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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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Prove that Cauchy-Riemann Equations are satisfied.

Prove that $f(x)$ satisfies the Cauchy-Riemann Equations at the point $z=0$ $$f(z) = \begin{cases} \dfrac{\overline{z}^3}{|z|^2}, & \text{if $z$ is not 0} \\ 0, & \text{if $z$ = 0} \end{...
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2answers
78 views

Complex function only differentiable in $y=x^2$

Find a function $F: \mathbb{C} \rightarrow \mathbb{C}$ that is differentiable in the parabola $y=x^2$ and not differentiable in the rest of the complex plane. Let $F(x,y) = u(x,y) + i v(x,y)$. If $F$ ...
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1answer
44 views

$f$ is $\mathbb{R}$ - differentiable iff $Re(f)$ and $Im(f)$ are $\mathbb{R}$ - differentiable

I just read that a sufficient condition for a function $f:A \rightarrow \mathbb{C},f(z) = u(z)+ i v(z)$ to be holomorphic is: $A$ open. f is $\mathbb{R}$ - differentiable in $A$. The Cauchy Riemann ...
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1answer
41 views

If $u(x,y)$ is not a harmonic function, can a harmonic conjugate $v(x,y)$ be found?

If $u(x,y)$ is not a harmonic function, i.e. does not satisfy Laplace equation $$\Delta u(x,y) = \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0$$ (and perhaps is not continously ...
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1answer
35 views

Is $f(z)=k\cdot\frac{e^z}{z}$ analytic?

Let's say we have the following complex function $$f(z)=k\cdot\frac{e^z}{z}$$ Is it analytic in some domain? When substituting with $z=x+iy$, using trigonometry and deriving the function on the form ...
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2answers
36 views

Harmonic conjugate and resulting analytic function

I can't quite figure out the last part of this problem... Find a harmonic conjugate of the harmonic function $u(x,y)= x^3-3xy^2$. Write the resulting analytic function in terms of the complex ...
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3answers
48 views

why is this complex function not analytic anywhere?

I know for $f(z)=z\bar z$, where $\bar z$ means the conjugate of $z$. Cauchy-Riemann equations are satisfied at $(0,0)$. Also, partial derivatives of U and V exist and are continuous everywhere. so ...
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1answer
29 views

True/false regarding C-R equation

Is the following statement is True /false ? The function $f$ defined by $$f(z) = \begin{cases} \frac {Im(z^2)}{\bar z} \ \text{if } z\neq 0 \\ 0 \ \text {if z=0} \end{cases}$$ satisfies the ...
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3answers
60 views

$\sqrt{|xy|}$ verifies the Cauchy-Riemann equations at $(0,0)$, but is not differentiable at $(0,0)$

Show that the function $f(z)=\sqrt{|xy|}$ verifies the Cauchy-Riemann equations at point zero, but it is not differentiable at that point. $u(x,y)=\sqrt{|xy|}$ and $v(x,y)=0$ since the imaginary part ...
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1answer
29 views

Proving complex analytic: Unsolvable Cauchy-Riemann Equations

I need to show that $f(z)$ is nowhere analytic, where $$ f(z) = e^{z\operatorname{Re}(z)}.$$ Letting $z = x + yi$, $$ f(x,y)=e^{x^2}e^{xyi}=e^{x^2}(\cos(xy)+i\sin(xy))$$ and letting $$u(x,y)=e^{x^2}...
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1answer
71 views

Using sequences prove that $f(z)=Re(z)$ and $g(z)=Im(z)$ are continuous.

I've found a few proofs showing the continuity with the Cauchy-Riemann equations but am unsure as to whether they are proved using sequences which the question I'm attempting requires. I may be wrong ...
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1answer
68 views

Finding the harmonic conjugate of a function

The problem: Find the harmonic conjugate of $G(x,y)= 2v^2(x,y)-2u^2(x,y)$ My attempt to solving it I know that "If two given functions u and v are harmonic in a domain D and their first-order ...
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1answer
27 views

Find a and b values so a given function is harmonic

The Problem: Let $f(z)=f(x+iy)=u(x,y)+iv(x,y)$ be an entire function. If $g(z)=au^2(x,y) - bv^2(x,y)$ find values for a and b so $g(z)$ is a harmonic function. My attempt to find a solution: Well ...
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1answer
62 views

Why is a complex function's derivative with respect to the complex conjugate zero

Here's what I'm thinking about: if a function has a nonzero derivative it's dependant on its input variable. Likewise the contrapositive: if it's independent of the variable it's derivative with ...
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3answers
35 views

Find all points where f(z) is differentiable.

I have $f(z)=2x+ixy^2$. I want to see if what I've done is correct or not. I've found the Cauchy-Riemann equations: $2=2xy$ and $0=-y^2$. Obviously the function is differentiable for all $y=0$. ...
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1answer
29 views

Composite of Holomorphic Functions is Holomorphic

Let $f(z)$ and $g(w)$ be holomorphic functions. I want to prove that the composite $g(f(z))$ is also holomorphic by using Cauchy-Riemann Equation directly. Let $g(f(z))=s(u(x,y), v(x,y))\,+\,i t(u(x,...
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1answer
43 views

Does this question about the sufficiency of the analyticity of f(z) make sense [duplicate]

Before I ask the question, I know for sure this following question makes sense: Prove that a necessary condition for $ f(z) $ to be analytic is that the Cauchy-Riemann equations are satisfied. This ...
2
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1answer
55 views

Cauchy-Riemann equations for $z=x+iy$ and $f(z)=R(x,y)e^{i\theta(x,y)}$

I know the following forms of the Cauchy-Riemann equations (please correct me if I made a mistake): $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$, $\frac{\partial u}{\partial y} = -...
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0answers
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Is this PDE (Poisson/Laplace equation) well-posed considering I have a very degenerate domain (picture included)?

In case (1) in the following picture we have the standard interior Poisson equation in 2D with Neumann boundary conditions on some smooth domain $\Omega$, subject to a point source at position $y$. I ...
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2answers
65 views

Differentiability in the complex plane and in $\Bbb R^2$.

Are the differentiability in the complex plane and differentiability in $\Bbb R^2$ different concepts? Consider the linear operator $T$ on $\Bbb R^2$ defined by $T(x,y) = (x+y,x-y),\ x,y \in \Bbb R$....
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1answer
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My friend's question about Cauchy Riemann eq. [closed]

For $u_{tt}=u_{xx}+u_{yy}$ , the complex variable $z=x+iy$ and $u(x,y,t)$ are given.Then can you formulate new Cauchy Riemann Eq for $u(x,y,t)$ My firend ask about this to me and we think about it ...
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0answers
62 views

Proposition regarding the Cauchy-Riemann equations

How do I prove the following proposition? I'm fine with the first one but the second equation is where I got confused. The text book that I'm using says $$ f '(z_0) = \frac{1}{2} \left(\frac{\partial ...
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1answer
43 views

Cauchy - Riemann equation conclusion

I've learnd that, given an open $\mathbb{\Omega}$, if a function $f : \mathbb{\Omega} \subset \mathbb{C} \longrightarrow \mathbb{C} $ is holomorphic in some open $A \subset \mathbb{\Omega}$ the ...
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0answers
36 views

Why does orientation preserving maps respect CR?

From this post it is written that "orientation preserving conformal maps respect CR. The matrix of the map must be a constant multiplied by some matrix of rotation that has a positive determinant. ...
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1answer
327 views

Prove that f satisfies Cauchy-Riemann Equations but is not differentiable

Prove that $f$ defined by $f(z) = \frac{z^5}{|z|^4} | (z\neq 0), f(0)=0$ satisfies the Cauchy-Riemann equations at $z=0$ but is not differentiable there. So I know that $u_x = v_x$. Therefore let $...
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1answer
76 views

Cauchy-Riemann equations & harmonic

Let f be holomorphic in an open set Ω ⊂ $\mathbb{C}$. Show that if u = Re f and v = Im f then u and v satisfy the Cauchy-Riemann equations. Show further that u is harmonic. [You may assume without ...
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2answers
26 views

find holomorphic function $f(z)$ for $u (x,y) = \cosh(ax)\sin(by)$

Question: [For which positive real numbers $a$ and $b$ is $u(x,y) = \cosh(ax)\sin(by)$ harmonic? When $a$ and $b$ satisfy this condition find a holomorphic function $f(z)$ such that $\Re f = u$] I ...
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2answers
35 views

Show $f$ is constant given $g=\overline{f}$

I am trying to show that if $f$ and $g=\overline{f}$ are both differentiable in a domain, then $f$ is constant on that domain. My attempt: Let $$f(z)=u(x,y)+iv(x,y)$$ $$g(z)=r(x,y)+is(x,y)$$ Equating ...
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2answers
62 views

Show $|f(z)|=1$ is constant

Suppose that $\Omega$ is a domain that $f:\Omega\rightarrow\mathbb{C}$ is analytic in $\Omega$, and that $|f(z)|=1 \ \forall z\in\Omega$. By using the CR equations, show that $f$ is constant. My ...
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2answers
149 views

How is $f(z)=1_{xy \ne 0}$ not complex differentiable? I know $f(x,y)=1_{xy \ne 0}$ isn't real-differentiable

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 2.18 This is for complex analysis, but I have a feeling this might be answered mostly ...
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1answer
115 views

Which one of the following are true?

Consider the function $e^{-z^{-4}}$ for $z≠0$ and $f(0)=0$. Then, (A) $f$ is not analytic. (B)$f$ is not differentiable at $z=0.$ (c)$f$ does not satisfy the C-R(Cauchy-Riemann) ...
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2answers
105 views

Proving an analytic function is constant [duplicate]

Let $f(z)$, and $g(z)$ be two analytic functions defined on a region $D\subset\mathbb{C}$. Suppose there exist a constant $r>0$ such that $|f(z)|^2+|g(z)|^2=r$ for every $z$ in $D$. Show that both ...
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1answer
110 views

Can we use the polar form of Cauchy-Riemann equations at zero?

Suppose that we want to show that the function $$f(z)=\begin{cases} \frac{z^5}{|z|^4},z\neq0\\ 0,z=0 \end{cases}$$ satisfies the Cauchy-Riemann equations at $z=0$. We see that $$\frac{z^5}{|z|^4}=\...
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1answer
241 views

Cauchy-Riemann equations for a holomorphic function given in polar form

Suppose $f(re^{i\theta})=R(r,\theta)e^{i\Theta(r,\theta)}$ describes a holomorphic function. Is there a Cauchy-Riemann system that $R(r,\theta)$ and $\Theta(r,\theta)$ must satisfy? I've seen this ...
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1answer
202 views

Clarification of Cauchy-Riemann equations

Theorem: Suppose that $$f(z)=u(x,y)+i\,v(x,y)$$ and that $f’(z)$ exists at a point $z_0=x_0+i\,y_0$. Then the first order partial derivatives of $u$ and $v$ must exists at $(x_0,y_0)$, and they must ...
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1answer
136 views

Using Cauchy Riemann equations

If i had some function $g(f(z)) $ and wanted to use the Cauchy Riemann equations to check if it's entire how would I get $ g(f(z))$ in the form $u(x,y)+iv(x,y)$ if the functions were real we would ...
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1answer
36 views

Finding the $z$ in $\mathbb{C}$ where the Cauchy Riemann equations hold for $f.$

I'm new to complex analysis and struggling with the application of Cauchy Riemann equation. I've been given the following question. Given $z=x+iy$ and $f(z) = x^3 + 3xy^2 + i(y^3 + 3x^2y)$ For which ...
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1answer
71 views

Cauchy–Riemann equations and complex differentiability at origin

Consider the function defined by $$f(z)=\begin{cases}0&\text{ if }\operatorname{Re}(z)=0\vee\operatorname{Im}(z)=0\\1&\text{ otherwise.}\end{cases}$$ I would like to show that the real and ...
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2answers
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Prove that $f(z) = |z|^2$ is differentiable only at the origin

Question: I need to show that $f(z) = |z|^2$ is differentiable only at the origin Sufficient condition for differentiability: The sufficient condition for a function $f(z) = u + iv$ to be analytic at ...
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1answer
148 views

Cauchy-Riemann contradiction?

Where is $f(z) = 2ixy$ holomorphic. I would work this out using $u(x,y) = 0, v(x,y) = 2xy$ $$u_x = 0 \qquad u_y = 0 \qquad v_x = 2y \qquad v_y = 2x$$ Since $u_x = v_y \Rightarrow x = 0$ and since $...
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1answer
994 views

Cauchy-Riemann Equations

Let $f(z)=\frac{x^{3}(1+i)-y^{3}(1+i)}{x^{2}+y^{2}}$ at $z$ not equal to zero and $f(z)=0$ at $z=0$. I want to show that the Cauchy-Riemann equations are satisfied at the origin but not analytic. ...
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1answer
2k views

Cauchy Riemann conditions

I am a bit confused about differentiability in complex analysis. We showed in class that if a function $f$ is differentiable at $z_0$, then the Cauchy-Riemann equations hold at $z_0$. We also showed ...
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1answer
1k views

Cauchy-Riemann implies analyticity [duplicate]

Possible Duplicate: When is a function satisfying the Cauchy-Riemann equations holomorphic? If real the functions $u(x,y)$ and $v(x,y)$ satisfy the Cauchy-Riemann equations and have continuous ...
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3answers
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Is there any connection between Green's Theorem and the Cauchy-Riemann equations?

Green's Theorem has the form: $$\oint P(x,y)dx = - \iint \frac{\partial P}{\partial x}dxdy , \oint Q(x,y)dy = \iint \frac{\partial Q}{\partial y}dxdy $$ The Cauchy-Riemann equations have the ...