21. POWERS OF I 



Writing 2 = re '^ 



2" = r"e'"^ = r'^(cos n6l + £ sin n6) 



by [20c]. If 71 is an integer, positive or negative, s " is single-valued; in particular, 2^ = 1, as 

 for a real number. For nonintegral n, 2" is many-valued, because of the ambiguity of 0. 

 For example: 



3I/2 ^ ^i/ifcos/i^V £ sin/-l 0^1 



But, if d is replaced hy 9 + 2mn where m is an integer 



cosf _i + mn) + i sin(_ 6 + mn\ 



,1/2 



rl/2 



If m is even, the expression in brackets reduces to cos (1/2 6) + i sin (1/2 0) and the same 

 value of 2^/^ is obtained as before. If, however, m is an odd integer, positive or negative, 



,1/2 _ 



.1/2 



[cos^l^j+.sin^l^)] 



Thus, as for a real number, z^^^ has two values, each the negative of the other; see Figure 21. 



Similarly, z^^^ has three values, with applitudes spaced 27r/3 radians apart; and, in 

 general, if ^ is a positive integer, z^^ ^ has k different values with amplitudes spaced 2n/k 

 apart. If n is not a rational number, that is, the ratio of two integers, 2" has an infinite number 

 of values. 



In working with many-valued functions 

 such as In 2 or 2", the value that is to be 

 employed for amp 2 must be clearly established. 

 If possible, amp z is usually so chosen that 

 the given function varies continuously as 2 

 is allowed to vary through such sets of values 

 as are of interest, and is also continuous with 

 the same function as ordinarily understood when 

 2 becomes real and positive. Thus, for real 

 2 > 0, In 2 is made to become the ordinary real 

 In 2, and 2" is real and positive. 



Special care is needed when a more 

 complicated function of 2 is involved, as, for 

 example, in In [(2 + a)/ (2 + ft)]. Every sum or 

 difference represents a new entity for which a 

 special rule must be adopted for the determina- 

 tion of its amplitude. Algebraic changes are 



Figure 21 - The various values of 2^^^ and 2^/^. 



37 



