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# 10 the function f x defined by πf x pπ q p i log x

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10. The function f ( x ) defined by πf ( x ) = + ( q - p ) I { log( x - 1) } + ( r - q ) I (log x )

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[X : 222] EXPONENTIAL, AND CIRCULAR FUNCTIONS 505 is equal to p when x > 1, to q when 0 < x < 1, and to r when x < 0. 11. For what values of z is (i) log z (ii) any value of Log z ( a ) real or ( b ) purely imaginary? 12. If z = x + iy then Log Log z = log R + i (Θ + 2 k 0 π ), where R 2 = (log r ) 2 + ( θ + 2 ) 2 and Θ is the least positive angle determined by the equations cos Θ : sin Θ : 1 :: log r : θ + 2 : p (log r ) 2 + ( θ + 2 ) 2 . Plot roughly the doubly infinite set of values of Log Log(1 + i 3), indicating which of them are values of log Log(1 + i 3) and which of Log log(1 + i 3). 222. The exponential function. In Ch. IX we defined a function e y of the real variable y as the inverse of the function y = log x . It is naturally suggested that we should define a function of the complex variable z which is the inverse of the function Log z . Definition. If any value of Log z is equal to ζ , we call z the exponen- tial of ζ and write z = exp ζ. Thus z = exp ζ if ζ = Log z . It is certain that to any given value of z correspond infinitely many different values of ζ . It would not be unnatural to suppose that, conversely, to any given value of ζ correspond infinitely many values of z , or in other words that exp ζ is an infinitely many-valued function of ζ . This is however not the case, as is proved by the following theorem. Theorem. The exponential function exp ζ is a one-valued function of ζ . For suppose that z 1 = r 1 (cos θ 1 + i sin θ 1 ) , z 2 = r 2 (cos θ 2 + i sin θ 2 ) are both values of exp ζ . Then ζ = Log z 1 = Log z 2 ,
[X : 225] THE GENERAL THEORY OF THE LOGARITHMIC, 506 and so log r 1 + i ( θ 1 + 2 ) = log r 2 + i ( θ 2 + 2 ) , where m and n are integers. This involves log r 1 = log r 2 , θ 1 + 2 = θ 2 + 2 nπ. Thus r 1 = r 2 , and θ 1 and θ 2 differ by a multiple of 2 π . Hence z 1 = z 2 . Corollary. If ζ is real then exp ζ = e ζ , the real exponential function of ζ defined in Ch. IX . For if z = e ζ then log z = ζ , i.e. one of the values of Log z is ζ . Hence z = exp ζ . 223. The value of exp ζ . Let ζ = ξ + and z = exp ζ = r (cos θ + i sin θ ) . Then ξ + = Log z = log r + i ( θ + 2 ) , where m is an integer. Hence ξ = log r , η = θ + 2 , or r = e ξ , θ = η - 2 ; and accordingly exp( ξ + ) = e ξ (cos η + i sin η ) . If η = 0 then exp ξ = e ξ , as we have already inferred in § 222 . It is clear that both the real and the imaginary parts of exp( ξ + ) are continuous functions of ξ and η for all values of ξ and η . 224. The functional equation satisfied by exp ζ . Let ζ 1 = ξ 1 + 1 , ζ 2 = ξ 2 + 2 . Then exp ζ 1 × exp ζ 2 = e ξ 1 (cos η 1 + i sin η 1 ) × e ξ 2 (cos η 2 + i sin η 2 ) = e ξ 1 + ξ 2 { cos( η 1 + η 2 ) + i sin( η 1 + η 2 ) } = exp( ζ 1 + ζ 2 ) . The exponential function therefore satisfies the functional relation f ( ζ 1 + ζ 2 ) = f ( ζ 1 ) f ( ζ 2 ), an equation which we have proved already ( § 205 ) to be true for real values of ζ 1 and ζ 2 .

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[X : 225] EXPONENTIAL, AND CIRCULAR FUNCTIONS 507 225. The general power a ζ . It might seem natural, as exp ζ = e ζ when ζ is real, to adopt the same notation when ζ is complex and to drop the notation exp ζ altogether. We shall not follow this course because we
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