4 Elementary Functions Hyperbolic Functions 4.28 Definitions and Periodicity 4.30 Elementary Properties

§4.29 Graphics

Referenced by:: §4.37(ii)
Permalink:: http://dlmf.nist.gov/4.29

§4.29(i) Real Arguments
§4.29(ii) Complex Arguments

§4.29(i) Real Arguments

Notes:: These graphs were produced at NIST.
Keywords:: hyperbolic functions, inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.29.i

Figure 4.29.1: $\mathop{\sinh\/}\nolimits x$ and $\mathop{\cosh\/}\nolimits x$ .

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function and : real variable
A&S Ref:: AMS55 (Figure 4.6)
Permalink:: http://dlmf.nist.gov/4.29.F1
Encodings:: pdf, png

Figure 4.29.2: Principal values of $\mathop{\mathrm{arcsinh}\/}\nolimits x$ and $\mathop{\mathrm{arccosh}\/}\nolimits x$ . ( $\mathop{\mathrm{arccosh}\/}\nolimits x$ is complex when

Symbols:: $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\mathop{\mathrm{arcsinh}\/}\nolimits z$ : inverse hyperbolic sine function and : real variable
A&S Ref:: AMS55 (Figure 4.8)
Permalink:: http://dlmf.nist.gov/4.29.F2
Encodings:: pdf, png

Figure 4.29.3: $\mathop{\tanh\/}\nolimits x$ and $\mathop{\coth\/}\nolimits x$ .

Symbols:: $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function and : real variable
A&S Ref:: AMS55 (Figure 4.6)
Permalink:: http://dlmf.nist.gov/4.29.F3
Encodings:: pdf, png

Figure 4.29.4: Principal values of $\mathop{\mathrm{arctanh}\/}\nolimits x$ and $\mathop{\mathrm{arccoth}\/}\nolimits x$ . ( $\mathop{\mathrm{arctanh}\/}\nolimits x$ is complex when

, and $\mathop{\mathrm{arccoth}\/}\nolimits x$ is complex when

Symbols:: $\mathop{\mathrm{arccoth}\/}\nolimits z$ : inverse hyperbolic cotangent function, $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function and : real variable
A&S Ref:: AMS55 (Figure 4.8)
Permalink:: http://dlmf.nist.gov/4.29.F4
Encodings:: pdf, png

Figure 4.29.5: $\mathop{\mathrm{csch}\/}\nolimits x$ and $\mathop{\mathrm{sech}\/}\nolimits x$ .

Symbols:: $\mathop{\mathrm{csch}\/}\nolimits z$ : hyperbolic cosecant function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function and : real variable
A&S Ref:: AMS55 (Figure 4.6)
Permalink:: http://dlmf.nist.gov/4.29.F5
Encodings:: pdf, png

Figure 4.29.6: Principal values of $\mathop{\mathrm{arccsch}\/}\nolimits x$ and $\mathop{\mathrm{arcsech}\/}\nolimits x$ . ( $\mathop{\mathrm{arcsech}\/}\nolimits x$ is complex when

and

Symbols:: $\mathop{\mathrm{arccsch}\/}\nolimits z$ : inverse hyperbolic cosecant function, $\mathop{\mathrm{arcsech}\/}\nolimits z$ : inverse hyperbolic secant function and : real variable
A&S Ref:: AMS55 (Figure 4.8)
Permalink:: http://dlmf.nist.gov/4.29.F6
Encodings:: pdf, png

§4.29(ii) Complex Arguments

Keywords:: hyperbolic functions, inverse hyperbolic functions
Permalink:: http://dlmf.nist.gov/4.29.ii

The conformal mapping $w=\mathop{\sinh\/}\nolimits z$ is obtainable from Figure 4.15.7 by rotating both the -plane and the -plane through an angle $\frac{1}{2}\pi$ , compare (4.28.8).

The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. They can be visualized with the aid of equations (4.28.8)–(4.28.13).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 4.28 Definitions and Periodicity 4.30 Elementary Properties

§4.29 Graphics

Contents

§4.29(i) Real Arguments

§4.29(ii) Complex Arguments