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.
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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 ...
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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 ...
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2 answers
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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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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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Given the harmonic function u(x,y), find its harmonic conjugate v(x,y) then express the analytics function f(z) in terms of z

I was given the harmonic function $u(x,y) = \frac{y}{x^2+y^2}$. Using the cauchy riemann equations i was able to deduce that $v(x,y) = \frac{x}{x^2+y^2}$. I now have the expression $f(z) = \frac{y}{x^...
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Complex functions that is not quite analytic

A complex-valued function $f(z)=u(x,y)+iv(x,y)$ is analytic if it satisfies the Cauchy-Riemann equations: $\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}=0$ $\frac{\partial u}{\partial ...
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Cauchy-Riemann equations and holomorphic functions

Suppose $f(z) = u(x,y) + iv(x,y)$ is holomorphic. Prove that if $f(\overline{z})$ is holomorphic as well, then $f$ is a constant function. I'm having trouble showing that the partial derivatives $u_x, ...
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Is this function complex differentiable?

Suppose $f(z)=u(x,y)+iv(x,y)$, where $z=x+iy$, for $x,y \in \mathbb{R}$. If u and v are real functions which are differentiable at all $(x,y) \in \mathbb{R}^2$. Then either show that it follows that $...
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Is $f(x + iy) = \exp(iy)$ analytic?

I know that if $|f| = c$ and $f$ is analytic in $D$ then $f = k$ in $D$. If I take $f(z) = f(x + iy) = \exp(iy)$ then $|f| = 1$ in the complex plain. But $f(z)$ is not constant, does this means that $...
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If f(z) is analytic then u(x,y)=Re(f(x+iy)) is harmonic [duplicate]

We have the given statement If f(z) is analytic then u(x,y)=Re(f(x+iy)) is harmonic. In order to prove/disprove it, I use the Cauchy-Riemann equations, since they define analycity: $u_x=v_y$ $u_y=-v_x$...
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Rapid procedure to construct an analytic function from a real part

I have this rapid procedure to construct an analytic function from a real part: \begin{equation} u(x,y)=e^x(cosy-siny) \end{equation} We have the partial derivatives: \begin{equation} \begin{array} ...
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If $f = u +iv$ is differentiable, then how to prove that its components $u$ and $v$ are differentiable?

I know that there is a sufficient condition theorem which states that that the partial derivatives have to continuous and $f$ should satisfy CR equation. But how to prove the opposite of the statement?...
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Cauchy Riemann Equations Sufficient Condition Proof, continuity not required at all partial derivatives?

Theorem: Suppose that $$f(z)=u(x,y)+i\,v(x,y)$$ and for $z_0 = x_0 + y_0i$, the first order partial derivatives of $u$ and $v$ all exist and are continuous at a point $(x_0,y_0)$. Then if $u$ and $v$ ...
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2 answers
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How can I find all holomorphic functions $f:\mathbb C \to \mathbb C$ having $f=u+iv$ with $v(x,y)=u^2(x,y)$

My current thought process is this: In order for f=u+iv to be analytic, it must differentiable. Therefore, it must satisfy the Cauchy-Riemann equation.The Cauchy-Riemann equation is equivalent to: $...
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Is $e^{\sin z}$ an analytic function?

Here is my work. $$ e^{\sin z} = e^{\sin(x+iy)} = e^{\sin x \cos(iy) + \cos x \sin(iy) } = e^{\sin x \cosh y + i\cos x \sinh y} $$ So, $u = e^{\sin x \cosh y}$ and $v = e^{\cos x \sinh y}$. Then, $$ \...
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Prove that if $f:\Bbb C\rightarrow C$ is a complex function whose components are of class $C^1$ then $f$ is differentiable.

If $f:\Bbb C\rightarrow\Bbb C$ is a complex function then there exist two real scalar function $u,v:\Bbb R^2\rightarrow \Bbb R$ such that $$ f(x+yi)=u(x,y)+v(x,y)i $$ for any $(x+yi)\in\Bbb C$. Now ...
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2 votes
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Complex Analysis - Cauchy-Riemann exercise

Let $f(z): \mathbb{C}\rightarrow \mathbb{C}$ be a complex function such that $f(z) = \begin{Bmatrix} \frac{z^5}{|z|^4},z\neq 0\\ 0, z=0 \end{Bmatrix}$. Prove that Cauchy-Riemann equations(in x and y ...
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Proving whether a complex function is analytical help!

I am currently struggling to prove whether a complex function is analytical. I understand that I must employ the Cauchy-Riemann relations to do this. However, the answer I get is one that I can't ...
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Find harmonic conjugate $v$ of $u(x,y)$ $=$ $e^{-2xy} \sin(x^2 - y^2) $

I hope someone can help me solve this problem, it is supplementary problem 2.53d from Schaum's book: Schaum's Outline of Complex Variables, 2nd ed. I'm sure $u(x,y)$ is harmonic because I checked $\...
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Analytic function with constant imaginary part

While learning some consequences of the Cauchy-Riemann theorem. we learned that An analytic function with constant imaginary (or real) part is constant. and in addition, Sum of analytic functions ...
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Cauchy's integral theorem and Green's Theorem clarification

I was reading this page on Wikipedia: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem In the very end, it says that "we therefore find that both integrands (and hence their integrals) ...
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Complex Analysis: Analytical Function $ f(z) ={z^{2}}/({\mathrm{e}^{x} \cos y+i \mathrm{e}^{x} \sin y})\,$?

Problem In which region of the complex plane is the following function analytic? $$f(z) = \dfrac{z^{2}}{\mathrm{e}^{x} \cos y+i \mathrm{e}^{x} \sin y}$$ If the function has a derivative over its ...
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Suppose that $f(z) = \frac{z}{1-z}$. Use Cauchy-Riemann Equations to determine the analyticity of $f(z)$. Then find the derivative of $f(z)$. [closed]

can anyone help me in solving this question? I have tried expressing $z=x+iy$ and multiplying the conjugate to numerator and denominator. However, the expansion and differentiation are too tedious. Is ...
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What is the radius of convergence of $\frac{\sinh(z)}z$ about $z=\pi$

I have a question that asks to find the radius of convergence for $\frac{sinh(z)}z$ about $z=\pi$. I attempted to solve this as: $$\frac{d}{dx} \frac{\sinh(z)}z = \frac{\cosh(z)}z - \frac{\sinh(z)}{z^...
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The Cauchy-Riemann conditions in curvilinear coordinates

Could you please help me with obtaining the Cauchy-Riemann conditions in curvilinear coordinates? Assume we have a smooth curve $L$ in 2D-plane, and the local (orthogonal) coordinate system are $(v_1, ...
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1 vote
1 answer
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How I must say about Cauchy-Riemann equation of $f(z)=z|z|$?

In $C\setminus \{0\}$, Since $\frac{\partial u}{\partial x} = \frac{2x^{2}+y^{2}}{\sqrt{ x^{2}+y^{2}}}$ and $ \frac{\partial u}{\partial y}=\frac{xy}{\sqrt{ x^{2}+y^{2}}}$, $\frac{\partial v}{\partial ...
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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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1 answer
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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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Prove that if $u(x,y)$ is harmonic then there is a $v(x,y)$ such that $f(z) = u + iv$ is analytic.

The problem has the following hint: show that $F = (-u_y, u_x)$ is a gradient field. So I took the hint and used $\nabla \times F = \hat{k}(u_{xx} + u_{yy})=0 $ Therefore there exists some function $f(...
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1 answer
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Doubt in the complex form of Cauchy-Riemann condition

I am reading complex number from mathematical methods point of view. There comes Cauchy-Riemann condition in different form. Suppose $f:\mathbb C\to\mathbb C$. As $z=x+iy$ So, $\displaystyle f(z)=u(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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1 vote
1 answer
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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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2 votes
1 answer
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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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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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1 answer
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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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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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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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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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5 votes
1 answer
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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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2 votes
1 answer
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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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