# 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.

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### Describe where the following function satisfy the Cauchy-Riemann equations, and where the function is differentiable. f(z) = z|z| [closed]

enter image description here I wanna know the solution of it. Thank you.
1 vote
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### Given $f(z) = |z|^{z_1}$, what is $f'{(z)}$.

Context I am working on the series solution of a second-order, homogeneous, linear, ordinary differential equation. Irrespective of the two roots of the indical equation, one of the two linearly ...
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### Verify that $\sqrt z$ is analytic using the Cauchy–Riemann equations

For a Mathematical Methods class I am taking this year, we started off with Complex Analysis. In one of the problem sets we were tasked to show that the complex-valued function $f(z)= \sqrt z$ ...
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### Question about Cauchy Conditions from History Section of Remmert's Textbook

Remmert's Theory of Complex Functions book is great for plenty of commentary on the history behind the mathematics introduced in each section. The picture below is of the section which talks about ...
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### Find the continuous function $f$ with domain $\mathbb C$ such that it's differentiable on $x^2-y^2=1$ but is not differentiable at other points

Find the continuous function $f$ with domain $\mathbb C$ such that it's differentiable on $x^2-y^2=1$ but is not differentiable at other points of $\mathbb C$. I assume $f=u(x,y)+iv(x,y)$, all points ...
1 vote
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### Determine at what points the complex function $f\left(z\right)=e^{2x}\cos3x+ie^{3x}\sin2y$ is differentiable.

Determine at what points the complex function $$f(z)=e^{2x}\cos3y+ie^{3x}\sin2y$$ is differentiable. The function is differentiable at $z_0 \in \mathbb C$ if $f$ is defined on a neighborhood of $z_0$ ...
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### Finding all differentiable complex functions satisfying given conditions.

The question is to find all differentiable functions $f:u+iv:\Omega \rightarrow \mathbb{C}$, $\Omega$ an open connected set, such that $u=3v$ and $f(0)=3+i$. I know Cauchy-Riemann conditions have to ...
1 vote
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### Finding the analytic function of $z$, given the real part

Problem. The real part of an analytic function $f(z)$ is given by $3x^2y-y^3$. Find the imaginary part. Find the analytic function of $z$. Attempt. I begin by applying the Cauchy-Riemann Equations. ...
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### Let $f$ be holomorphic on connected open set $D$. Prove $u^4+v^4$ being constant implies $f=u+iv$ being constant.

Differentiating $u^4+v^4$, we have $$4u^3u_x + 4v^3v_x = 0$$ $$4u^3u_y + 4v^3v_y = 0$$ Then we have $$u^3v^3 u_x v_y = u^3v^3 u_y v_x.$$ If $uv\neq 0$, then we can apply Cauchy-Riemann condition ...
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### complex function continuous but not differentiable on an open set

Could one possibly come up with an example of a complex-valued function such that $f(z)$ is continuous on the open set $D=\{ z \in \mathbb{C}:z \neq 0 \}$ and $\lim\limits_{z\to 0} f(z) = \infty$ but ...
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### When does 𝑅𝑒∫𝑓(𝑧)𝑑𝑧=∫𝑅𝑒(𝑓(𝑧))𝑑𝑧? [closed]

When does $Re(\int_Cf(z)dz) = \int_c Re(f(z)dz)$? I genuinely have no idea where to even begin with this. I thought about letting $f$ be a function, such as $f = \frac{1}{z}$ but don't know how that ...
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1 vote
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### Show that complex function satisfies Cauchy-Riemann equations at z=1 but is not differentiable there.

Question is here $$f(z) = \frac{|z-1|^4}{(z-1)^3} | (z\neq 1), f(1)=0$$ I have no idea how to partially differentiate a function with $|z-1|^4$ in the numerator and I can't find examples of similar ...
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### about the sufficient conditions for complex differentiability via Cauchy-Riemann:

I just noticed in the rule about the sufficient conditions for complex differentiability via Cauchy-Riemann: When we're considering as to whether or not $g: \mathbb C \to \mathbb C$ is differentiable ...
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### Determine derivative wherever the derivative exists of $-i(1-y^2)+(2x-y)(y)$

Is this correct? (Edit: I'm just going to outline the steps and post the rest as an answer.) $g: \mathbb C \to \mathbb C, g(z) = -i(1-y^2)+(2x-y)(y)$ Step 1. $g$ is differentiable only on $\{y=x\}$. ...
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### Modulus of the complex derivative in terms of partial derivatives

In Ahlfors Complex Analysis, the author writes the equation $$|f'(z)|^2=\left( \frac{\partial u}{\partial x}\right)^2+\left( \frac{\partial v}{\partial y}\right)^2$$ I am wondering why this is ture. I ...
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### Cauchy-Riemann Equation satisfies at $z=0$

Let $f(z)=\begin{cases} \frac{z^5}{\left | z \right |^4} & \text{ if } z\neq 0 \\ 0 & \text{ if } z=0 \end{cases}$ I could show this is continuous on $\mathbb{C}$. And, I would like to show ...
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### Need help with this proof from Axler's Harmonic Function Theory

A free, legal copy is available here. On pp. 81-82, I have two questions: Why do the Cauchy-Riemann equations imply that all complex derivatives except the $m^{th}$ vanish at the origin? If anything, ...
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### generalization of Cauchy-Riemann conditions

Esteemed experts, Please excuse the ignorance and language of a poor physicist. As we know, the real $u(x,y)$ and imaginary $v(x,y)$ parts of an analytic function (in some domain) satisfy the Cauchy-...
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I am working through Arfken, Weber, and Harris' s "Mathematical Methods for Physicists" $7^{th}$ edition. Example 12.7.1 is: "Prove Jensen's theorem (that $\left| F(z) \right|^2$ can ...
How to know a function like for example $f(z)=\frac{1}{z}$ is analytic without turning it into $\frac{x-iy}{x^2+y^2}$ and then taking the partial and stuff?