Exercise 12.  Show that  [Graphics:Images/ComplexFunExponentialModHome_gr_366.gif]  is analytic for all  z  by showing that its real and imaginary parts

satisfy the Cauchy-Riemann sufficient conditions for differentiability.  

Solution 12.

See text and/or instructor's solution manual.

Solution.   

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_367.gif]

          [Graphics:../Images/ComplexFunExponentialModHome_gr_368.gif],   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_369.gif]     so that

[Graphics:../Images/ComplexFunExponentialModHome_gr_370.gif],   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_371.gif],   

[Graphics:../Images/ComplexFunExponentialModHome_gr_372.gif],   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_373.gif].   


The  Cauchy Riemann equations are  

        [Graphics:../Images/ComplexFunExponentialModHome_gr_374.gif],    and

        [Graphics:../Images/ComplexFunExponentialModHome_gr_375.gif],  

which hold for all  z.  

The partials are continuous everywhere, so   

[Graphics:../Images/ComplexFunExponentialModHome_gr_376.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_377.gif],  

for all  [Graphics:../Images/ComplexFunExponentialModHome_gr_378.gif].  

Indeed, we have   [Graphics:../Images/ComplexFunExponentialModHome_gr_379.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_380.gif],   and  

[Graphics:../Images/ComplexFunExponentialModHome_gr_381.gif],

[Graphics:../Images/ComplexFunExponentialModHome_gr_382.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_383.gif],

[Graphics:../Images/ComplexFunExponentialModHome_gr_384.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_385.gif].  

We are done.   

Aside.  We can let Mathematica double check our work.

[Graphics:../Images/ComplexFunExponentialModHome_gr_386.gif]
[Graphics:../Images/ComplexFunExponentialModHome_gr_387.gif]
[Graphics:../Images/ComplexFunExponentialModHome_gr_388.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_389.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_390.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_391.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_392.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_393.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_394.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_395.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_396.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_397.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_398.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_399.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_400.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_401.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_402.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_403.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_404.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_405.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_406.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_407.gif]


[Graphics:../Images/ComplexFunExponentialModHome_gr_408.gif]

[Graphics:../Images/ComplexFunExponentialModHome_gr_409.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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