Example 5.5.  Show that the identity  [Graphics:Images/ComplexFunLogarithmMod_gr_87.gif]  is not always valid.  

Explore Solution 5.5.

Enter  [Graphics:../Images/ComplexFunLogarithmMod_gr_93.gif]  and take their logarithms.

[Graphics:../Images/ComplexFunLogarithmMod_gr_94.gif]




[Graphics:../Images/ComplexFunLogarithmMod_gr_95.gif]

 

So we find in this counter example that it is possible for  [Graphics:../Images/ComplexFunLogarithmMod_gr_96.gif],

this is in contrast to the multivalued logarithm for which  .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell