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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
15 views

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 ...
strugglingStudent's user avatar
2 votes
1 answer
53 views

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 ...
Joan S. Guillamet F.'s user avatar
0 votes
0 answers
59 views

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 ...
Lowie's user avatar
  • 1
0 votes
0 answers
32 views

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\...
Ben Bockhoff's user avatar
0 votes
1 answer
64 views

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 ...
Joan S. Guillamet F.'s user avatar
0 votes
0 answers
48 views

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}{...
Ciara Zheng's user avatar
0 votes
0 answers
19 views

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 ...
Sachin's user avatar
  • 21
1 vote
0 answers
45 views

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 ...
hasManyStupidQuestions's user avatar
0 votes
0 answers
44 views

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 ...
strugglingStudent's user avatar
0 votes
0 answers
44 views

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 ...
JohnMcSwag's user avatar
0 votes
0 answers
55 views

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}{\...
SSD's user avatar
  • 580
2 votes
1 answer
75 views

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'...
ekorel's user avatar
  • 57
0 votes
4 answers
147 views

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: ...
Elise9's user avatar
  • 67
0 votes
0 answers
53 views

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 $\...
Jianing Song's user avatar
  • 1,298
4 votes
3 answers
369 views

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 ...
QIRUN CONG's user avatar
2 votes
2 answers
81 views

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 $v = 0$ ...
oakley09's user avatar
1 vote
3 answers
71 views

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 ...
A A's user avatar
  • 561
1 vote
2 answers
206 views

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 ...
Leox's user avatar
  • 7,899
1 vote
1 answer
123 views

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 \...
Carlos Adir's user avatar
  • 1,292
0 votes
0 answers
32 views

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 ...
KUSH JUVEKAR's user avatar
0 votes
1 answer
180 views

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{\...
MKS's user avatar
  • 720
-2 votes
1 answer
93 views

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 ...
Jackie Arika's user avatar
2 votes
1 answer
60 views

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}=...
vinicio's user avatar
  • 61
0 votes
1 answer
32 views

$ 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 $. ...
AsaMitaka's user avatar
2 votes
2 answers
129 views

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 ...
Daniel Muñoz's user avatar
0 votes
2 answers
74 views

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 $u$ ...
Florian Huo's user avatar
3 votes
2 answers
171 views

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 ...
Travis Willse's user avatar
0 votes
1 answer
50 views

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} {\...
Inocenciaa's user avatar
1 vote
0 answers
40 views

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? ...
GuPe's user avatar
  • 7,318
0 votes
1 answer
57 views

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 ...
gemini's user avatar
  • 29
2 votes
2 answers
336 views

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 ...
Parinn's user avatar
  • 525
2 votes
0 answers
52 views

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.
Marcos R's user avatar
0 votes
0 answers
59 views

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 ...
user avatar
2 votes
1 answer
110 views

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) = \...
syphracos's user avatar
  • 486
0 votes
1 answer
53 views

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
Fairy Tail's user avatar
0 votes
1 answer
76 views

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 $$...
Astronomical's user avatar
0 votes
1 answer
37 views

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, ...
Oscar Zhou's user avatar
4 votes
3 answers
99 views

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 ...
user avatar
2 votes
1 answer
44 views

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 ...
Vera Leighton's user avatar
-1 votes
2 answers
98 views

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*} \...
WilliamMastin's user avatar
2 votes
1 answer
76 views

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 ...
Tomm's user avatar
  • 71
2 votes
2 answers
90 views

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 ...
user avatar
0 votes
2 answers
68 views

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 ...
user avatar
2 votes
1 answer
129 views

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 $...
Teo LC's user avatar
  • 47
2 votes
1 answer
183 views

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 ...
math-physicist's user avatar
0 votes
1 answer
63 views

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 ...
Serafina's user avatar
  • 470
2 votes
2 answers
134 views

(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 ...
Gab's user avatar
  • 87
3 votes
0 answers
63 views

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 $(0, \infty) \times (-\pi, \pi]$....
Yathiraj Sharma's user avatar
1 vote
1 answer
46 views

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: $\{{0,-1, e^{i\pi/3}, e^{-i\pi/3}}\}$ ; $\{{0,1, e^{i\pi/3}, e^{-i\pi/3}}\}$ ; $\{{0,-1, e^{2i\pi/3}, e^{-2i\pi/3}}\}$ ...
平常心是道's user avatar
0 votes
2 answers
48 views

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 ...
agaminon's user avatar
  • 337

1
2 3 4 5 6