# hyperbolic

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##### 1: 4.37 Inverse Hyperbolic Functions
###### §4.37 Inverse Hyperbolic Functions
The principal values of the inverse hyperbolic cosecant, hyperbolic secant, and hyperbolic tangent are given by …
##### 2: 36.2 Catastrophes and Canonical Integrals
###### Normal Forms for Umbilic Catastrophes with Codimension $K=3$
36.2.3 $\Phi^{(\mathrm{H})}\left(s,t;\mathbf{x}\right)=s^{3}+t^{3}+zst+yt+xs,$ $\mathbf{x}=\{x,y,z\}$,
(hyperbolic umbilic).
##### 4: 4.29 Graphics
###### §4.29(ii) Complex Arguments
The conformal mapping $w=\sinh z$ is obtainable from Figure 4.15.7 by rotating both the $w$-plane and the $z$-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. …
##### 5: 4.28 Definitions and Periodicity
###### Relations to Trigonometric Functions
As a consequence, many properties of the hyperbolic functions follow immediately from the corresponding properties of the trigonometric functions.
###### Periodicity and Zeros
The functions $\sinh z$ and $\cosh z$ have period $2\pi i$, and $\tanh z$ has period $\pi i$. …
##### 6: 4.1 Special Notation
 $k,m,n$ integers. …
The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. … ; the hyperbolic trigonometric (or just hyperbolic) functions $\sinh z$, $\cosh z$, $\tanh z$, $\operatorname{csch}z$, $\operatorname{sech}z$, $\coth z$; the inverse hyperbolic functions $\operatorname{arcsinh}z$, $\operatorname{Arcsinh}z$, etc. …
##### 7: 4.32 Inequalities
###### §4.32 Inequalities
4.32.3 $|\cosh x-\cosh y|\geq|x-y|\sqrt{\sinh x\sinh y},$ $x>0$, $y>0$,
For these and other inequalities involving hyperbolic functions see Mitrinović (1964, pp. 61, 76, 159) and Mitrinović (1970, p. 270).
##### 8: 4.31 Special Values and Limits
###### §4.31 Special Values and Limits
4.31.1 $\lim_{z\to 0}\frac{\sinh z}{z}=1,$
4.31.2 $\lim_{z\to 0}\frac{\tanh z}{z}=1,$
4.31.3 $\lim_{z\to 0}\frac{\cosh z-1}{z^{2}}=\frac{1}{2}.$
##### 9: 4.41 Sums
###### §4.41 Sums
For sums of hyperbolic functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §43), Prudnikov et al. (1986a, §5.3), and Zucker (1979).
##### 10: 4.34 Derivatives and Differential Equations
###### §4.34 Derivatives and Differential Equations
4.34.4 $\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{csch}z=-\operatorname{csch}z\coth z,$
4.34.5 $\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sech}z=-\operatorname{sech}z\tanh z,$
With $a\neq 0$, the general solutions of the differential equations …
4.34.11 $w=A\cosh\left(az\right)+B\sinh\left(az\right),$