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

5 (b).  [Graphics:Images/ComplexFunExponentialModHome_gr_169.gif].

Solution 5 (b).

See text and/or instructor's solution manual.

Answer.   [Graphics:../Images/ComplexFunExponentialModHome_gr_170.gif],   where n is an integer.

Solution Method I.   Use the relation  [Graphics:../Images/ComplexFunExponentialModHome_gr_171.gif],  and observe that

          [Graphics:../Images/ComplexFunExponentialModHome_gr_172.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_173.gif].

Using polar coordinates  [Graphics:../Images/ComplexFunExponentialModHome_gr_174.gif]  we have  [Graphics:../Images/ComplexFunExponentialModHome_gr_175.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_176.gif]  calculate  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_177.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_178.gif].

Thus,   [Graphics:../Images/ComplexFunExponentialModHome_gr_179.gif],   where n is an integer.

Solution Method II.   Start with

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_180.gif]   

Then,     [Graphics:../Images/ComplexFunExponentialModHome_gr_181.gif],   where n is an integer.

Solution Method III.   Use equation (5-9)  [Graphics:../Images/ComplexFunExponentialModHome_gr_182.gif].  

Start with  [Graphics:../Images/ComplexFunExponentialModHome_gr_183.gif],  and compute  [Graphics:../Images/ComplexFunExponentialModHome_gr_184.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_185.gif],  and then  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_186.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_187.gif].  

We are done.   

Aside.  We can let Mathematica double check our work.

[Graphics:../Images/ComplexFunExponentialModHome_gr_188.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_189.gif]
[Graphics:../Images/ComplexFunExponentialModHome_gr_190.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_191.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_192.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_193.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_194.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_195.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_196.gif]

 

[Graphics:../Images/ComplexFunExponentialModHome_gr_197.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_198.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_199.gif]

                                                            Solutions to the equation   [Graphics:../Images/ComplexFunExponentialModHome_gr_200.gif].  Where the image point is  [Graphics:../Images/ComplexFunExponentialModHome_gr_201.gif],  
                                                            and the principal solution value is  [Graphics:../Images/ComplexFunExponentialModHome_gr_202.gif],   
                                                            and some of the points  [Graphics:../Images/ComplexFunExponentialModHome_gr_203.gif],   where n is an integer. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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