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4 Elementary FunctionsNotation

§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 lnz, Lnz; the exponential expz, ez; the circular trigonometric (or just trigonometric) functions sinz, cosz, tanz, cscz, secz, cotz; the inverse trigonometric functions arcsinz, Arcsinz, etc.; the hyperbolic trigonometric (or just hyperbolic) functions sinhz, coshz, tanhz, cschz, sechz, cothz; the inverse hyperbolic functions arcsinhz, Arcsinhz, etc.

Sometimes in the literature the meanings of ln and Ln are interchanged; similarly for arcsinz and Arcsinz, etc. Sometimes “arc” is replaced by the index “1”, e.g. sin1z for arcsinz and Sin1z for Arcsinz.