# §4.1 Special Notation

(For other notation see Notation for the Special Functions.)

$k,m,n$ integers. real or complex constants. real variables. complex variable. base of natural logarithms.

It is assumed the user is familiar with the definitions and properties of elementary functions of real arguments $x$. The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$.

The main functions treated in this chapter are the logarithm $\mathop{\ln\/}\nolimits z$, $\mathop{\mathrm{Ln}\/}\nolimits z$; the exponential $\mathop{\exp\/}\nolimits z$, $e^{z}$; the circular trigonometric (or just trigonometric) functions $\mathop{\sin\/}\nolimits z$, $\mathop{\cos\/}\nolimits z$, $\mathop{\tan\/}\nolimits z$, $\mathop{\csc\/}\nolimits z$, $\mathop{\sec\/}\nolimits z$, $\mathop{\cot\/}\nolimits z$; the inverse trigonometric functions $\mathop{\mathrm{arcsin}\/}\nolimits z$, $\mathop{\mathrm{Arcsin}\/}\nolimits z$, etc.; the hyperbolic trigonometric (or just hyperbolic) functions $\mathop{\sinh\/}\nolimits z$, $\mathop{\cosh\/}\nolimits z$, $\mathop{\tanh\/}\nolimits z$, $\mathop{\mathrm{csch}\/}\nolimits z$, $\mathop{\mathrm{sech}\/}\nolimits z$, $\mathop{\coth\/}\nolimits z$; the inverse hyperbolic functions $\mathop{\mathrm{arcsinh}\/}\nolimits z$, $\mathop{\mathrm{Arcsinh}\/}\nolimits z$, etc.

Sometimes in the literature the meanings of $\mathop{\ln\/}\nolimits$ and $\mathop{\mathrm{Ln}\/}\nolimits$ are interchanged; similarly for $\mathop{\mathrm{arcsin}\/}\nolimits z$ and $\mathop{\mathrm{Arcsin}\/}\nolimits z$, etc. Sometimes “arc” is replaced by the index “$-1$”, e.g. ${\mathop{\sin\/}\nolimits^{-1}}z$ for $\mathop{\mathrm{arcsin}\/}\nolimits z$ and $\mathrm{Sin}^{-1}\;z$ for $\mathop{\mathrm{Arcsin}\/}\nolimits z$.