§4.1 Special Notation

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

$k,m,n$	integers.
$a,c$	real or complex constants.
$x,y$	real variables.
$z=x+iy$	complex variable.
$e$	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 $\mathop{\mathrm{Arcsin}\/}\nolimits z$ .