Example 3.6.  Show that the function defined by

                    [Graphics:Images/CauchyRiemannMod_gr_375.gif]  

is not differentiable at the point  [Graphics:Images/CauchyRiemannMod_gr_376.gif]  even though the Cauchy-Riemann equations (3-16) are satisfied at the point  [Graphics:Images/CauchyRiemannMod_gr_377.gif].  

 

Explore Solution 3.6.

 

Solution.   We must use limits to calculate the partial derivatives at  [Graphics:../Images/CauchyRiemannMod_gr_401.gif].  

                    [Graphics:../Images/CauchyRiemannMod_gr_402.gif],

                    [Graphics:../Images/CauchyRiemannMod_gr_403.gif],
                    
                    [Graphics:../Images/CauchyRiemannMod_gr_404.gif],   

                    [Graphics:../Images/CauchyRiemannMod_gr_405.gif].

Thus, we can see that   

                    [Graphics:../Images/CauchyRiemannMod_gr_406.gif],     and     [Graphics:../Images/CauchyRiemannMod_gr_407.gif].  

Hence the Cauchy-Riemann equations (3-16) hold at the point  [Graphics:../Images/CauchyRiemannMod_gr_408.gif].  

        We now use Equation (3-1),   [Graphics:../Images/CauchyRiemannMod_gr_409.gif],   from Section 3.1,

and show that  [Graphics:../Images/CauchyRiemannMod_gr_410.gif]  is not differentiable at the point  [Graphics:../Images/CauchyRiemannMod_gr_411.gif].   We do this by choosing two paths

that go through the origin and compute the limit of the difference quotient along each path.

First, let [Graphics:../Images/CauchyRiemannMod_gr_412.gif] approach  [Graphics:../Images/CauchyRiemannMod_gr_413.gif]  along the [Graphics:../Images/CauchyRiemannMod_gr_414.gif]-axis, given by the parametric equations  [Graphics:../Images/CauchyRiemannMod_gr_415.gif],  then  

                    [Graphics:../Images/CauchyRiemannMod_gr_416.gif]  

Second, let [Graphics:../Images/CauchyRiemannMod_gr_417.gif] approach  [Graphics:../Images/CauchyRiemannMod_gr_418.gif]  along the line  [Graphics:../Images/CauchyRiemannMod_gr_419.gif],  given by the parametric equations  [Graphics:../Images/CauchyRiemannMod_gr_420.gif],  then  

                    [Graphics:../Images/CauchyRiemannMod_gr_421.gif]  

        The limits along the two paths are different, so there is no possible value for the right side of Equation (3-1).  

Therefore,   [Graphics:../Images/CauchyRiemannMod_gr_422.gif]  is not differentiable at the point   [Graphics:../Images/CauchyRiemannMod_gr_423.gif].

 

We are done.

 

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_424.gif] and [Graphics:../Images/CauchyRiemannMod_gr_425.gif] can assist us in finding the partial derivatives.  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_426.gif]

[Graphics:../Images/CauchyRiemannMod_gr_427.gif]


[Graphics:../Images/CauchyRiemannMod_gr_428.gif]

[Graphics:../Images/CauchyRiemannMod_gr_429.gif]


[Graphics:../Images/CauchyRiemannMod_gr_430.gif]

[Graphics:../Images/CauchyRiemannMod_gr_431.gif]


[Graphics:../Images/CauchyRiemannMod_gr_432.gif]

[Graphics:../Images/CauchyRiemannMod_gr_433.gif]


[Graphics:../Images/CauchyRiemannMod_gr_434.gif]

[Graphics:../Images/CauchyRiemannMod_gr_435.gif]


[Graphics:../Images/CauchyRiemannMod_gr_436.gif]

[Graphics:../Images/CauchyRiemannMod_gr_437.gif]


[Graphics:../Images/CauchyRiemannMod_gr_438.gif]

[Graphics:../Images/CauchyRiemannMod_gr_439.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_440.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_441.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_442.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_443.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_444.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_445.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_446.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_447.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_448.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_449.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_450.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_451.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_452.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_453.gif]

Unfortunately, all of these partial derivatives,  [Graphics:../Images/CauchyRiemannMod_gr_454.gif],  are not defined at the point [Graphics:../Images/CauchyRiemannMod_gr_455.gif],  and from

the looks of these formulas there is no magical way to guess the values   [Graphics:../Images/CauchyRiemannMod_gr_456.gif].

Therefore, it is necessary to use the limit definition of partial derivatives to properly determine these values.  

 

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_457.gif] and [Graphics:../Images/CauchyRiemannMod_gr_458.gif] can assist us in finding the limits.  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_459.gif]

[Graphics:../Images/CauchyRiemannMod_gr_460.gif]


[Graphics:../Images/CauchyRiemannMod_gr_461.gif]

[Graphics:../Images/CauchyRiemannMod_gr_462.gif]


[Graphics:../Images/CauchyRiemannMod_gr_463.gif]

[Graphics:../Images/CauchyRiemannMod_gr_464.gif]


[Graphics:../Images/CauchyRiemannMod_gr_465.gif]

[Graphics:../Images/CauchyRiemannMod_gr_466.gif]


[Graphics:../Images/CauchyRiemannMod_gr_467.gif]

[Graphics:../Images/CauchyRiemannMod_gr_468.gif]


[Graphics:../Images/CauchyRiemannMod_gr_469.gif]

[Graphics:../Images/CauchyRiemannMod_gr_470.gif]

This shows that   [Graphics:../Images/CauchyRiemannMod_gr_471.gif].

Thus, Mathematica has shown that the Cauchy-Riemann equations hold at the origin.   

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_472.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_473.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_474.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_475.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_476.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_477.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_478.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_479.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_480.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_481.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_482.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_483.gif]

This shows that   [Graphics:../Images/CauchyRiemannMod_gr_484.gif].

Thus, Maple has shown that the Cauchy-Riemann equations hold at the origin.   

 

        Now use Equation (3-1),   [Graphics:../Images/CauchyRiemannMod_gr_485.gif],   from Section 3.1,

and show that  [Graphics:../Images/CauchyRiemannMod_gr_486.gif]  is not differentiable at  [Graphics:../Images/CauchyRiemannMod_gr_487.gif].   

We do this by choosing two paths that go through the origin and compute the limit of the difference quotient along each path.

First, let [Graphics:../Images/CauchyRiemannMod_gr_488.gif] approach  [Graphics:../Images/CauchyRiemannMod_gr_489.gif]  along the [Graphics:../Images/CauchyRiemannMod_gr_490.gif]-axis, given by the parametric equations  [Graphics:../Images/CauchyRiemannMod_gr_491.gif].  

Second, let [Graphics:../Images/CauchyRiemannMod_gr_492.gif] approach  [Graphics:../Images/CauchyRiemannMod_gr_493.gif]  along the line  [Graphics:../Images/CauchyRiemannMod_gr_494.gif],  given by the parametric equations  [Graphics:../Images/CauchyRiemannMod_gr_495.gif].  

 

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_496.gif]

[Graphics:../Images/CauchyRiemannMod_gr_497.gif]


[Graphics:../Images/CauchyRiemannMod_gr_498.gif]

[Graphics:../Images/CauchyRiemannMod_gr_499.gif]


[Graphics:../Images/CauchyRiemannMod_gr_500.gif]

[Graphics:../Images/CauchyRiemannMod_gr_501.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_502.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_503.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_504.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_505.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_506.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_507.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_508.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_509.gif]

 

        The limits along the two paths are different, so there is no possible value for the right side of Equation (3-1).  

Therefore,   [Graphics:../Images/CauchyRiemannMod_gr_510.gif]  is not differentiable at the point   [Graphics:../Images/CauchyRiemannMod_gr_511.gif].

 

We are really done.

 

Aside.  Figure E.6 a, shows the graphs of   [Graphics:../Images/CauchyRiemannMod_gr_512.gif]   for   [Graphics:../Images/CauchyRiemannMod_gr_513.gif]   and   [Graphics:../Images/CauchyRiemannMod_gr_514.gif]   for   [Graphics:../Images/CauchyRiemannMod_gr_515.gif] .  

The partial derivatives of  [Graphics:../Images/CauchyRiemannMod_gr_516.gif],  for  [Graphics:../Images/CauchyRiemannMod_gr_517.gif],  are  

                    [Graphics:../Images/CauchyRiemannMod_gr_518.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_519.gif],   

and the partial derivatives of  [Graphics:../Images/CauchyRiemannMod_gr_520.gif],  for  [Graphics:../Images/CauchyRiemannMod_gr_521.gif],  are

                    [Graphics:../Images/CauchyRiemannMod_gr_522.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_523.gif].  

Notice that at the point   [Graphics:../Images/CauchyRiemannMod_gr_524.gif],   we have   [Graphics:../Images/CauchyRiemannMod_gr_525.gif]   and   [Graphics:../Images/CauchyRiemannMod_gr_526.gif],   and these partial derivatives

appear along the edges of the surfaces for  [Graphics:../Images/CauchyRiemannMod_gr_527.gif]  at the point  [Graphics:../Images/CauchyRiemannMod_gr_528.gif].

Similarly,  at the point   [Graphics:../Images/CauchyRiemannMod_gr_529.gif],   we have   [Graphics:../Images/CauchyRiemannMod_gr_530.gif]   and   [Graphics:../Images/CauchyRiemannMod_gr_531.gif]   and these partial derivatives

appear along the edges of the surfaces for  [Graphics:../Images/CauchyRiemannMod_gr_532.gif]  at the point  [Graphics:../Images/CauchyRiemannMod_gr_533.gif].

 

[Graphics:../Images/CauchyRiemannMod_gr_534.gif]     [Graphics:../Images/CauchyRiemannMod_gr_535.gif]  

                    [Graphics:../Images/CauchyRiemannMod_gr_536.gif]  for  [Graphics:../Images/CauchyRiemannMod_gr_537.gif].                                         [Graphics:../Images/CauchyRiemannMod_gr_538.gif]  for  [Graphics:../Images/CauchyRiemannMod_gr_539.gif].
                                                                                  Figure E.3.6 a

 

[Graphics:../Images/CauchyRiemannMod_gr_540.gif]     [Graphics:../Images/CauchyRiemannMod_gr_541.gif]  

                    [Graphics:../Images/CauchyRiemannMod_gr_542.gif]  for  [Graphics:../Images/CauchyRiemannMod_gr_543.gif],                                         [Graphics:../Images/CauchyRiemannMod_gr_544.gif]  for  [Graphics:../Images/CauchyRiemannMod_gr_545.gif],  

                     at [Graphics:../Images/CauchyRiemannMod_gr_546.gif] we have [Graphics:../Images/CauchyRiemannMod_gr_547.gif].                                          at [Graphics:../Images/CauchyRiemannMod_gr_548.gif] we have  [Graphics:../Images/CauchyRiemannMod_gr_549.gif].  
                                                                                  Figure E.3.6 b

 

[Graphics:../Images/CauchyRiemannMod_gr_550.gif]     [Graphics:../Images/CauchyRiemannMod_gr_551.gif]  

                    [Graphics:../Images/CauchyRiemannMod_gr_552.gif]  for  [Graphics:../Images/CauchyRiemannMod_gr_553.gif],                                         [Graphics:../Images/CauchyRiemannMod_gr_554.gif]  for  [Graphics:../Images/CauchyRiemannMod_gr_555.gif],  

                    at [Graphics:../Images/CauchyRiemannMod_gr_556.gif] we have [Graphics:../Images/CauchyRiemannMod_gr_557.gif].                                           at [Graphics:../Images/CauchyRiemannMod_gr_558.gif] we have  .  
                                                                                  Figure E.3.6 c

 

                                        For the function   [Graphics:../Images/CauchyRiemannMod_gr_560.gif]we see that

                                        [Graphics:../Images/CauchyRiemannMod_gr_561.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_562.gif].  

                                                                                  Figure E.3.6

 

Conclusion.  This function   [Graphics:../Images/CauchyRiemannMod_gr_563.gif]is a classic example of a function for which

the Cauchy-Riemann equations hold at the point  [Graphics:../Images/CauchyRiemannMod_gr_564.gif],  yet  [Graphics:../Images/CauchyRiemannMod_gr_565.gif]  is not differentiable at the point  [Graphics:../Images/CauchyRiemannMod_gr_566.gif].

 

Remark.  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