Question

COMPLEX VARIABLES: how do you show that the function is nowhere analytic?
solution is on first page of the attached pdf number 2

Answer 1

what does it mean to be ANALYTIC????

Answer 2

A function is analytic if it obeys the Cauchy-Riemann equations.

Answer 3

so we just prove that the Cauchy riemann equation does not fit in the equation?

Answer 4

in the solution how do they come up with the equation where it is differentiable??

Answer 5

It is differentiable where the CR equations are satisfied. That is, if
\[f = u(x,y) + i\cdot v(x,y) \]
where u and v are real, differentiable functions,
\[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \]
and
\[ \frac{\partial u}{ \partial y} = -\frac{\partial v}{\partial x} \]

Answer 6

In your case,
\[ f = xy + i\cdot y\]
so u = xy and v = y. The first equation says
\[ y = 1 \]
and the second says
\[x = 0 \]
The only possible place where this function is differentiable is therefore when x= 0 and y = 1 .... that is, the point z= i . A function cannot be analytic at a single point, so it's nowhere analytic.

Answer 7

how do you know to set y = 1?

Answer 8

That's the first CR equation.

Answer 9

oh i get it now, thanks!!!

Answer 10

Just to clarify, for a function to be analytic it must satisfy the CR equations over some open domain -- meaning that if the equations are satisfied only at a point, but not in the immediate neighborhood of that point, the function is not analytic.

Answer 11

ok thanks!