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

5 (d).  [Graphics:Images/ComplexFunExponentialModHome_gr_239.gif].

Solution 5 (d).

See text and/or instructor's solution manual.

Answer.   [Graphics:../Images/ComplexFunExponentialModHome_gr_240.gif],   where n is an integer.

Solution Method I.   Use the relation  [Graphics:../Images/ComplexFunExponentialModHome_gr_241.gif],  and observe that

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_242.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_243.gif].

Using polar coordinates  [Graphics:../Images/ComplexFunExponentialModHome_gr_244.gif]  we have  [Graphics:../Images/ComplexFunExponentialModHome_gr_245.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_246.gif]  calculate  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_247.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_248.gif].

Thus,   [Graphics:../Images/ComplexFunExponentialModHome_gr_249.gif],   where n is an integer.

Solution Method II.   Start with

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_250.gif]  

Then,   [Graphics:../Images/ComplexFunExponentialModHome_gr_251.gif],   where n is an integer.

Solution Method III.   Use equation (5-9)  [Graphics:../Images/ComplexFunExponentialModHome_gr_252.gif].  

Start with  [Graphics:../Images/ComplexFunExponentialModHome_gr_253.gif],  and compute  [Graphics:../Images/ComplexFunExponentialModHome_gr_254.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_255.gif],  and then  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_256.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_257.gif].  

We are done.   

Aside.  We can let Mathematica double check our work.

[Graphics:../Images/ComplexFunExponentialModHome_gr_258.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_259.gif]
[Graphics:../Images/ComplexFunExponentialModHome_gr_260.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_261.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_262.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_263.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_264.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_265.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_266.gif]

 

[Graphics:../Images/ComplexFunExponentialModHome_gr_267.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_268.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_269.gif]

                                                            Solutions to the equation   [Graphics:../Images/ComplexFunExponentialModHome_gr_270.gif].  Where the image point is  [Graphics:../Images/ComplexFunExponentialModHome_gr_271.gif],   
                                                            and the principal solution value is  [Graphics:../Images/ComplexFunExponentialModHome_gr_272.gif],   
                                                            and some of the points  [Graphics:../Images/ComplexFunExponentialModHome_gr_273.gif],   where n is an integer. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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