Exercise 1.  Find the principal value of  

1 (c).  [Graphics:Images/ComplexFunComplexPowerModHome_gr_19.gif].  

Solution 1 (c).

See text and/or instructor's solution manual.

Answer.   The principal value is   [Graphics:../Images/ComplexFunComplexPowerModHome_gr_20.gif].

Solution.   Use  [Graphics:../Images/ComplexFunComplexPowerModHome_gr_21.gif],  where  [Graphics:../Images/ComplexFunComplexPowerModHome_gr_22.gif].   Calculation reveals  

                    [Graphics:../Images/ComplexFunComplexPowerModHome_gr_23.gif]  

We are done.   

Aside.  We can let Mathematica double check our work.

[Graphics:../Images/ComplexFunComplexPowerModHome_gr_24.gif]

[Graphics:../Images/ComplexFunComplexPowerModHome_gr_25.gif]



[Graphics:../Images/ComplexFunComplexPowerModHome_gr_26.gif]

[Graphics:../Images/ComplexFunComplexPowerModHome_gr_27.gif]

                              [Graphics:../Images/ComplexFunComplexPowerModHome_gr_28.gif]

          Some of the points  [Graphics:../Images/ComplexFunComplexPowerModHome_gr_29.gif]  (the principal value  [Graphics:../Images/ComplexFunComplexPowerModHome_gr_30.gif]  is in green).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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