CONCEPTUAL TOOLS

By:  Carl H. Durney and Neil E. Cotter

Complex Analysis

 

 

Basic math

 

 

Re[]

 

 

Example 2
 
 
 

 

Ex:            Find , (i.e., find the real part) where "x" is real.
Ans:         1.5 cos(x + π/2)
Sol'n:       We may take one of several different approaches to convert the quantity inside the brackets into the form a + jb (where a is our final answer).  We'll take the approach of rationalizing the fraction.
                       
                                     
                                     
                  We now use Euler's formula to expand the complex exponential:
                                     
                                     
                  Our final answer is the real part, which we may express in several ways.
                         or
                       
Note:       A curious feature of this problem is that the fraction consisting of complex numbers is purely imaginary.  We now examine this symbolically.
                       
                  Whenever the numerator and denominator of a fraction have the above pattern, we will find that the result is purely imaginary.  Note the necessary minus sign.