Exercise 9.  Explain why  

9 (b).  [Graphics:Images/ComplexFunExponentialModHome_gr_326.gif]   is nowhere analytic.  

Solution 9 (b).

See text and/or instructor's solution manual.

Solution.   

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_327.gif]    


          [Graphics:../Images/ComplexFunExponentialModHome_gr_328.gif],      [Graphics:../Images/ComplexFunExponentialModHome_gr_329.gif]     so that

[Graphics:../Images/ComplexFunExponentialModHome_gr_330.gif],   [Graphics:../Images/ComplexFunExponentialModHome_gr_331.gif],   [Graphics:../Images/ComplexFunExponentialModHome_gr_332.gif],   [Graphics:../Images/ComplexFunExponentialModHome_gr_333.gif].   

The  Cauchy Riemann equations are  

        [Graphics:../Images/ComplexFunExponentialModHome_gr_334.gif],  

        [Graphics:../Images/ComplexFunExponentialModHome_gr_335.gif].  

Hence we obtain  [Graphics:../Images/ComplexFunExponentialModHome_gr_336.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_337.gif],  which hold if and only if both  [Graphics:../Images/ComplexFunExponentialModHome_gr_338.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_339.gif],  which is impossible.

Therefore,  [Graphics:../Images/ComplexFunExponentialModHome_gr_340.gif]  is nowhere differentiable.  

We are done.   

Aside.  Another reason why f(z) is not analytic is given in Exercise 16 in Section 3.2.   

The complex form of the Cauchy-Riemann equations is  [Graphics:../Images/ComplexFunExponentialModHome_gr_341.gif].

The complex form of the Cauchy-Riemann equations fails to hold because  [Graphics:../Images/ComplexFunExponentialModHome_gr_342.gif].  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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