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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Conformal mapping is holomorphic
Let $\Omega \subseteq \mathbb{R}^2$ open, $f : \Omega \rightarrow \mathbb{R}^2$ be orientation preserving and map curves intersecting orthogonally to curves intersecting orthogonally. Then $f$ is ...
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Find parameters such that $f(z)=\log(x^2+y^2)-a(\arg z)^2+ib\arg\bar z+\dfrac{c}{\bar z}$ is analytic
The problem asks you for what values of $a,b,c\in\mathbb R$ you can find a domain in $\mathbb C$ where
$$f(z)=\log(x^2+y^2)-a(\arg z)^2+ib\arg\bar z+\dfrac{c}{\bar z}$$
is analytic.
Due to the nature ...
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Equivalence of Cauchy-Riemann equations
You have the CR equations in polar and cartesian coördinates that are equivalent if f(z) is complex differentiable because you can use the chain rule on u and v but if f isn't differentiable you can ...
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Complex Differentials and Cauchy Riemann Differentials
I'm trying to solve this question:
Given $f = u + iv$ is complex differentiable. Show that:
$f' = \frac{\partial u}{\partial x} - i\frac{\partial u}{\partial y} = \frac{\partial v}{\partial y} + i\...
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Where's $\dfrac{\sinh(\sin z)}{z^2+9}$ analytic?
The exercise pretty much asks to find the domain on which the function above,
$$f(z)=\dfrac{\sinh(\sin z)}{z^2+9},$$
is analytic.
The numerator, $\sinh(\sin z)$, is an entire function so we shall only ...
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How do you explain the following equations geometrically?
Let D be the field in C, $f=u+iv \in C^{1}(D)$. Proof:
$$\begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y}\newline
\frac{\partial v}{\partial x} & \frac{\partial v}{...
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Definition of the Complex Derivative and Cauchy Riemann Equations
This is from my notes on the derivation of the Cauchy Riemann equations. My question is about going from the limit as z->z_0 to the lim as Δx->0. Why is f'(z_0) the same as the partial ...
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3D / nD version of "amplitwist" (Drehstreckung) property of holomorphic functions?
Note: This question is probably a duplicate but difficult to Google, so please feel free to close and link to the original question if this is a duplicate. /Note
Question 1: Is there a term for ...
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Weaker version of Goursats theroem
I have a homework problem that states:
Let $\Omega \subseteq \mathbb{C}$ open. Suppose that $f:\Omega \rightarrow \mathbb{C}$ is holomorphic and $C^1$. Show that:
$\int_{T}f dz = 0$
Where $T$ is an ...
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Can you use the Cauchy-Riemann equations to show that a complex function $f(z)$ has an analytical antiderivative?
In complex analysis the fundamental theorem of calculus is only applicable to functions that have analytical antiderivatives. Since it may not be easy to know if that is the case, there are ways to ...
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Cauchy Riemann using polar form of $f(z)$? (NOT polar form of $z$)
Every time I search for a polar form of the Cauchy-Riemann equations, I find answers relating the derivatives of the real and complex parts of $f(z)$ to each other. That is: $\dfrac{\partial u}{\...
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Must every Cauchy-Riemann condition be fulfilled simultaneously?
Working through problems in my complex analysis book, and I have to determine where the derivative exists for a function. I know that the derivative can exist only along a certain curve, however I don'...
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Let function $f=u +iv$ is analytic on $D$ and for some $a,b,c \in \mathbb R$, $a^2+b^2 \neq 0$ and $au+bv=c$ on $D$. Prove that $f$ is constant on $D$
Let function $f=u +iv$ be analytic on some domain $D$. Let $a,b,c \in \mathbb R$ such that $a^2+b^2 \neq 0$ and $au+bv=c$ on $D$. Prove that $f$ is constant on $D$.
What I have done is following: ...
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Must Cauchy–Riemann equation hold if we only know that the partial derivatives exist?
Suppose that $f$ is a holomorphic function function in a neighborhood of $z_0$, $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$ exist at $z_0$. Must Cauchy–Riemann equation $\...
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Prove an entire function is constant on complex plane
Let $f(z)$, $F(z)$ be two analytic functions on $\Bbb C$ satisfies $f(z)=F(\overline{f(z)})$. Here $\overline{f(z)}$ is the complex conjugate of $f(z)$. Prove that $f(z)$ is constant on $\Bbb C$.
I ...
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Is this function holomorphic in $z = 0$?
Consider the function
$$
f(z) = \frac{\mathrm{Im}(z^2)}{|z|^2} \qquad (z \neq 0)
$$
with $f(z) = 0$ for $z = 0$. Using $z = x+iy$ we find
$$
f(z) = \frac{2xy}{x^2+y^2} \ ,
$$
so $u = f$ and $v = 0$ ...
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What holomorphic functions $f = u + iv$ satisfy $u(x, y) = x^3 + xg(y)$, where $g$ is a twice continuously differentiable function?
I'm trying to determine the analytic functions $f = u + iv$ for which $u(x, y) = x^3 + xg(y)$, where $g$ is a
twice continuously differentiable function.
What I've tried is to figure out what v must ...
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Іs there any relationship between $v(y)$ and $u(y)$ where $f(iy) = u(y) + iv(y)$?
Let $f(z)$ be a differentiable function of the complex variable $z$ and let $f(iy) = u(y) + iv(y)$ for $z = iy$ where $y$ is a real variable.
Question. Is there any relationship between the real and ...
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Minimum and maximum of $|f(z)|$, analytic but doesn't satisfy Cauchy-Riemann
Maximum modulus principle
Let $D\subset \mathbb{C}$ be a limited domain (connected open set) and $f(z)$ a complex holomorphic function in $D$, i.e. $f:\bar{D} \to \mathbb{C}$ such that
$$ z = x + i y \...
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Differentiabilty of Complex Function
My professor was discussing this question. After finding out that CR equations are satisified at origin for the function he used the definition of derivative to check for differentiabilty. Also, he ...
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Find conjugate harmonic function of $u=x^2+y^2$?
Find conjugate harmonic function of $u=x^2+y^2$?
By Cauchy-Riemann equations,
$u_x=v_y$ and $u_y=-v_x$.
Now, $v_x=-u_y=-2y$ and $v_y=u_x=2x$.
We have $dv=\frac{\partial v}{\partial x} dx+\frac{\...
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Can a) and b) real components of an complex equation? [closed]
I have a problem somewhere ... I missed a point..
So given are following equations: (with $C \to R$)
(i) $a(z) = x^2 - y^2$
(ii) $b(z) = x^2 + y^2$
And I am supposed to find out if those ...
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1
answer
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Proving Cauchy Riemann Equations for General Functions
How would one prove that Cauchy-Riemann equations hold for any holomorphic functions? I used the definition of complex differentiability to achieve the second equation ( $\frac{\partial u}{\partial y}=...
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$ f(z) = z^2 $ show that image curves of lines $x=a$ and $y=b$ are perpendicular to each other.
Little sidenote: Im new to function theory, so I'm inexperienced and don't have many tools regarding this topic.
$\mathbb{C^{*}} =$ { $z \in \mathbb{C} | z \neq 0$ } with the function $ f(z) = z^2 $.
...
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A set of "Cauchy-Riemann" conditions for quaternion function?
I was trying to get a "Cauchy-Riemann" conditions for a quaternion function $f:\Bbb H\to\Bbb H$, $f(q)=m(q)+n(q)i+o(q)j+p(q)k$ and $m, n, o, p:\Bbb H\to\Bbb R$. Defining the quaternion ...
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Question about the C-R equation and the Analytic functions
I am currently reading Complex Analysis by Stein and found the follow theorem (Theorem 2.4 on Page 13):
Suppose that $f = u + iv$ is a complex-valued function defined on an open set $\Omega$. If $u$ ...
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Which holomorphic functions have constant argument on rays from the origin? On circles centered at the origin?
Multiples $$f(z) = c z, \qquad c \in \Bbb C \setminus \{0\},$$ of the identity function on $\Bbb C \setminus \{0\}$ trivially all satisfy the following special condition:
Condition A: All points on a ...
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Function / question about Cauchy-Riemann equations
$$f(x+iy) = \sin^2(x+y) + i\cos^2(x+y)$$
I have calculated
$$\frac{\partial u} {\partial x} = 2\sin(x+y)\cos(x+y)$$
$$\frac{\partial u} {\partial y} = 2\sin(x+y)\cos(x+y)$$
$$\frac{\partial v} {\...
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Are pointwise solutions also weak?
Suppose the partials of $u, v :\Bbb R^2\to \Bbb R$ exist and satisfy the Cauchy-Riemann equations everywhere, with $u$ and $v$ locally integrable. Is it true that $u$ and $v$ are also weak solutions? ...
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How do you get the real part of function $f(z)$ given the imaginary
I have a function $f(z)$ with imaginary part $v(x,y) = \frac x{x^2 + y^2}$. How do I find the real part of this function?
I am trying to solve this using the Cauchy-Riemann equations. I have found ...
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Question about $f(z)=\exp (-\frac{1}{z^4})$
Let $f(z)=\exp (-\frac{1}{z^4})$ for $z\neq 0$ and $f(0)=0$.
I know this is a famous example and got asked a lot
However my question is not about the origin, but what is the best way to actually show ...
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Where is $f(z) = z^k \Re(z)$ complex differentiable? [closed]
I'm trying to figure out where $f(z) = z^k \Re(z)$ is complex differentiable. I've tried to express $z$ as $z=x+yi$, but I can't figure out how to apply the Cauchy-Riemann equations.
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Cauchy-Riemann: total or partial derivative with respect to $\bar z$
The Cauchy-Riemann conditions for a function $f$ are respected if:
$$\frac{\partial f}{\partial \bar z} = 0$$
But is it a partial or a total derivative actually ?
If it is partial, then I can safely ...
user1107523
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Show that $f(z) = \frac{\sinh (\sqrt z)}{\sqrt z}$ is holomorphic on $\mathbb C$
I have to show that $f(z) = \frac{\sinh (\sqrt z)}{\sqrt z}$ is holomorphic on $\mathbb C$. My attempt is the following:
Let $f(z) = \frac{g(z)}{h(z)}$, where $g(z) = \sinh (\sqrt z)$ and $h(z) = \...
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If f=u+iv and u and v satify the Cauchy-Riemann equations does that imply that f is analytic? or do we need more conditions [closed]
If f=u+iv and u and v satisfy the Cauchy-Riemann equations does that imply that f is analytic? or do we need more conditions
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I need help evaluating the given function
I was given the following problem and told to evaluate where C is the circle $|z|=2$ :
$$ \int_{|z|=2} \frac{1}{z^2-1} dz$$
I've tried solving using Cauchy's Integral Formula and I got the answers $$...
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Components of a 2D vector field satisfy Cauchy Riemann conditions implies the finite transformation is holomorphic
I am reading M. Schottenloher's book on Conformal field theory.
https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/01_978-3-540-68625-5_Ch01_23-08-08.pdf
On page 19 after Proposition 1.12, ...
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Prove Cauchy-Riemann respected with $\frac{\partial f}{\partial \bar z} = 0$
I struggle a lot with complex analysis currently.
I need to find the domain of analycity of $f(z) =: u(x, y) + iv(x, y)$, so the function needs to be continuous at $z$, its partial derivates must ...
user1107523
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Finding all functions making $u + iu$ holomorphic
I wish to find all possible $C^{1}$ functions $u: \mathbb{R}^2 \rightarrow \mathbb{R} $ s.t. $f(x + iy) = u(x,y) + iu(x,y)$ is analytic/holomorphic/complex differentiable. This would occur if and only ...
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Why does path-independence of limits hold in the complex numbers but not in $\mathbb{R}^2$?
In my studies of complex differentiation, I've come across the following paradox concerning real partial derivatives:
In proofs of the Cauchy-Riemann equations, the fact that the limit
\begin{align*}
\...
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how does matrix theory yield the Laplace equation from the Cauchy-Riemann equations
In her paper "How I became a torchbearer for matrix theory" Taussky writes:
In the late thirties Isuddenly realized that the Cauchy-Riemann
equations and the fact that they imply the ...
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2
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Using Cauchy-Riemann equations on holomorphic sets
I am given $\frac{1}{e^z}$ and asked to state the subsets of $\mathbb C$ on which the following function is holomorphic, and to calculate the derivative on its domains of holomorphicity. My first step ...
user1092087
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Cauchy-Riemann on holomorphic sets
I am given $e^{z^2+3z+4}$ and asked to state the subsets of $\mathbb C$ on which the following function is holomorphic, and to calculate the derivative on its domains of holomorphicity. My first step ...
user1092087
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What does it mean when the second Wirtinger derivative is sometimes zero?
I have to show that $f(z) =\sin (\bar z)$ is not analytic anywhere.
One way is to check the CR equations by letting $\sin (z) = \sin(x+iy)$ and do some algebra. From the CR equations I obtained that $...
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Prove that if $f=u+iv$ is conformal in $G\subseteq\mathbb C$, then the Cauchy-Riemann Equations hold for $u$ and $v$.
I was reading Conway's Complex Analysis text and found the proof of the following theorem confusing.
Prove that if $f=u+iv$ is conformal in $G\subseteq\mathbb C$, then the Cauchy-Riemann Equations ...
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Verify that $u, \; v$ are continuous in a neighborhood of $z=0$ and satisfy the Cauchy-Riemann Eqns at $z=0$. Show that $f'(0)$ does not exist.
This is a question from a previous complex analysis qualifying exam that I'm working through to study for my own upcoming qual. I'm really struggling to know where to go with it and any help would be ...
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(complex variables)Let $D \subset C$ be an open connected...
Let $D \subset \mathbb{C}$ be an open connected.
a) Use the Cauchy-Riemann equations to prove that if $F: D \to \mathbb{R}$ is holomorphic, then $F$ is constant.
b)Let $f, g: D \to \mathbb{C}$ be ...
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Analytic if CR equations in polar form satisfied?
Consider the following theorem.
Theorem: Suppose $u(r, \theta)$ and $v(r, \theta)$ (which we shall denote by $u$ and $v$ here afterwards) are real valued functions on $(0, \infty) \times (-\pi, \pi]$....
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How do I go about finding the set of points in $\Bbb C$ where $f(z)=(z^4 + z)e^{\bar{z}}$ is differentiable
$$f(z) = {(z^4 + z)}e^{\bar{z}}$$
In the exercise there are $6$ possible answers:
$\{{0,-1, e^{i\pi/3}, e^{-i\pi/3}}\}$ ; $\{{0,1, e^{i\pi/3}, e^{-i\pi/3}}\}$ ; $\{{0,-1, e^{2i\pi/3}, e^{-2i\pi/3}}\}$ ...
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How to calculate this integral using Cauchy's theorem? [duplicate]
The integral is:
$$\int_{0}^{2\pi}e^{-\sin\theta}\cos(\theta+\cos\theta)d\theta$$
My first thoughts were to use the Cauchy-Riemann equations in polar coordinates to show that the function is regular ...
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