Two questions, one on harmonic functions

[nacho-man]

Could I get some hints on how to evaluate these question.
The question asking to find where $f(re^{i\theta})$ is differentiable doesn't seem to involved,
however would I use C-R equations, or would it just be for wherever $r \neq 0$. Although that is given in the domain, so I'm assuming they want us to use C-R equations.
If i were to use C-R equations, then I would have to convert to cartesian coordinates, correct?
How would I do that.As for the second question,
could I get any hints on what theorems may be relevant. I am not sure how to approach this.

Thanks

 

[alyafey22]

In the first question you are asked to find the domain of differentiability of the function

\(\displaystyle f(z)=\log(z^2) \)

Hint : choose a suitable branch cut .

 

[chisigma]

Could I get some hints on how to evaluate these question.
The question asking to find where $f(re^{i\theta})$ is differentiable doesn't seem to involved,
however would I use C-R equations, or would it just be for wherever $r \neq 0$. Although that is given in the domain, so I'm assuming they want us to use C-R equations.
If i were to use C-R equations, then I would have to convert to cartesian coordinates, correct?
How would I do that.As for the second question,
could I get any hints on what theorems may be relevant. I am not sure how to approach this.

Thanks

Question 1: with a little of patience You can derive the C-R equations in polar coordinates...

$\displaystyle \frac{\partial{u}}{\partial{r}} = \frac{1}{r}\ \frac {\partial {v}}{\partial{\theta}} $

$\displaystyle \frac{\partial{v}}{\partial{r}} = - \frac{1}{r}\ \frac {\partial {u}}{\partial{\theta}} \ (1)$

Here $\displaystyle u(r, \theta) = \ln r^{2}$ and $v(r, \theta) = 2\ \theta$, so that is...

$\displaystyle \frac{\partial{u}}{\partial{r}} = \frac{2}{r},\ \frac{\partial{u}}{\partial{\theta}} = 0,\ \frac{\partial{v}}{\partial{r}} = 0,\ \frac{\partial{v}}{\partial{\theta}} = 2$ so that the (1) are satisfied everywhere with the only exception of the point r = 0... Kind regards $\chi$ $\sigma$

 

[chisigma]

Could I get some hints on how to evaluate these question.
The question asking to find where $f(re^{i\theta})$ is differentiable doesn't seem to involved,
however would I use C-R equations, or would it just be for wherever $r \neq 0$. Although that is given in the domain, so I'm assuming they want us to use C-R equations.
If i were to use C-R equations, then I would have to convert to cartesian coordinates, correct?
How would I do that.As for the second question,
could I get any hints on what theorems may be relevant. I am not sure how to approach this.

Thanks

Question 2: a function u(x,y) is said to be harmonic if its second derivatives are continuous and is...

$\displaystyle u_{xx} = - u_{yy}\ (1)$

An important theorem extablishes that if u is harmonic in a simply connected domain G, then it exist a function v(x,y) so that f(x + i y) = u(x,y) + i v(x,y) is analytic in G [v is said to be the harmonic coniugate of u...].

If we suppose that G is a disk centered in w with radious r and call $\gamma$ its contour, the the Cauchy Integral Formula extablishes that...

$\displaystyle f(w) = \frac{1}{2\ \pi\ i}\ \int_{\gamma} \frac{f (z)}{z - w}\ d z\ (2)$

With the substitution $z - w = r\ e^{i\ \theta}$ the (2) becomes... $\displaystyle f(w) = \frac{1}{2\ \pi}\ \int_{0}^{2\ \pi} f(w + r\ e^{i\ \theta})\ d \theta\ (3)$

... and, taking the real part of f we have...

$\displaystyle u(w) = \frac{1}{2\ \pi}\ \int_{0}^{2\ \pi} u (w + r\ e^{i\ \theta})\ d \theta\ (4)$

Kind regards

$\chi$ $\sigma$

 

[blue_raver22]

for your questions! It looks like you are on the right track for the first question. Yes, you would use the Cauchy-Riemann equations to determine where $f(re^{i\theta})$ is differentiable. Remember that the Cauchy-Riemann equations are a set of necessary conditions for complex differentiability, so if they are satisfied, then the function is differentiable at that point. You are correct that you would need to convert to Cartesian coordinates to use the Cauchy-Riemann equations. To do this, you can use the identity $e^{i\theta}=\cos\theta+i\sin\theta$ and then substitute this into the given function. From there, you can use the Cauchy-Riemann equations to determine where the function is differentiable.

For the second question, it would be helpful to know what the specific question is asking for. However, some relevant theorems for harmonic functions include the Mean Value Property, the Maximum Modulus Principle, and the Cauchy Integral Theorem. It may also be helpful to consider the definition of a harmonic function and think about how it relates to these theorems. Good luck with your evaluations!