20. SOME COMMON FUNCTIONS OF 2 



Functions constructed by means of algebraic processes, perhaps with the addition of 

 the process of taking a limit, can be taken over at once from the field of real numbers into the 

 complex field. It is convenient to begin with certain transcendental functions defined by 

 means of series. 



In the defining series 



2 3 4 



e' = I + X +-2_ +^ +-?_ + . . . [20a] 



2! 3! 4! 



substitution of iO for x gives 



l + id--^- i2-+l- [20b] 



2! 3! 4! 



Comparison with the series for sin and cos 0, which are stated in Equations [33b, c] shows 

 that the following important formula holds: 



e"^ = cos (9 + i sin (9 [20c] 



Thus 



It follows that any complex number can be written in the alternative forms 



z = X + iy = r(cos 6 + i sin d) = re'^ 

 Its conjugate is 



2* = X - iy = r(cos d - i sin 6) = re~'^ 



Two other useful functions are the hyperbolic sine and cosine: 



sinh a; = 1 (e* - g-*) = 3;+-^+^+— + ... [20d] 



2^ -' 3! 5! 7! 



cosh a; = l(e^ +e-^) = 1 +-^ +^+^ + ... [20e] 



2 2 ! 4 ! 6 ! 



From the series it is easily verified that 



sin {iy) = i sinh y, cos {iy) = cosh y, 



sinh {iy) - i sin y, cosh {iy) = cos y 



Finally, writing z = re'^ and In for the natural logarithm. 



In 2 = In r + z^ = 2 In {x^ ^- y^) -v i tan ~ ^ — 



Here In r or ln(a;^ + y^) is to be interpreted as the ordinary real logarithm. Thus In z is 

 many-valued. Its imaginary part has an infinite number of values spaced 2n^' apart, namely, 

 written in terms of any one of them id, id + Ivi, id + ^ni . . ., id - 2ni, id - 4:7ri . . . Even if 

 2 = X and is real and positive, for complete generality In 2 ^ In a? = (In x) real + 2ffm, where 

 n is any integer, positive or negative. 



36 



