Exercise 15.  Explain why no branch of the logarithm is defined when  [Graphics:Images/ComplexFunLogarithmModHome_gr_1207.gif].  

Solution 15.

See text and/or instructor's solution manual.

Solution.   Any branch of the logarithm is defined as an inverse of the exponential.  

Since there is no value z for which  [Graphics:../Images/ComplexFunLogarithmModHome_gr_1208.gif],  there can be no branch of the logarithm that is defined at  [Graphics:../Images/ComplexFunLogarithmModHome_gr_1209.gif].  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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