Exercise 1.  Using Definition 5.1, explain why  [Graphics:Images/ComplexFunExponentialModHome_gr_1.gif].  

Solution 1.

See text and/or instructor's solution manual.

Solution.   Recall that   [Graphics:../Images/ComplexFunExponentialModHome_gr_2.gif]   is compact notation for   [Graphics:../Images/ComplexFunExponentialModHome_gr_3.gif],   and that   [Graphics:../Images/ComplexFunExponentialModHome_gr_4.gif],   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_5.gif] for all  [Graphics:../Images/ComplexFunExponentialModHome_gr_6.gif].    

Then, by definition  [Graphics:../Images/ComplexFunExponentialModHome_gr_7.gif],  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_8.gif]  so that  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_9.gif]

We are done.   

Aside.  The reason we split off the constant term  [Graphics:../Images/ComplexFunExponentialModHome_gr_10.gif]  is to avoid the unpleasant computation  [Graphics:../Images/ComplexFunExponentialModHome_gr_11.gif].  

What is the mathematical definition of  "[Graphics:../Images/ComplexFunExponentialModHome_gr_12.gif] ?  In calculus we call it an indeterminate form."

For the computer software Mathematica it results in the following error message:

[Graphics:../Images/ComplexFunExponentialModHome_gr_13.gif]
[Graphics:../Images/ComplexFunExponentialModHome_gr_14.gif]
[Graphics:../Images/ComplexFunExponentialModHome_gr_15.gif]

For the computer software [Graphics:../Images/ComplexFunExponentialModHome_gr_16.gif] it results in the following calculation:

[>  0^0;

                           1

In Section 5.2 we will see that  log(z) is defined,  for  [Graphics:../Images/ComplexFunExponentialModHome_gr_17.gif].  In Section 5.3 we will define  [Graphics:../Images/ComplexFunExponentialModHome_gr_18.gif],  for  [Graphics:../Images/ComplexFunExponentialModHome_gr_19.gif].  

If you are curious about the "[Graphics:../Images/ComplexFunExponentialModHome_gr_20.gif] controversy you can read the paragraph Zero to the zero power" on Wikipedia.org ,
the URL is:  http://en.wikipedia.org/wiki/Exponentiation#Powers_of_zero.  
Another paragraph Zero to the zero power is found on the page http://en.wikipedia.org/wiki/Defined_and_undefined

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.