ON THE IMAGINARY POWER OF A NUMBER 321 



3. Under the definition follows (for real numbers) the construction 

 of the rational and irrational (by partition and by sequence) and the nega- 

 tive number completing the real number continuum. In the usual way 

 the function a^ (a positive, x real) and the exponential 



e' = l + x + ~+ ■■■ (6) 



where e = 2.71 . . . , are constructed. The ordinary laws for the con- 

 vergence of infinite series and products established. The inversion of the 

 square leads to the imaginary iy and thence the complex or integral num- 

 ber X + iy. The operations of addition, multiplication, involution and 

 their inverses shown to lead to a complex number. The convergence of 

 series of powers and their products established. Then finally arises the 

 question of the imaginary power of a real number and the necessity for 

 its construction as an integral number if possible. This brings us to 

 the point of constructing a'" or preferably e" where e is the constant nape- 

 rian number 2.71 . . . and i = V — 1. 



With respect to the s3Tnbol e"'' Chrystal says, Algebra, Part II, p. 264: 



It seems to be forgotten by some writers that the e in e'^ is a mere nominis umbra 



— a contraction for the name of a function and not 2.71828 Oblivion of 



this fact has led to some strange pieces of mathematical logic. 



• And in the preface (vii) he says : 



Some expounders of the theory of the exponential function of an imaginary argu- 

 ment seem ever to have forgotten the obvious truism that one can have no property 

 of a function which has not been defined. 



It is in the attempt to remove the "shadow" from this fundamental 

 symbol that this note is undertaken. 



4. The construction e", when x is real, constructs the real continuum 

 of positive numbers as x varies from — oo to -f- 0° , this construction 

 being given by (6). The construction e'^ is defined as an integral number 

 in (3) by being subject, as part of its definition, to the laws of the integer 



gix giy = e'(^+y', e^'/e'^ = e't^-y', {e'"")'^ = e--^^ (7) 



The construction of e''' as a possible complex number presents itself 

 as a problem. To find two real number constructions C(x) and S(x) such 

 that under the defining laws of e''', we shall have 



e^^ = C(.t) +iS{x). ' (8a) 



