1550251515-Classical_Complex_Analysis__Gonzalez_

'Elementary Functions 273

1. e^2 krri = 1 ( k an integer)


2. ezez' = ez+z' (additivity of exponents)


3. lezl = ex,argez = y + 2k7r


4. ez f:. 0 for every z


5. ez+2k1ri = ez

Property 5 shows that the fm:1.ction w = ez is periodic with imaginary

periods 2k7ri. The smallest nontrivialt periods of ez in absolute value are

2ni and -2ni. We usually consider 2ni as the fundamental or primitive

period, in the sense that all other periods are integral multiples of 2ni. It is

clear that ez does not admit a nontrivial period smaller in absolute value

than 2ni, since e^2 >.7l'i = cos 2.h + i sin 2.h f:. 1 for 0 < ,\ < 1. In addition,

suppose that w = p + qi f:. 0 is a period of ez. Then ez+w = ez for every

z, and it follows that ew = 1, which implies that iewl = eP = 1, sop= 0.

Hence ew = eiq = cos q + i sin q = 1, giving q = 2kn and w = 2kni.

In view of the periodicity of the exponential function, to discuss geo-

metrically the mapping defined by w = ez it suffices to consider the values

assumed by w in any strip Sk = {z: (2k - l)n < Imz S (2k + l)n, k an

integer}, in particular in the strip

S 0 ={z: -n<Imzsn}

which will be called the fundamental strip (Fig. 5.23). We note that the

upper boundary y = 7r is regarded as belonging to S 0 , while the lower

boundary y = -'Tr does not, and similarly for the other strips.

To the horizontal line

L = {z: z = x + im, -oo < x < +oo, -'Tr < m S 7r}

y

• 
  
  
  
  • 
    
    
    
    • -....?:rri - - - - --
    
    
    
    
  
  
  
  

Tri p

{===y====m===l~==:::+::z===L

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  • 
    
    
    
    • 
      
      
      
      • --37Ti - - - - - - -
      
      
      
      
    
    
    
    
  
  
  
  

Fig. 5.23

L' v

u

tzero, which is a period of every function, is called a trivial period, and it is

disregarded in discussing periodic functions.