Exercise 17.  Use the fact that  [Graphics:Images/ComplexFunExponentialModHome_gr_510.gif]  is analytic to show that  [Graphics:Images/ComplexFunExponentialModHome_gr_511.gif]  is a harmonic function.  

Solution 17.

See text and/or instructor's solution manual.

Answer.   Show that   [Graphics:../Images/ComplexFunExponentialModHome_gr_512.gif]  is the imaginary part of  [Graphics:../Images/ComplexFunExponentialModHome_gr_513.gif],  and therefore harmonic by Theorem 3.8 in Section 3.3.  

Solution.   

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_514.gif]    

Theorem 3.8 in Section 3.3  states that if  [Graphics:../Images/ComplexFunExponentialModHome_gr_515.gif] is analytic then both  [Graphics:../Images/ComplexFunExponentialModHome_gr_516.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_517.gif]  are harmonic functions.  

Therefore  [Graphics:../Images/ComplexFunExponentialModHome_gr_518.gif]  is a harmonic function.  

We are done.   

Aside.  We can let Mathematica double check our work.

[Graphics:../Images/ComplexFunExponentialModHome_gr_519.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_520.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_521.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_522.gif]

Aside.  It is always possible to see if Laplace's equation holds.

[Graphics:../Images/ComplexFunExponentialModHome_gr_523.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_524.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_525.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_526.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_527.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_528.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_529.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_530.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_531.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_532.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c)