Exercise 20.  Many texts give an alternative definition for  [Graphics:Images/ComplexFunExponentialModHome_gr_643.gif],  starting with (5-1) as the definition for  [Graphics:Images/ComplexFunExponentialModHome_gr_644.gif].  

20 (d).  Now use the Cauchy-Riemann equations to conclude from part (c) that

                    [Graphics:Images/ComplexFunExponentialModHome_gr_687.gif].  

Solution 20 (d).

See text and/or instructor's solution manual.

Solution.   From part (c)

          [Graphics:../Images/ComplexFunExponentialModHome_gr_688.gif],      [Graphics:../Images/ComplexFunExponentialModHome_gr_689.gif]     so that

[Graphics:../Images/ComplexFunExponentialModHome_gr_690.gif],   [Graphics:../Images/ComplexFunExponentialModHome_gr_691.gif],   [Graphics:../Images/ComplexFunExponentialModHome_gr_692.gif],   [Graphics:../Images/ComplexFunExponentialModHome_gr_693.gif].   

The  Cauchy Riemann equations are  

        [Graphics:../Images/ComplexFunExponentialModHome_gr_694.gif],  

        [Graphics:../Images/ComplexFunExponentialModHome_gr_695.gif].  

Hence we obtain    [Graphics:../Images/ComplexFunExponentialModHome_gr_696.gif].   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2008 John H. Mathews, Russell W. Howell