Exercise 5.  Find all values of  z  for which the following equations hold.  

5 (a).  [Graphics:Images/ComplexFunExponentialModHome_gr_136.gif].

Solution 5 (a).

See text and/or instructor's solution manual.

Answer.   [Graphics:../Images/ComplexFunExponentialModHome_gr_137.gif],   where n is an integer.

Solution Method I.   Use the relation  [Graphics:../Images/ComplexFunExponentialModHome_gr_138.gif],  and observe that

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_139.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_140.gif].

Using polar coordinates  [Graphics:../Images/ComplexFunExponentialModHome_gr_141.gif]  we have  [Graphics:../Images/ComplexFunExponentialModHome_gr_142.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_143.gif]  calculate  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_144.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_145.gif].

Thus,   [Graphics:../Images/ComplexFunExponentialModHome_gr_146.gif],   where n is an integer.

Solution Method II.   Start with

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_147.gif]    

Then,   [Graphics:../Images/ComplexFunExponentialModHome_gr_148.gif],   where n is an integer.

Solution Method III.   Use equation (5-9)  [Graphics:../Images/ComplexFunExponentialModHome_gr_149.gif].  

Start with  [Graphics:../Images/ComplexFunExponentialModHome_gr_150.gif],  and compute  [Graphics:../Images/ComplexFunExponentialModHome_gr_151.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_152.gif],  and then  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_153.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_154.gif].  

We are done.   

Aside.  We can let Mathematica double check our work.

[Graphics:../Images/ComplexFunExponentialModHome_gr_155.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_156.gif]
[Graphics:../Images/ComplexFunExponentialModHome_gr_157.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_158.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_159.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_160.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_161.gif]

 

[Graphics:../Images/ComplexFunExponentialModHome_gr_162.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_163.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_164.gif]

                                                            Solutions to the equation   [Graphics:../Images/ComplexFunExponentialModHome_gr_165.gif].  Where the image point is  [Graphics:../Images/ComplexFunExponentialModHome_gr_166.gif],     
                                                            and the principal solution value is  [Graphics:../Images/ComplexFunExponentialModHome_gr_167.gif],   
                                                            and some of the points  [Graphics:../Images/ComplexFunExponentialModHome_gr_168.gif],   where n is an integer.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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