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

5 (c).  [Graphics:Images/ComplexFunExponentialModHome_gr_204.gif].

Solution 5 (c).

See text and/or instructor's solution manual.

Answer.   [Graphics:../Images/ComplexFunExponentialModHome_gr_205.gif],   where n is an integer.

Solution Method I.   Use the relation  [Graphics:../Images/ComplexFunExponentialModHome_gr_206.gif],  and observe that

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_207.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_208.gif].

Using polar coordinates  [Graphics:../Images/ComplexFunExponentialModHome_gr_209.gif]  we have  [Graphics:../Images/ComplexFunExponentialModHome_gr_210.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_211.gif]  calculate  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_212.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_213.gif].

Thus,   [Graphics:../Images/ComplexFunExponentialModHome_gr_214.gif],   where n is an integer.

Solution Method II.   Start with

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_215.gif]  

Then,   [Graphics:../Images/ComplexFunExponentialModHome_gr_216.gif],   where n is an integer.  

Solution Method III.   Use equation (5-9)  [Graphics:../Images/ComplexFunExponentialModHome_gr_217.gif].  

Start with  [Graphics:../Images/ComplexFunExponentialModHome_gr_218.gif],  and compute  [Graphics:../Images/ComplexFunExponentialModHome_gr_219.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_220.gif],  and then  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_221.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_222.gif].  

We are done.   

Aside.  We can let Mathematica double check our work.

[Graphics:../Images/ComplexFunExponentialModHome_gr_223.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_224.gif]
[Graphics:../Images/ComplexFunExponentialModHome_gr_225.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_226.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_227.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_228.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_229.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_230.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_231.gif]

 

[Graphics:../Images/ComplexFunExponentialModHome_gr_232.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_233.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_234.gif]

                                                            Solutions to the equation   [Graphics:../Images/ComplexFunExponentialModHome_gr_235.gif].  Where the image point is  [Graphics:../Images/ComplexFunExponentialModHome_gr_236.gif],   
                                                            and the principal solution value is  [Graphics:../Images/ComplexFunExponentialModHome_gr_237.gif],   
                                                            and some of the points  [Graphics:../Images/ComplexFunExponentialModHome_gr_238.gif],   where n is an integer. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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