Exercise 15.  Show that   [Graphics:Images/ComplexFunExponentialModHome_gr_438.gif]   converges for   [Graphics:Images/ComplexFunExponentialModHome_gr_439.gif].   

Solution 15.

See text and/or instructor's solution manual.

Solution.   [Graphics:../Images/ComplexFunExponentialModHome_gr_440.gif].  

This is a geometric series  of the form   [Graphics:../Images/ComplexFunExponentialModHome_gr_441.gif]   where   [Graphics:../Images/ComplexFunExponentialModHome_gr_442.gif]   and converges for  [Graphics:../Images/ComplexFunExponentialModHome_gr_443.gif]  by Theorem 4.12 in Section 4.3.

We know that   [Graphics:../Images/ComplexFunExponentialModHome_gr_444.gif].

Taking the modulus we have  [Graphics:../Images/ComplexFunExponentialModHome_gr_445.gif].   Then

[Graphics:../Images/ComplexFunExponentialModHome_gr_446.gif]    iff    [Graphics:../Images/ComplexFunExponentialModHome_gr_447.gif]    iff    [Graphics:../Images/ComplexFunExponentialModHome_gr_448.gif]    iff    [Graphics:../Images/ComplexFunExponentialModHome_gr_449.gif]   iff    [Graphics:../Images/ComplexFunExponentialModHome_gr_450.gif]    iff     [Graphics:../Images/ComplexFunExponentialModHome_gr_451.gif].   

Therefore,     [Graphics:../Images/ComplexFunExponentialModHome_gr_452.gif]   converges for   [Graphics:../Images/ComplexFunExponentialModHome_gr_453.gif].   

We are done.   

Aside.  We can see what Mathematica does

[Graphics:../Images/ComplexFunExponentialModHome_gr_454.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_455.gif]

We are really done.   

Warning.  Mathematica has plugged  [Graphics:../Images/ComplexFunExponentialModHome_gr_456.gif]  into  [Graphics:../Images/ComplexFunExponentialModHome_gr_457.gif].  

There has not been a check to see if  [Graphics:../Images/ComplexFunExponentialModHome_gr_458.gif] !

Do not blindly trust computer answer !

The point   [Graphics:../Images/ComplexFunExponentialModHome_gr_459.gif]  satisfies  [Graphics:../Images/ComplexFunExponentialModHome_gr_460.gif].

        If you substitute  [Graphics:../Images/ComplexFunExponentialModHome_gr_461.gif]  into  [Graphics:../Images/ComplexFunExponentialModHome_gr_462.gif],  then the you will get  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_463.gif].  

        The point  [Graphics:../Images/ComplexFunExponentialModHome_gr_464.gif]  has   [Graphics:../Images/ComplexFunExponentialModHome_gr_465.gif]   and does not satisfy  [Graphics:../Images/ComplexFunExponentialModHome_gr_466.gif].

If you substitute  [Graphics:../Images/ComplexFunExponentialModHome_gr_467.gif] into  [Graphics:../Images/ComplexFunExponentialModHome_gr_468.gif],  then the series does not converge.

[Graphics:../Images/ComplexFunExponentialModHome_gr_469.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_470.gif]
[Graphics:../Images/ComplexFunExponentialModHome_gr_471.gif]

If it is difficult to imagine that  [Graphics:../Images/ComplexFunExponentialModHome_gr_472.gif]  diverges, then just look at the terms  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_473.gif]  

We are really really done.   

Caveat.  If we are only interested in the formula  [Graphics:../Images/ComplexFunExponentialModHome_gr_474.gif]  the image of the right half-plane  [Graphics:../Images/ComplexFunExponentialModHome_gr_475.gif]  is easy to find.

However, if we are only interested in the formula  [Graphics:../Images/ComplexFunExponentialModHome_gr_476.gif]  the image of the right left-plane  [Graphics:../Images/ComplexFunExponentialModHome_gr_477.gif]  is also easy to find.

Would you be shocked to find out that these images regions are the same ?  Hold on to your hats !

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_478.gif]          [Graphics:../Images/ComplexFunExponentialModHome_gr_479.gif]


                    
                    [Graphics:../Images/ComplexFunExponentialModHome_gr_480.gif]          [Graphics:../Images/ComplexFunExponentialModHome_gr_481.gif]

                                        The composite mapping  [Graphics:../Images/ComplexFunExponentialModHome_gr_482.gif],  where   [Graphics:../Images/ComplexFunExponentialModHome_gr_483.gif],   [Graphics:../Images/ComplexFunExponentialModHome_gr_484.gif],   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_485.gif].  

 

 

                                        [Graphics:../Images/ComplexFunExponentialModHome_gr_486.gif]          [Graphics:../Images/ComplexFunExponentialModHome_gr_487.gif]


                                        
                                        [Graphics:../Images/ComplexFunExponentialModHome_gr_488.gif]          [Graphics:../Images/ComplexFunExponentialModHome_gr_489.gif]

                                        The composite mapping  [Graphics:../Images/ComplexFunExponentialModHome_gr_490.gif],  where   [Graphics:../Images/ComplexFunExponentialModHome_gr_491.gif],   [Graphics:../Images/ComplexFunExponentialModHome_gr_492.gif],   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_493.gif].  

Aside.  This can be viewed as a source and sink.  We will learn more about sources and sinks in Section 11.11.

 

Aside. How good is your eye? Did you notice the subtle difference in the above graphs. Look again in the W-planes.

 

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_479.gif]                    [Graphics:../Images/ComplexFunExponentialModHome_gr_487.gif]

 

 Always remember that the exponential function is periodic.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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