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.
0
votes
1answer
21 views
Prove that Cauchy-Riemann Equations are satisfied.
Prove that $f(x)$ satisfies the Cauchy-Riemann Equations at the point $z=0$
$$f(z) =
\begin{cases}
\dfrac{\overline{z}^3}{|z|^2}, & \text{if $z$ is not 0} \\
0, & \text{if $z$ = 0}
\end{...
1
vote
2answers
78 views
Complex function only differentiable in $y=x^2$
Find a function $F: \mathbb{C} \rightarrow \mathbb{C}$ that is differentiable in the parabola $y=x^2$ and not differentiable in the rest of the complex plane.
Let $F(x,y) = u(x,y) + i v(x,y)$. If $F$ ...
3
votes
1answer
44 views
$f$ is $\mathbb{R}$ - differentiable iff $Re(f)$ and $Im(f)$ are $\mathbb{R}$ - differentiable
I just read that a sufficient condition for a function $f:A \rightarrow \mathbb{C},f(z) = u(z)+ i v(z)$ to be holomorphic is:
$A$ open.
f is $\mathbb{R}$ - differentiable in $A$.
The Cauchy Riemann ...
0
votes
1answer
41 views
If $u(x,y)$ is not a harmonic function, can a harmonic conjugate $v(x,y)$ be found?
If $u(x,y)$ is not a harmonic function, i.e. does not satisfy Laplace equation
$$\Delta u(x,y) = \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0$$
(and perhaps is not continously ...
0
votes
1answer
35 views
Is $f(z)=k\cdot\frac{e^z}{z}$ analytic?
Let's say we have the following complex function
$$f(z)=k\cdot\frac{e^z}{z}$$
Is it analytic in some domain? When substituting with $z=x+iy$, using trigonometry and deriving the function on the form ...
0
votes
2answers
36 views
Harmonic conjugate and resulting analytic function
I can't quite figure out the last part of this problem...
Find a harmonic conjugate of the harmonic function $u(x,y)= x^3-3xy^2$. Write the resulting analytic function in terms of the complex ...
0
votes
3answers
48 views
why is this complex function not analytic anywhere?
I know for $f(z)=z\bar z$, where $\bar z$ means the conjugate of $z$. Cauchy-Riemann equations are satisfied at $(0,0)$. Also, partial derivatives of U and V exist and are continuous everywhere. so ...
0
votes
1answer
29 views
True/false regarding C-R equation
Is the following statement is True /false ?
The function $f$ defined by $$f(z) = \begin{cases} \frac {Im(z^2)}{\bar z} \ \text{if } z\neq 0 \\ 0 \ \text {if z=0} \end{cases}$$
satisfies the ...
1
vote
3answers
60 views
$\sqrt{|xy|}$ verifies the Cauchy-Riemann equations at $(0,0)$, but is not differentiable at $(0,0)$
Show that the function $f(z)=\sqrt{|xy|}$ verifies the Cauchy-Riemann equations at point zero, but it is not differentiable at that point.
$u(x,y)=\sqrt{|xy|}$ and $v(x,y)=0$ since the imaginary part ...
1
vote
1answer
29 views
Proving complex analytic: Unsolvable Cauchy-Riemann Equations
I need to show that $f(z)$ is nowhere analytic, where $$ 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}...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
-1
votes
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 ...
0
votes
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 $y=0$. ...
0
votes
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,...
0
votes
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
votes
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} = -...
1
vote
0answers
52 views
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 ...
1
vote
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 $T(x,y) = (x+y,x-y),\ x,y \in \Bbb R$....
0
votes
1answer
43 views
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 ...
0
votes
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
$$
f '(z_0) = \frac{1}{2} \left(\frac{\partial ...
0
votes
1answer
43 views
Cauchy - Riemann equation conclusion
I've learnd that, given an open $\mathbb{\Omega}$, if a function $f : \mathbb{\Omega} \subset \mathbb{C} \longrightarrow \mathbb{C} $ is holomorphic in some open $A \subset \mathbb{\Omega}$ the ...
1
vote
0answers
36 views
Why does orientation preserving maps respect CR?
From this post it is written that "orientation preserving conformal maps respect CR. The matrix of the map must be a constant multiplied by some matrix of rotation that has a positive determinant. ...
0
votes
1answer
327 views
Prove that f satisfies Cauchy-Riemann Equations but is not differentiable
Prove that $f$ defined by $f(z) = \frac{z^5}{|z|^4} | (z\neq 0), f(0)=0$ satisfies the Cauchy-Riemann equations at $z=0$ but is not differentiable there.
So I know that $u_x = v_x$.
Therefore let $...
0
votes
1answer
76 views
Cauchy-Riemann equations & harmonic
Let f be holomorphic in an open set Ω ⊂ $\mathbb{C}$. Show that if u = Re f and v = Im f then u and v satisfy the Cauchy-Riemann equations. Show further that u is harmonic.
[You may assume without ...
0
votes
2answers
26 views
find holomorphic function $f(z)$ for $u (x,y) = \cosh(ax)\sin(by)$
Question: [For which positive real numbers $a$ and $b$ is
$u(x,y) = \cosh(ax)\sin(by)$
harmonic? When $a$ and $b$ satisfy this condition find a holomorphic function $f(z)$ such that $\Re f = u$]
I ...
1
vote
2answers
35 views
Show $f$ is constant given $g=\overline{f}$
I am trying to show that if $f$ and $g=\overline{f}$ are both differentiable in a domain, then $f$ is constant on that domain.
My attempt:
Let $$f(z)=u(x,y)+iv(x,y)$$
$$g(z)=r(x,y)+is(x,y)$$
Equating ...
1
vote
2answers
62 views
Show $|f(z)|=1$ is constant
Suppose that $\Omega$ is a domain that $f:\Omega\rightarrow\mathbb{C}$ is analytic in $\Omega$, and that $|f(z)|=1 \ \forall z\in\Omega$. By using the CR equations, show that $f$ is constant.
My ...
1
vote
2answers
149 views
How is $f(z)=1_{xy \ne 0}$ not complex differentiable? I know $f(x,y)=1_{xy \ne 0}$ isn't real-differentiable
A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 2.18
This is for complex analysis, but I have a feeling this might be answered mostly ...
2
votes
1answer
115 views
Which one of the following are true?
Consider the function $e^{-z^{-4}}$ for $z≠0$ and $f(0)=0$. Then,
(A) $f$ is not analytic.
(B)$f$ is not differentiable at $z=0.$
(c)$f$ does not satisfy the C-R(Cauchy-Riemann) ...
2
votes
2answers
105 views
Proving an analytic function is constant [duplicate]
Let $f(z)$, and $g(z)$ be two analytic functions defined on a region $D\subset\mathbb{C}$. Suppose there exist a constant $r>0$ such that $|f(z)|^2+|g(z)|^2=r$ for every $z$ in $D$. Show that both ...
2
votes
1answer
110 views
Can we use the polar form of Cauchy-Riemann equations at zero?
Suppose that we want to show that the function
$$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}=\...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
2
votes
2answers
2k views
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 ...
2
votes
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 $...
0
votes
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. ...
1
vote
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 ...
2
votes
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 ...
39
votes
3answers
3k views
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 ...
