# 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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### 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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### "Jensen's Theorem" from a mathematical physics textbook

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 ...
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### How to know a complex function is analytic without going to the whole process of checking Cauchy-Riemann condition?

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?
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### Key step in a proof involving complex variables

This question is within the context of Cauchy-Riemann equations. The text I am reading lists a theorem regarding the existence of the derivative of a function $f$ at point $z_0$. My question concerns ...
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### What is the general setting for the Cauchy-Riemann equations and the triholomorphic equations?

Let $U \subseteq \mathbb{R}^n$ be an open subset, let $M_n(\mathbb{R})$ be the algebra of real $n \times n$ matrices, and let $B \subseteq M_n(\mathbb{R})$ be a real subalgebra. Assume the coordinates ...
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### Complex analysis of $f(z) = 2 y^2 \sin x − i y^3 \cos x$ [closed]

I want to find the domain of definition, the domain of derivability and the domain of analyticity of the function $f(z) = 2 y^2 \sin x − i y^3 \cos x$ The domain of definition is $\mathbb C$ but I ...
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### Cauchy Equation and Continuity

Suppose that $f$ is a real-valued function defined on all of $R$ and satisfying the identity $$f(x+y)=f(x) f(y)$$ for all $x, y$ in $\mathbb{R}$. Prove that, if $f$ is continuous at $x=0$, then $f$ ...
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### Applying the dbar operator to differentiate a complex function

I'm trying to differentiate this complex function $$f(z) = (|z|-1)^2$$ in order to determine where it satisfies the Cauchy-Riemann equation $\frac{\partial f}{\partial \bar{z}} = 0$. So I first ...
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### Proof Detail: Cauchy-Riemann Equations Imply Holomorphy

I'm working through the proof of Theorem 2.4 in Chapter 1 of Stein/Shakarchi's Complex Analysis. I'm looking for clarification on where exactly we use the hypothesis that $u$ and $v$ are continuously ...
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### State a primitive of f and use this to evaluate and find the length of path L(s,w)

Let x, y, p, q $\varepsilon$ R let s = p + iq , and let w = x + iy , and let f(z) = $e^z$ State a primitive of f and use this to evaluate $\int_{l(s,w)} {f(z) dz}$ I have worked out that a ...
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### Solutions of Cauchy-Riemann equations have limit at infinity

Let $f:\mathbb{R}^n\to \mathbb{R}^n$, $n\geq 3$ be a $\mathcal{C}^3$, sense preserving function which is a solution to the Cauchy-Riemann system $$Df^T(x)Df(x)=J_f^{2/n}(x)I,$$ where $Df$ is the ...
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### Inverse Function Theorem for Complex Functions

I want to prove the inverse function theorem for complex functions using a planar mapping approach. All that's left for me to do is to show that: $$g(w) = \xi(u,v)+i\ \eta(u,v)$$ satisfies the C-R ...
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### Holomorphy and polynomials: Cauchy-Riemann equations

I'm having some trouble proving this assertion: Let $\mathcal{P}\in \mathbb{C}[x,y]$ (where $x$ and $y$ are real variables) and $z=x+iy$. Prove that $\mathcal{P}(x,y)$ satisfies the Cauchy-Riemann ...
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### Checking holomorphic domain

Let $z=x+iy$ and $f(z) = 3 x y^2 + i y^3$ let $\frac{\partial}{\overline{\partial z}} = \frac{1}{2} (\frac{\partial}{\partial x} + i \frac{\partial}{\partial y})$ Computing the Cauchy–Riemann ...
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### Cauchy-Riemann Equations

I have a question. If I have a complex function and I know that for $r=1$, $f(z=e^{iθ})=\cos(θ)\sin(θ)$ And I know that $f(z)$ is entire. Now, I need to calculate $f'(z)$ for every $z$ in $r=1$, ...
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### Determine where the Cauchy Riemann conditions are satisfied for $w=\overline{z}$ (the complex conjugate of $z$)
I have a question in my complex analysis worksheet. Determine where the Cauchy Riemann conditions are satisfied for $w=\overline{z}$ (the complex conjugate of $z$). As far as I know the function has ...