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 (c).  Using a similar procedure for  [Graphics:Images/ComplexFunExponentialModHome_gr_669.gif],  show we wind up getting a pair of solutions

[Graphics:Images/ComplexFunExponentialModHome_gr_670.gif],  and  [Graphics:Images/ComplexFunExponentialModHome_gr_671.gif]  where [Graphics:Images/ComplexFunExponentialModHome_gr_672.gif] and [Graphics:Images/ComplexFunExponentialModHome_gr_673.gif] are functions of y alone.

Solution 20 (c).

See text and/or instructor's solution manual.

Solution.   The result in part (a) implies that  [Graphics:../Images/ComplexFunExponentialModHome_gr_674.gif].

This means  [Graphics:../Images/ComplexFunExponentialModHome_gr_675.gif]  is constant with respect to x, so  [Graphics:../Images/ComplexFunExponentialModHome_gr_676.gif],  where [Graphics:../Images/ComplexFunExponentialModHome_gr_677.gif] is a function of y alone.

Details.   From part (a)   [Graphics:../Images/ComplexFunExponentialModHome_gr_678.gif],  hence

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_679.gif]   

This means  [Graphics:../Images/ComplexFunExponentialModHome_gr_680.gif]  is constant with respect to x, so  [Graphics:../Images/ComplexFunExponentialModHome_gr_681.gif],  where [Graphics:../Images/ComplexFunExponentialModHome_gr_682.gif] is a function of y alone.

Combining this with the result of part (b) we see that

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_683.gif],    and  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_684.gif],

where [Graphics:../Images/ComplexFunExponentialModHome_gr_685.gif] and [Graphics:../Images/ComplexFunExponentialModHome_gr_686.gif] are functions of y alone.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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