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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32 views

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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50 views

Analytic complex function with $u=v^2$

I must prove that if $f = u+iv$ is analytic with $v=u^2$ then $f$ is constant. My attempt is $$f(x,y) = u(x,y)+iv(x,y) = u(x,y)+iu(x,y)^2$$ and write $$\frac{\partial v}{\partial y} = \frac{\partial }{...
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1answer
46 views

Cauchy-Riemann equations and required continuity of derivatives

So I just read, that for any analytic function, the Cauchy-Riemann equations will hold. However, the reverse, i.e. Cauchy-Riemann equations hold -> function is analytic, is supposedly only true if ...
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25 views

Show that $u$ is constant for some positive function $u$ on $\Bbb R^2$ [duplicate]

Suppose $u_{xx}+u_{yy} = 0$ for some positive function $u$ on $\Bbb R^2$. Show that $u$ is constant. I am not sure how to show this but I think this has something to do with the Cauchy Riemann ...
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37 views

Show that $f^*$ is analytic on $\overline {G}.$

Let $G \subseteq \mathbb C$ be a region and $f : G \longrightarrow \mathbb C$ be an analytic function. Let $\overline G : = \left \{z\ |\ \overline {z} \in G \right \}.$ Then by C-R equations show ...
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1answer
51 views

Constant function in $\mathbb{C}$

The following five problems (Ahlfors Theorem 11 chapter 3) are trivial consequences of the Open mapping theorem. (a) Show that a holomorphic function $f$ on a domain $\Omega$ whose derivative vanishes ...
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34 views

Polar form of CR equation

We know that, by applying a coordinate change, we have the polar form of CR-equation which is given by: $$ \begin{split} \frac{\partial u}{\partial r} &= \frac{1}{r}\frac{\partial v}{\partial \...
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51 views

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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113 views

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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50 views

"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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29 views

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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40 views

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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2answers
36 views

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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1answer
29 views

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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40 views

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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60 views

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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60 views

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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44 views

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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2answers
74 views

Complex function $\operatorname{Log}(\frac{z-a}{z-b})$ and Cauchy-Riemann-equation [closed]

I need a proof that the following complex function satisfies the Cauchy-Riemann equations $$\operatorname{Log}\left(\frac{z-a}{z-b}\right),$$ where $z\in\mathbb{C}$ and $a,b\in\mathbb{R}$ such that $a&...
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85 views

Is it analytic ? $f(z)=z^2\bar{z}$

Please find the question and its solution. My answer gone wrong dont know why, I think the answer marked in the question is wrong. My try: I solved for CR equations and replaced x=y=0 and found CR ...
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1answer
36 views

Proving an equation relating to the Cauchy-Riemann Equations.

Here is what the question asks: Let $\Omega \subset \mathbb{C}$ be a domain (meaning that it's a connected and open), and let $f: \Omega \rightarrow \mathbb{C}$ be holomorphic. Prove that if $f(z) = ...
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40 views

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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1answer
25 views

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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1answer
38 views

The solution of this Cauchy–Riemann equations.

・$f(z)=f(x+iy)=e^xu(y)+ie^xv(y)$. ・$f(z)$ is regular. ・$f(x)=e^x.$ (When $y=0$, $f(z)=e^x$.) Under this condition, determine $u(y), v(y).$ The answer is following. From Cauchy–Riemann equations, $u(y)=...
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1answer
46 views

If $v$ is an incompressible, irrotational fluid satisfying Euler's equation, show $\overline{v}$ is holomorphic

This feels like it'll be simple once I see it, but I've been stuck for a while now. The setup is from Sijue Wu's 1997 paper Well-posedness in Sobolev spaces of the full water wave problem in 2D. $v=(...
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1answer
141 views

Cauchy-Riemann Equation for Surface Equations

Suppose we are given smooth functions $W,V:\mathbb{R}^2\times\mathbb{C}\to\mathbb{C}^2$ and are asked to determine if the function $Q(W(x,y,z(x,y)),V(x,y,z(x,y))$ solving $$ Q(W(x,y,z(x,y)),V(x,y,z(x,...
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1answer
181 views

Showing the Polar form of the Cauchy-Riemann Equations imply the Cartesian Form

Often times, when the Cauchy-Riemann equations $$u_x = v_y; \ u_y = -v_x$$ are introduced, it's an exercise to show that one can arrive at the "polar form" of those equations. That is, the ...
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25 views

Complex differentiation, Cauchy Riemann

By using the definition of complex differentiation and the complex derivative, how can we show that $f'(z_0)= \frac{1}{2} (\frac{\partial f}{\partial x}(z_0)-i\frac{\partial f}{\partial y}(z_0)). $ I ...
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1answer
34 views

Why $\dfrac{\partial v}{\partial y}=0$ which also gives us $\dfrac{\partial v}{\partial x}=0$?

I passed through this answer here and I don't understand the following. We are at the moment: If $u^2+v^2 \neq 0$, then $\dfrac{\partial v}{\partial y}=0$ which also gives us $\dfrac{\partial v}{\...
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1answer
38 views

For which $z \in \mathbb{C}$ does $f(z) = |z|^2 +2z$ satisfy Cauchy-Riemann equations?

The task is as follows: $$ \text{Check, for which } z\in\mathbb{C} \text{ function } f(z) = |z|^2 +2z \text{ satisfies the Cauchy-Riemann equations.} $$ I figured I need to transform $f(z)$ to $f(x,y) ...
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2answers
84 views

show $f$ complex analytic is constant

$U\subseteq \mathbb{C}$ open and connected, $f : U \rightarrow \mathbb{C}$ complex analytic function. Suppose $f(z) = u(x, y) + iv(x, y)$, $z = x + iy$, and $u$ and $v$ are real-valued functions. ...
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1answer
20 views

Find a real-valued function which satisfied Cauchy-Riemann equations

Question: Find a real-valued function $v(x,y)$ such that $v(0,0)=1$ and together, $u$ and $v$ satisfy the Cauchy-Riemann equations in the entire complex plane. Let $v(x,y)=M(x,y)+i.0$ which is a real-...
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1answer
40 views

Cauchy-Riemann conditions implied by harmonicity and orthogonal gradients?

If I have two functions $\eta:\mathbb{R}^2\to\mathbb{R}$ and $\xi:\mathbb{R}^2\to\mathbb{R}$ for which the following relations hold: $$ \begin{align} \left( \frac{\partial\xi}{\partial x} \right)^2 + \...
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1answer
90 views

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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24 views

Show that the inverse function also satisfies C-R equation.

Suppose that a smooth function $f=\left(f_{1}, f_{2}\right): \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ satisfies the following property: $$ \frac{\partial f_{1}}{\partial x}=\frac{\partial f_{2}}{\...
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1answer
59 views

Prove $\int_1^{\infty}\psi(x)(x^\frac{s}{2}+ x^\frac{1-s}{2})\frac{dx}{x} $ is Entire

Define $$f(s)= \int_1^{\infty}\psi(x)(x^\frac{s}{2}+ x^\frac{1-s}{2})\frac{dx}{x} $$ where $ \psi(x)= \sum_{n=1}^{\infty}e^{-n^2\pi x}$ is the Jacobi Theta function. Claim- $f(s)$ is Entire My Try :- ...
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1answer
84 views

Question regarding analyticity, differentiability and the Cauchy-Riemann Equations ( Complex Analysis )

For non-analyticity, we can show that the Cauchy-Riemann equations don't hold anywhere. But here we can see that the Cauchy-Riemann equations hold when $y=2x$. I understand what the book did here but ...
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1answer
30 views

How can I take a partial derivative of an equation like $2u+v=5$?

I know that $\frac{du}{dx} = 0$ and $\frac{du}{dy} = 0$. But I don't undestand how? how can you take a derivative of this when you don't know the value of either u or v?
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67 views

Let $f(z) = u(x,y) + iv(x,y)$ be an holomorphic function and $2u(x,y) + v(x,y) = 5$. Show that $f$ is constant

I understand that it suffices to show that $f'(z)=0$ What I don't understand though, is how to take a partial derivative from $2u(x,y) + v(x,y) = 5$ ... Another solution I thought of is, If $2u(x,y) + ...
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22 views

Are conjugate complex functions independent of each other when differentiating if the Cauchy Riemann conditions are satisfied?

Let $f(z)$ be a function of a complex variable that satisfies the Cauchy-Riemann conditions (i.e. is analytic). Let $f^*(z)$ be its complex conjugate (such that for any value of z, the complex ...
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75 views

Does this piecewise function satisfy the Cauchy Riemann equations?

We were asked if the function: $$ f(z) = \left\{ \begin{array}{ll} e^{-1/z^4} & \quad z \neq 0 \\ 0 & \quad z = 0 \end{array} \right. $$ satisfies ...
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62 views

Why does an analytic function $z^2$, which satisfies the Cauchy-Riemann equations, not satisfy the condition of being harmonic?

Why does an analytic function $z^2$, which satisfies the Cauchy-Riemann equations, not satisfy the condition of being harmonic? We have $u = x^2-y^2$ and $v = 2xy$ \begin{align} du/dx &= 2x, &...
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1answer
15 views

Proof that harmonic functions $u,v$ satisfy this condition.

Question: Suppose $u,v$ are harmonic and satisfy the Cauchy-Riemann equations in $\mathbb{R}^2$. Show that $f = u + iv$ satisfies $f'(x) = u_x(x,0) - iu_y(x,0)$ for real $x$. My attempts: I considered ...
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1answer
41 views

Where is $e^z(z-\bar{z})^2$ differentiable?

Question: Where is $f(z) = e^z(z-\bar{z})^2$ differentiable? My attempts: By letting $z = x+iy$ and by using the Cauchy Riemann Equations, I was able to get to a point where $\sin y = 0$ and $\cos y = ...
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1answer
185 views

Derivative of a Complex Function as a $\mathbb{C}$-linear Transformation

I'm trying to show that the derivative of a differentiable complex function is a $\mathbb{C}$-linear mapping of $\mathbb{C}$ to itself, and since every $\mathbb{C}$-linear map is of the form $$ \...
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1answer
65 views

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 ...
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1answer
253 views

Prove if a function is holomorphic, then its complex conjugate is holomorphic, by Cauchy-Riemann Equations

Given a function $f(z)$ is holomorphic on a disc $D(0,R)$, we want to prove that $g(z)= \overline {f(\bar z)}$ is holomorphic. Below is my proof. $\lim_{h\to 0} \frac{\overline {f(\overline{z+h})}-\...
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2answers
113 views

Wirtinger derivative form of Cauchy–Riemann equations

I'm trying to understand the Cauchy-Riemann equations using the traditional $u, v$ form and the Wirtinger derivative form. Taking $\ln|z|$ as an example function, for the normal $u, v$ form I have: $$\...