Example 3.4.  We know that   [Graphics:Images/CauchyRiemannMod_gr_231.gif]   is differentiable and that   [Graphics:Images/CauchyRiemannMod_gr_232.gif].   

Furthermore, the Cartesian coordinate form for  [Graphics:Images/CauchyRiemannMod_gr_233.gif]  is

                    [Graphics:Images/CauchyRiemannMod_gr_234.gif].  

Use the Cartesian coordinate form of the Cauchy-Riemann equations and find  [Graphics:Images/CauchyRiemannMod_gr_235.gif].  

Solution.  It is easy to verify that Cauchy-Riemann equations (3-16) are indeed satisfied:

                    [Graphics:Images/CauchyRiemannMod_gr_236.gif]     and     [Graphics:Images/CauchyRiemannMod_gr_237.gif].  

Using Equations (3-14) and (3-15), respectively, to compute  [Graphics:Images/CauchyRiemannMod_gr_238.gif]  gives

                    [Graphics:Images/CauchyRiemannMod_gr_239.gif],     

                    and   

                    [Graphics:Images/CauchyRiemannMod_gr_240.gif],
                    
as expected.

 

Explore Solution 3.4.

 

Solution.  It is easy to verify that Cauchy-Riemann equations (3-16) are indeed satisfied:

                    [Graphics:../Images/CauchyRiemannMod_gr_241.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_242.gif].  

Using Equations (3-14) and (3-15), respectively, to compute  [Graphics:../Images/CauchyRiemannMod_gr_243.gif]  gives

                    [Graphics:../Images/CauchyRiemannMod_gr_244.gif],     

                    and   

                    [Graphics:../Images/CauchyRiemannMod_gr_245.gif],
                    
as expected.

 

We are done.

 

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_246.gif] and [Graphics:../Images/CauchyRiemannMod_gr_247.gif] can assist us in finding the partial derivatives.  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_248.gif]

[Graphics:../Images/CauchyRiemannMod_gr_249.gif]


[Graphics:../Images/CauchyRiemannMod_gr_250.gif]

[Graphics:../Images/CauchyRiemannMod_gr_251.gif]


[Graphics:../Images/CauchyRiemannMod_gr_252.gif]

[Graphics:../Images/CauchyRiemannMod_gr_253.gif]


[Graphics:../Images/CauchyRiemannMod_gr_254.gif]

[Graphics:../Images/CauchyRiemannMod_gr_255.gif]


[Graphics:../Images/CauchyRiemannMod_gr_256.gif]

[Graphics:../Images/CauchyRiemannMod_gr_257.gif]


[Graphics:../Images/CauchyRiemannMod_gr_258.gif]

[Graphics:../Images/CauchyRiemannMod_gr_259.gif]


[Graphics:../Images/CauchyRiemannMod_gr_260.gif]

[Graphics:../Images/CauchyRiemannMod_gr_261.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_262.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_263.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_264.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_265.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_266.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_267.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_268.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_269.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_270.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_271.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_272.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_273.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_274.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_275.gif]

 

The Cauchy-Riemann equations hold  all points   [Graphics:../Images/CauchyRiemannMod_gr_276.gif]   in the complex plane,

therefore   [Graphics:../Images/CauchyRiemannMod_gr_277.gif]   is an entire function.  

Verify that the derivative can be calculated with either of the formulas:

(3-14)              [Graphics:../Images/CauchyRiemannMod_gr_278.gif],     

                        or  

(3-15)              [Graphics:../Images/CauchyRiemannMod_gr_279.gif].  

 

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_280.gif]

[Graphics:../Images/CauchyRiemannMod_gr_281.gif]


[Graphics:../Images/CauchyRiemannMod_gr_282.gif]

[Graphics:../Images/CauchyRiemannMod_gr_283.gif]


[Graphics:../Images/CauchyRiemannMod_gr_284.gif]

[Graphics:../Images/CauchyRiemannMod_gr_285.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_286.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_287.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_288.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_289.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_290.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_291.gif]

Therefore, both Mathematica and Maple have shown that if  

                    [Graphics:../Images/CauchyRiemannMod_gr_292.gif],   

then the derivative is

                    [Graphics:../Images/CauchyRiemannMod_gr_293.gif],   

as expected.

 

We are really done.

 

Aside.  Figure E.3.4 a, shows the graphs of   [Graphics:../Images/CauchyRiemannMod_gr_294.gif]   and   [Graphics:../Images/CauchyRiemannMod_gr_295.gif].  

The partial derivatives of  [Graphics:../Images/CauchyRiemannMod_gr_296.gif]  are  

                    [Graphics:../Images/CauchyRiemannMod_gr_297.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_298.gif],   

and the partial derivatives of  [Graphics:../Images/CauchyRiemannMod_gr_299.gif]  are

                    [Graphics:../Images/CauchyRiemannMod_gr_300.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_301.gif].  
        
At the point   [Graphics:../Images/CauchyRiemannMod_gr_302.gif],   we have   [Graphics:../Images/CauchyRiemannMod_gr_303.gif]   and   [Graphics:../Images/CauchyRiemannMod_gr_304.gif],   and these partial derivatives appear along

the edges of the surfaces for  [Graphics:../Images/CauchyRiemannMod_gr_305.gif]  at the points   [Graphics:../Images/CauchyRiemannMod_gr_306.gif]   and   [Graphics:../Images/CauchyRiemannMod_gr_307.gif],   respectively.

Similarly,  at the point   [Graphics:../Images/CauchyRiemannMod_gr_308.gif],   we have   [Graphics:../Images/CauchyRiemannMod_gr_309.gif]   and   [Graphics:../Images/CauchyRiemannMod_gr_310.gif]   and these partial derivatives appear

along the edges of the surfaces for [Graphics:../Images/CauchyRiemannMod_gr_311.gif] at the points [Graphics:../Images/CauchyRiemannMod_gr_312.gif] and  [Graphics:../Images/CauchyRiemannMod_gr_313.gif],  respectively.

 

[Graphics:../Images/CauchyRiemannMod_gr_314.gif]     [Graphics:../Images/CauchyRiemannMod_gr_315.gif]  

                    [Graphics:../Images/CauchyRiemannMod_gr_316.gif].                                                                          [Graphics:../Images/CauchyRiemannMod_gr_317.gif].
                                                                                  Figure E.3.4 a

 

[Graphics:../Images/CauchyRiemannMod_gr_318.gif]     [Graphics:../Images/CauchyRiemannMod_gr_319.gif]  

                    [Graphics:../Images/CauchyRiemannMod_gr_320.gif],                                                                         [Graphics:../Images/CauchyRiemannMod_gr_321.gif],  

                    at [Graphics:../Images/CauchyRiemannMod_gr_322.gif] we have [Graphics:../Images/CauchyRiemannMod_gr_323.gif].                                                at [Graphics:../Images/CauchyRiemannMod_gr_324.gif] we have  [Graphics:../Images/CauchyRiemannMod_gr_325.gif].  
                                                                                  Figure E.3.4 b

 

[Graphics:../Images/CauchyRiemannMod_gr_326.gif]     [Graphics:../Images/CauchyRiemannMod_gr_327.gif]  

                    [Graphics:../Images/CauchyRiemannMod_gr_328.gif],                                                                         [Graphics:../Images/CauchyRiemannMod_gr_329.gif],  

                    at [Graphics:../Images/CauchyRiemannMod_gr_330.gif] we have [Graphics:../Images/CauchyRiemannMod_gr_331.gif].                                                at [Graphics:../Images/CauchyRiemannMod_gr_332.gif] we have  .  
                                                                                  Figure E.3.4 c

 

                                        For the function   [Graphics:../Images/CauchyRiemannMod_gr_334.gif]   we see that

                                        [Graphics:../Images/CauchyRiemannMod_gr_335.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_336.gif].  

                                                                                  Figure E.3.4

 

Remark 1.  You might wonder why we need the Cauchy-Riemann equations to differentiate a well known function.

We are preparing for Section 3.3 where the Cauchy-Riemann equations are used to construct a conjugate harmonic function.

 

Remark 2.  In this book the use of computers is optional.  

Hopefully this text will promote their use and understanding.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2011 John H. Mathews, Russell W. Howell