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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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Let $f, g: D \rightarrow \mathbb{C}$ be holomorphic functions on the domain $D \subseteq \mathbb{C}$ such that $f'(z) = g'(z)$ for each $z \in D$

(a) Given is the function $f: \mathbb{C} \rightarrow \mathbb{C}$ with the formula $ f(z) = \frac{1 - i}{4} \left( z^2 + \bar{z}^2 \right) + \frac{1 + i}{2} |z|^2. $ Is $f$ an entire function? If not, ...
Markus's user avatar
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Partial derivatives for functions with one complex variable

So far, suppose that we defined partial derivatives for functions $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$. In Cauchy-Riemann equations, for $f: \mathbb{C} \rightarrow \mathbb{C}$, we first view $f(...
Gunt Ryumet's user avatar
4 votes
2 answers
250 views

Holomorphic function preserving real and imaginary axis

Let $f$ be a holomorphic function such that $f(\mathbb{R})\subseteq \mathbb{R}$ and $f(i\mathbb{R})\subseteq i\mathbb{R}$. Prove that $f(-z)=-f(z)$. We want to prove $g(z)=f(-z)+f(z)$ is zero. Notice ...
Kadmos's user avatar
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Is the existence of continuous first-order partial derivatives necessary for the differentiability of a complex function?

The theorem for the sufficient conditions for differentiability states: Let the function $$f(z) = u(x, y) + i v(x, y)$$ be defined throughout some $\varepsilon$ neighbourhood of a point $z_0 = x_0 +...
A. Srivastava's user avatar
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1 answer
47 views

$f(x + iy) = u(x,y) + iv(x,y)$ holomorphic implies that $F(x,y) = (u(x,y), v(x,y))$ is differentiable

My question comes from section 1.6 of Complex Analysis (4th edition) by Serge Lang: Let $U$ be an open subset of $\mathbb{C}$ and let \begin{align*} f(x + iy) = u(x,y) + i v(x,y), \qquad x + iy \...
Leonidas's user avatar
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2 votes
2 answers
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Derivative notation in section 1.2 of Stein-Shakarchi's Complex Analysis

I'm reading about holomorphic functions in section 1.2 of Complex Analysis by Stein and Shakarchi, and I am pretty confused about the derivative notation that the authors employ. In this section the ...
Leonidas's user avatar
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0 answers
56 views

Confusions about proof of Cauchy-Riemann equations for $r$ and $\theta$

$f(z)=f(x+iy)=u(x,y)+iv(x,y)$ $f(z)=f(x+iy)=f(re^{i\theta})=f(r\cos\theta+ir\sin\theta)=u(r\cos\theta,r\sin\theta)+iv(r\cos\theta,r\sin\theta)=\varphi(r,\theta)+i\psi(r,\theta)$ $\varphi(r,\theta)=u(r\...
studyhard's user avatar
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1 vote
1 answer
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Finding real constants of the function which are analytic

Find the real constants a,b,c such that the function is analytic $f(z)=r^2a\cos2\theta+ br\sin\theta+i(cr^2 \sin\theta\cos\theta+ r\cos\theta)$ The cauchy-Rieman equations for polar coordinates are $\...
Jeonjungkook Kookie's user avatar
1 vote
1 answer
105 views

To find the constants of a function which is analytic

Find the real constants $a,b,c,d$ so that the function is analytic. $f(z)=x^2+axy+by^2+ i(cx^2+dxy+y^2)$ I know that,since the given function is analytic,we can use cauchy reimann equations to solve ...
Jeonjungkook Kookie's user avatar
2 votes
1 answer
96 views

How do I show that the Cauchy-Riemann Equations hold for any polynomial?

I'm self-studying complex analysis and trying to show that the Cauchy-Riemann Equations hold for any complex polynomial $$f(z) = a_n z^n + \dots + a_1 z + a_0$$ but I'm unsure how to actually get a ...
Keshav Balwant Deoskar's user avatar
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1 answer
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Determine all holomorphic functions $f :\mathbb{C} → \mathbb{C}$ with $Re(f(z))) = x^2 + axy + by^2$ for $x = Re(z), y = Im(z)$.

I am having problems solving this exercise: Let $a, b ∈ \mathbb{R}$ be given. Determine all holomorphic functions $f :\mathbb{C} → \mathbb{C}$ with $Re(f(z))) = x^2 + axy + by^2$ for $x = Re(z), y = ...
Marco Di Giacomo's user avatar
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3 answers
92 views

Find $f(z)=u+iv$ if $u=\frac{e^x( (x^2-y^2)\cos y+ 2xy\sin y)}{(x^2+y^2)^2}$

They ask me to find the analytic function $f(z)=u(x,y)+iv(x,y)$ if we know that $$u(x,y)=\frac{e^x( (x^2-y^2)\cos y+ 2xy\sin y)}{(x^2+y^2)^2}.$$ Am I supposed to find $v(x,y)$ satisfying the Cauchy-...
Conreu's user avatar
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2 votes
1 answer
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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 ...
Conreu's user avatar
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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 ...
student's user avatar
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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 ...
Conreu's user avatar
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0 answers
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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}{...
Ciara's user avatar
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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 ...
Sachin's user avatar
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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 ...
hasManyStupidQuestions's user avatar
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0 answers
50 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
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0 answers
48 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
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0 answers
72 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
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2 votes
1 answer
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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'...
ekorel's user avatar
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4 answers
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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: ...
Elise9's user avatar
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0 answers
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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 $\...
Jianing Song's user avatar
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4 votes
3 answers
459 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
85 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
84 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
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1 vote
2 answers
217 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
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1 vote
1 answer
134 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
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0 votes
1 answer
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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{\...
MKS's user avatar
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-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
66 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
  • 90
0 votes
1 answer
40 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
147 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
99 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
182 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
44 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
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0 votes
1 answer
110 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
656 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
  • 537
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
78 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
133 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
57 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
44 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
103 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
48 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
101 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
92 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
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