# hyperbolic functions

(0.007 seconds)

## 1—10 of 142 matching pages

##### 2: 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. …
##### 3: 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).
##### 4: 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 $\pi i$. …
##### 5: 4.29 Graphics
###### §4.29(i) Real Arguments Figure 4.29.6: Principal values of arccsch ⁡ x and arcsech ⁡ x . … Magnify
###### §4.29(ii) Complex Arguments
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. …
##### 6: 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 $\lim_{z\to 0}\frac{\tanh z}{z}=1,$
##### 8: 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).