Example 3.9. Show
that the function
is
differentiable at points that lie on the x
and y axes but analytic nowhere.
Explore Solution 3.9.
Enter the functions u[x,y] and v[x,y] and use them to form f[z]. Then determine where the Cauchy-Riemann equations hold.
![[Graphics:../Images/CauchyRiemannMod_gr_242.gif]](../Images/CauchyRiemannMod_gr_242.gif)
The Cauchy-Riemann equations hold only if x y =
0 which occurs along the coordinate
axes. Furthermore, all of the partial
derivatives
,
,
,
and
are
continuous. Hence,
is differentiable only when
, which
occurs at points that lie on the coordinate
axes. Therefore,
is
nowhere analytic.