Cauchy-Riemann $w = |z^2|$

Question

So for these types of questions, I can compute the partial differentials for Cauchy-Riemann but then I have trouble seeing/explaining where the function is differentiable?

For example with this question I end up with

$\partial u/\partial x = 2x$

$\partial v/\partial y = 0 $

$\partial u/\partial y = 2y $

$\partial v/\partial x = 0 $

Now what?

Could you include what you determined $u$ and $v$ to be? I got something different. — Michael Albanese, Jun 19 '14 at 05:35

tattwamasi amrutam · Answer 1 · 2014-06-19T06:42:25.677

0

For a function to be analytic at a point, partial derivatives must exist and they should satisfy cauchy-riemann equations, which says $u_x=v_y$ and $u_y=-v_x$.

Since you have got $u_x=2x$ and $v_y=0$, cauchy riemann equation gives $2x=0$ and hence $x=0$. Similarly $u_y=-v_x$ gives $2y=0$ and hence $y=0$.

So In here the function is only analytic at $(0,0)$.

edited Jun 19 '14 at 06:42

answered Jun 19 '14 at 05:36

tattwamasi amrutam

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I understand what it should satisfy, but I don't understand why it is only analytic at (0,0) from that? Could you explain perhaps? Thanks – MathHelp1 Jun 19 '14 at 05:46
@MathHelp1 now take a look – tattwamasi amrutam Jun 19 '14 at 06:43
1

Ah, now it makes sense! Thanks. – MathHelp1 Jun 19 '14 at 06:44

Cauchy-Riemann $w = |z^2|$

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