Digital Library of Mathematical Functions
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4 Elementary FunctionsHyperbolic Functions

§4.28 Definitions and Periodicity

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Notes:
See Hobson (1928, pp. 322–326), Levinson and Redheffer (1970, pp. 56–57).
Keywords:
hyperbolic functions
Permalink:
http://dlmf.nist.gov/4.28
4.28.1\mathop{\sinh\/}\nolimits z=\frac{e^{z}-e^{{-z}}}{2},
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Defines:
\mathop{\sinh\/}\nolimits z: hyperbolic sine function
Symbols:
e: base of exponential function and z: complex variable
A&S Ref:
4.5.1
Referenced by:
‣ §4.45(i), §4.45(ii)
Permalink:
http://dlmf.nist.gov/4.28.E1
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4.28.2\mathop{\cosh\/}\nolimits z=\frac{e^{z}+e^{{-z}}}{2},
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Defines:
\mathop{\cosh\/}\nolimits z: hyperbolic cosine function
Symbols:
e: base of exponential function and z: complex variable
A&S Ref:
4.5.2
Permalink:
http://dlmf.nist.gov/4.28.E2
Encodings:
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4.28.4\mathop{\tanh\/}\nolimits z=\frac{\mathop{\sinh\/}\nolimits z}{\mathop{\cosh\/% }\nolimits z},
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Defines:
\mathop{\tanh\/}\nolimits z: hyperbolic tangent function
Symbols:
\mathop{\cosh\/}\nolimits z: hyperbolic cosine function, \mathop{\sinh\/}\nolimits z: hyperbolic sine function and z: complex variable
A&S Ref:
4.5.3
Permalink:
http://dlmf.nist.gov/4.28.E4
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4.28.5\mathop{\mathrm{csch}\/}\nolimits z=\frac{1}{\mathop{\sinh\/}\nolimits z},
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Defines:
\mathop{\mathrm{csch}\/}\nolimits z: hyperbolic cosecant function
Symbols:
\mathop{\sinh\/}\nolimits z: hyperbolic sine function and z: complex variable
A&S Ref:
4.5.4
Permalink:
http://dlmf.nist.gov/4.28.E5
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4.28.6\mathop{\mathrm{sech}\/}\nolimits z=\frac{1}{\mathop{\cosh\/}\nolimits z},
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Defines:
\mathop{\mathrm{sech}\/}\nolimits z: hyperbolic secant function
Symbols:
\mathop{\cosh\/}\nolimits z: hyperbolic cosine function and z: complex variable
A&S Ref:
4.5.5
Permalink:
http://dlmf.nist.gov/4.28.E6
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4.28.7\mathop{\coth\/}\nolimits z=\frac{1}{\mathop{\tanh\/}\nolimits z}.
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Defines:
\mathop{\coth\/}\nolimits z: hyperbolic cotangent function
Symbols:
\mathop{\tanh\/}\nolimits z: hyperbolic tangent function and z: complex variable
A&S Ref:
4.5.6
Referenced by:
‣ §4.45(i), §4.45(ii)
Permalink:
http://dlmf.nist.gov/4.28.E7
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Relations to Trigonometric Functions

4.28.8\mathop{\sin\/}\nolimits\!\left(iz\right)=i\mathop{\sinh\/}\nolimits z,
4.28.9\mathop{\cos\/}\nolimits\!\left(iz\right)=\mathop{\cosh\/}\nolimits z,
4.28.10\mathop{\tan\/}\nolimits\!\left(iz\right)=i\mathop{\tanh\/}\nolimits z,
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Symbols:
\mathop{\tanh\/}\nolimits z: hyperbolic tangent function, \mathop{\tan\/}\nolimits z: tangent function and z: complex variable
A&S Ref:
4.5.9
Referenced by:
§22.10(ii)
Permalink:
http://dlmf.nist.gov/4.28.E10
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4.28.12\mathop{\sec\/}\nolimits\!\left(iz\right)=\mathop{\mathrm{sech}\/}\nolimits z,
4.28.13\mathop{\cot\/}\nolimits\!\left(iz\right)=-i\mathop{\coth\/}\nolimits z.

As a consequence, many properties of the hyperbolic functions follow immediately from the corresponding properties of the trigonometric functions.

Periodicity and Zeros

The functions \mathop{\sinh\/}\nolimits z and \mathop{\cosh\/}\nolimits z have period 2\pi i, and \mathop{\tanh\/}\nolimits z has period \pi i. The zeros of \mathop{\sinh\/}\nolimits z and \mathop{\cosh\/}\nolimits z are z=ik\pi and z=i\left(k+\frac{1}{2}\right)\pi, respectively, k\in\Integer.

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