# Questions tagged [cauchy-riemann-equations]

For questions on the solving/usage/applications of the Cauchy-Riemann equations. Use this tag alongside the complex analysis tag.

256 questions
Filter by
Sorted by
Tagged with
15 views

### Conformal mapping is holomorphic

Let $\Omega \subseteq \mathbb{R}^2$ open, $f : \Omega \rightarrow \mathbb{R}^2$ be orientation preserving and map curves intersecting orthogonally to curves intersecting orthogonally. Then $f$ is ...
53 views

### Find parameters such that $f(z)=\log(x^2+y^2)-a(\arg z)^2+ib\arg\bar z+\dfrac{c}{\bar z}$ is analytic

The problem asks you for what values of $a,b,c\in\mathbb R$ you can find a domain in $\mathbb C$ where $$f(z)=\log(x^2+y^2)-a(\arg z)^2+ib\arg\bar z+\dfrac{c}{\bar z}$$ is analytic. Due to the nature ...
59 views

### Equivalence of Cauchy-Riemann equations

You have the CR equations in polar and cartesian coördinates that are equivalent if f(z) is complex differentiable because you can use the chain rule on u and v but if f isn't differentiable you can ...
32 views

93 views

### Can a) and b) real components of an complex equation? [closed]

I have a problem somewhere ... I missed a point.. So given are following equations: (with $C \to R$) (i) $a(z) = x^2 - y^2$ (ii) $b(z) = x^2 + y^2$ And I am supposed to find out if those ...
60 views

183 views

### Prove that if $f=u+iv$ is conformal in $G\subseteq\mathbb C$, then the Cauchy-Riemann Equations hold for $u$ and $v$.

I was reading Conway's Complex Analysis text and found the proof of the following theorem confusing. Prove that if $f=u+iv$ is conformal in $G\subseteq\mathbb C$, then the Cauchy-Riemann Equations ...
63 views

### Verify that $u, \; v$ are continuous in a neighborhood of $z=0$ and satisfy the Cauchy-Riemann Eqns at $z=0$. Show that $f'(0)$ does not exist.

This is a question from a previous complex analysis qualifying exam that I'm working through to study for my own upcoming qual. I'm really struggling to know where to go with it and any help would be ...
134 views

### (complex variables)Let $D \subset C$ be an open connected...

Let $D \subset \mathbb{C}$ be an open connected. a) Use the Cauchy-Riemann equations to prove that if $F: D \to \mathbb{R}$ is holomorphic, then $F$ is constant. b)Let $f, g: D \to \mathbb{C}$ be ...
63 views

### Analytic if CR equations in polar form satisfied?

Consider the following theorem. Theorem: Suppose $u(r, \theta)$ and $v(r, \theta)$ (which we shall denote by $u$ and $v$ here afterwards) are real valued functions on $(0, \infty) \times (-\pi, \pi]$....
1 vote
### How do I go about finding the set of points in $\Bbb C$ where $f(z)=(z^4 + z)e^{\bar{z}}$ is differentiable
$$f(z) = {(z^4 + z)}e^{\bar{z}}$$ In the exercise there are $6$ possible answers: $\{{0,-1, e^{i\pi/3}, e^{-i\pi/3}}\}$ ; $\{{0,1, e^{i\pi/3}, e^{-i\pi/3}}\}$ ; $\{{0,-1, e^{2i\pi/3}, e^{-2i\pi/3}}\}$ ...
The integral is: $$\int_{0}^{2\pi}e^{-\sin\theta}\cos(\theta+\cos\theta)d\theta$$ My first thoughts were to use the Cauchy-Riemann equations in polar coordinates to show that the function is regular ...