# Check if the complex function is differentiable

by Fabio010
Tags: check, complex, differentiable, function
 P: 83 The question is to check where the following complex function is differentiable. $$w=z \left| z\right|$$ $$w=\sqrt{x^2+y^2} (x+i y)$$ $$u = x\sqrt{x^2+y^2}$$ $$v = y\sqrt{x^2+y^2}$$ Using the Cauchy Riemann equations $$\frac{\partial }{\partial x}u=\frac{\partial }{\partial y}v$$ $$\frac{\partial }{\partial y}u=-\frac{\partial }{\partial x}v$$ my results: $$\frac{x^2}{\sqrt{x^2+y^2}}=\frac{y^2}{\sqrt{x^2+y^2}}$$ $$\frac{x y}{\sqrt{x^2+y^2}}=0$$ solutions says that it's differentiable at (0,0). But doesn't it blow at (0,0)?
 P: 428 If you just plug in ##y=0## and ##x=0## you will get an indeterminate form which is meaningless. If you evaluate the limits, I think that you get all expressions equal to ##0##, but double check that.
 P: 170 Division by zero is not allowed in complex analysis, so your final equations are not defined at x=y=0. They are not equal.
P: 428

## Check if the complex function is differentiable

 Quote by FactChecker Division by zero is not allowed in complex analysis, so your final equations are not defined at x=y=0. They are not equal.
That is true, but this function is differentiable at ##z=0##. If you evaluate the two limits along the real and imaginary axes (with ##h\in\mathbb{R}##)

$\displaystyle\lim_{h\rightarrow0}\displaystyle\frac{(0+0i+h)\left|(0+0i +h)\right|}{h}=0$

$\displaystyle\lim_{h\rightarrow0}\displaystyle\frac{(0+0i+ih)\left|(0+0 i+ih)\right|}{ih}=0$

So the function is differentiable at ##0##. I don't remember enough from my complex analysis course (which had a number of students who had not taken real analysis, so it was a bit less rigorous than some courses) to reconcile this. My recollection is that the limits of the Cauchy-Riemann equations could be evaluated, but a quick look online showed that my recollection was incorrect. Perhaps, since the partial derivatives are undefined at 0 the Cauchy-Riemann equations are not applicable?
HW Helper
Thanks
P: 25,175
 Quote by DrewD That is true, but this function is differentiable at ##z=0##. If you evaluate the two limits along the real and imaginary axes (with ##h\in\mathbb{R}##) $\displaystyle\lim_{h\rightarrow0}\displaystyle\frac{(0+0i+h)\left|(0+0i +h)\right|}{h}=0$ $\displaystyle\lim_{h\rightarrow0}\displaystyle\frac{(0+0i+ih)\left|(0+0 i+ih)\right|}{ih}=0$ So the function is differentiable at ##0##. I don't remember enough from my complex analysis course (which had a number of students who had not taken real analysis, so it was a bit less rigorous than some courses) to reconcile this. My recollection is that the limits of the Cauchy-Riemann equations could be evaluated, but a quick look online showed that my recollection was incorrect. Perhaps, since the partial derivatives are undefined at 0 the Cauchy-Riemann equations are not applicable?
I would use the definition of the derivative as a difference quotient to show it's differentiable at z=0.
 P: 428 Yes, but ##w(0)=0##, so I left it out.