In this chapter we shall generally find it convenient

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In this chapter we shall generally find it convenient to write x + iy rather than x + yi . 495
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[X : 219] THE GENERAL THEORY OF THE LOGARITHMIC, 496 more restricted sense, or in other words to pick out, from the general class of complex functions of the two real variables x and y , a special class to which the expression shall be restricted. But if we were to attempt to explain how this selection is made, and what are the characteristic properties of the special class of functions selected, we should be led far beyond the limits of this book. We shall therefore not attempt to give any general definitions, but shall confine ourselves entirely to special functions defined directly. 218. We have already defined polynomials in z ( § 39 ), rational func- tions of z ( § 46 ), and roots of z ( § 47 ). There is no difficulty in extending to the complex variable the definitions of algebraical functions , explicit and implicit, which we gave ( §§ 26 27 ) in the case of the real variable x . In all these cases we shall call the complex number z , the argument ( § 44 ) of the point z , the argument of the function f ( z ) under consideration. The question which will occupy us in this chapter is that of defining and determining the principal properties of the logarithmic, exponential, and trigonometrical or circular functions of z . These functions are of course so far defined for real values of z only, the logarithm indeed for positive values only. We shall begin with the logarithmic function. It is natural to attempt to define it by means of some extension of the definition log x = Z x 1 dt t ( x > 0); and in order to do this we shall find it necessary to consider briefly some extensions of the notion of an integral. 219. Real and complex curvilinear integrals. Let AB be an arc C of a curve defined by the equations x = φ ( t ) , y = ψ ( t ) , where φ and ψ are functions of t with continuous differential coefficients φ 0 and ψ 0 ; and suppose that, as t varies from t 0 to t 1 , the point ( x, y ) moves along the curve, in the same direction, from A to B .
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[X : 220] EXPONENTIAL, AND CIRCULAR FUNCTIONS 497 Then we define the curvilinear integral Z C { g ( x, y ) dx + h ( x, y ) dy } , (1) where g and h are continuous functions of x and y , as being equivalent to the ordinary integral obtained by effecting the formal substitutions x = φ ( t ), y = ψ ( t ), i.e. to Z t 1 t 0 { g ( φ, ψ ) φ 0 + h ( φ, ψ ) ψ 0 } dt. We call C the path of integration . Let us suppose now that z = x + iy = φ ( t ) + ( t ) , so that z describes the curve C in Argand’s diagram as t varies. Further let us suppose that f ( z ) = u + iv is a polynomial in z or rational function of z . Then we define Z C f ( z ) dz (2) as meaning Z C ( u + iv )( dx + i dy ) , which is itself defined as meaning Z C ( u dx - v dy ) + i Z C ( v dx + u dy ) . 220. The definition of Log ζ . Now let ζ = ξ + be any complex number. We define Log ζ , the general logarithm of ζ , by the equation Log ζ = Z C dz z ,
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[X : 220] THE GENERAL THEORY OF THE LOGARITHMIC, 498 where C is a curve which starts from 1 and ends at ζ and does not pass through the origin. Thus ( Fig. 54 ) the paths ( a ), ( b ), ( c ) are paths such as are contemplated in the definition. The value of Log z
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