/4.31.T1
See also:
Annotations for
§4.31
and
Ch.4
4.31.1
lim
z
→
0
sinh
z
z
=
1
,
ⓘ
Symbols:
sinh
z
: hyperbolic sine function
and
z
: complex variable
Permalink:
http://dlmf.nist.gov/4.31.E1
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§4.31
and
Ch.4
4.31.2
lim
z
→
0
tanh
z
z
=
1
,
ⓘ
Symbols:
tanh
z
: hyperbolic tangent function
and
z
: complex variable
Permalink:
http://dlmf.nist.gov/4.31.E2
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§4.31
and
Ch.4
4.31.3
lim
z
→
0
cosh
z
-
1
z
2
=
1
2
.
ⓘ
Symbols:
cosh
z
: hyperbolic cosine function
and
z
: complex variable
Permalink:
http://dlmf.nist.gov/4.31.E3
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§4.31
and
Ch.4