# Thread: Approaching complex problem.....

1. ## Approaching complex problem.....

Lets say we have a function: $\displaystyle f(z) = x+i|y|$.

We want to find all point that is it differentiable. Right of the bat we know that it is differentiable for $\displaystyle y >0$. For $\displaystyle f(z) = x+iy$ is a "direct" function of $\displaystyle x+iy$. E.g. we do not have to appeal to CR equations.

Also it is not differentiable for $\displaystyle y<0$ since $\displaystyle f(z) = x-iy$ is not a "direct" function of $\displaystyle x+iy$.

Now for $\displaystyle y = 0$ can we use the same argument to show that $\displaystyle f(z) = x+i|y|$ is not differentiable there?

Because then we have $\displaystyle f(z) = x$. And this is not a "direct" function of $\displaystyle x+iy$.

2. I mean analytic takes care of everything. Its stronger then differentiability. Thats what "function of z" is.

3. Originally Posted by heathrowjohnny
Lets say we have a function: $\displaystyle f(z) = x+i|y|$.

We want to find all point that is it differentiable. Right of the bat we know that it is differentiable for $\displaystyle y >0$. For $\displaystyle f(z) = x+iy$ is a "direct" function of $\displaystyle x+iy$. E.g. we do not have to appeal to CR equations.

Also it is not differentiable for $\displaystyle y<0$ since $\displaystyle f(z) = x-iy$ is not a "direct" function of $\displaystyle x+iy$.

Now for $\displaystyle y = 0$ can we use the same argument to show that $\displaystyle f(z) = x+i|y|$ is not differentiable there?

Because then we have $\displaystyle f(z) = x$. And this is not a "direct" function of $\displaystyle x+iy$.
For $\displaystyle y=0$ you can use the definition of differenciable, $\displaystyle \lim_{z\to z_0} \frac{f(z)-f(z_0)}{z-z_0}$. Now $\displaystyle z=x+iy$ and $\displaystyle z_0 = x$. Therefore, $\displaystyle \lim_{(x,y)\to (x_0,0)}\frac{x+i|y| - x}{iy} = \lim_{y\to 0}\frac{y}{|y|}$.
And this limit does not exist.

4. so you can't just say that $\displaystyle f(z) = x$ is not an "analytic function" of $\displaystyle x+iy$?