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.
170
questions
0
votes
0answers
32 views
Cauchy-Riemann Equation satisfies at $z=0$
Let $f(z)=\begin{cases}
\frac{z^5}{\left | z \right |^4} & \text{ if } z\neq 0 \\ 0
& \text{ if } z=0
\end{cases} $
I could show this is continuous on $\mathbb{C}$.
And, I would like to show ...
0
votes
1answer
50 views
Analytic complex function with $u=v^2$
I must prove that if $f = u+iv$ is analytic with $v=u^2$ then $f$ is constant. My attempt is $$f(x,y) = u(x,y)+iv(x,y) = u(x,y)+iu(x,y)^2$$ and write $$\frac{\partial v}{\partial y} = \frac{\partial }{...
2
votes
1answer
46 views
Cauchy-Riemann equations and required continuity of derivatives
So I just read, that for any analytic function, the Cauchy-Riemann equations will hold. However, the reverse, i.e. Cauchy-Riemann equations hold -> function is analytic, is supposedly only true if ...
0
votes
0answers
25 views
Show that $u$ is constant for some positive function $u$ on $\Bbb R^2$ [duplicate]
Suppose $u_{xx}+u_{yy} = 0$ for some positive function $u$ on $\Bbb R^2$. Show that $u$ is constant.
I am not sure how to show this but I think this has something to do with the Cauchy Riemann ...
0
votes
1answer
37 views
Show that $f^*$ is analytic on $\overline {G}.$
Let $G \subseteq \mathbb C$ be a region and $f : G \longrightarrow \mathbb C$ be an analytic function. Let $\overline G : = \left \{z\ |\ \overline {z} \in G \right \}.$ Then by C-R equations show ...
0
votes
1answer
51 views
Constant function in $\mathbb{C}$
The following five problems (Ahlfors Theorem 11 chapter 3) are trivial consequences of the Open mapping theorem.
(a) Show that a holomorphic function $f$ on a domain $\Omega$ whose derivative vanishes ...
0
votes
0answers
34 views
Polar form of CR equation
We know that, by applying a coordinate change, we have the polar form of CR-equation which is given by:
$$
\begin{split}
\frac{\partial u}{\partial r} &= \frac{1}{r}\frac{\partial v}{\partial \...
0
votes
1answer
51 views
Need help with this proof from Axler's Harmonic Function Theory
A free, legal copy is available here.
On pp. 81-82, I have two questions:
Why do the Cauchy-Riemann equations imply that all complex derivatives except the $m^{th}$ vanish at the origin? If anything, ...
5
votes
1answer
113 views
generalization of Cauchy-Riemann conditions
Esteemed experts,
Please excuse the ignorance and language of a poor physicist.
As we know, the real $u(x,y)$ and imaginary $v(x,y)$ parts of an analytic function (in some domain) satisfy the Cauchy-...
2
votes
1answer
50 views
"Jensen's Theorem" from a mathematical physics textbook
I am working through Arfken, Weber, and Harris' s "Mathematical Methods for Physicists" $7^{th}$ edition. Example 12.7.1 is:
"Prove Jensen's theorem (that $\left| F(z) \right|^2$ can ...
0
votes
0answers
29 views
How to know a complex function is analytic without going to the whole process of checking Cauchy-Riemann condition?
How to know a function like for example $f(z)=\frac{1}{z}$ is analytic without turning it into $\frac{x-iy}{x^2+y^2}$ and then taking the partial and stuff?
2
votes
0answers
40 views
Key step in a proof involving complex variables
This question is within the context of Cauchy-Riemann equations.
The text I am reading lists a theorem regarding the existence of the derivative of a function $f$ at point $z_0$. My question concerns ...
4
votes
0answers
46 views
What is the general setting for the Cauchy-Riemann equations and the triholomorphic equations?
Let $U \subseteq \mathbb{R}^n$ be an open subset, let $M_n(\mathbb{R})$ be the algebra of real $n \times n$ matrices, and let $B \subseteq M_n(\mathbb{R})$ be a real subalgebra. Assume the coordinates ...
2
votes
2answers
36 views
Complex analysis of $f(z) = 2 y^2 \sin x − i y^3 \cos x$ [closed]
I want to find the domain of definition, the domain of derivability and the domain of analyticity of the function $f(z) = 2 y^2 \sin x − i y^3 \cos x$
The domain of definition is $\mathbb C$ but I ...
0
votes
1answer
29 views
Cauchy Equation and Continuity
Suppose that $f$ is a real-valued function defined on all of $R$ and satisfying the identity
$$ f(x+y)=f(x) f(y)$$
for all $x, y$ in $\mathbb{R}$.
Prove that, if $f$ is continuous at $x=0$, then $f$ ...
1
vote
0answers
40 views
Applying the dbar operator to differentiate a complex function
I'm trying to differentiate this complex function
$$ f(z) = (|z|-1)^2 $$
in order to determine where it satisfies the Cauchy-Riemann equation $ \frac{\partial f}{\partial \bar{z}} = 0$.
So I first ...
2
votes
0answers
47 views
Proof Detail: Cauchy-Riemann Equations Imply Holomorphy
I'm working through the proof of Theorem 2.4 in Chapter 1 of Stein/Shakarchi's Complex Analysis. I'm looking for clarification on where exactly we use the hypothesis that $u$ and $v$ are continuously ...
0
votes
0answers
60 views
State a primitive of f and use this to evaluate and find the length of path L(s,w)
Let x, y, p, q $\varepsilon$ R let s = p + iq , and let w = x + iy , and let f(z) = $e^z$
State a primitive of f and use this to evaluate
$\int_{l(s,w)} {f(z) dz}$
I have worked out that a ...
0
votes
0answers
60 views
Solutions of Cauchy-Riemann equations have limit at infinity
Let $f:\mathbb{R}^n\to \mathbb{R}^n$, $n\geq 3$ be a $\mathcal{C}^3$, sense preserving function which is a solution to the Cauchy-Riemann system $$Df^T(x)Df(x)=J_f^{2/n}(x)I,$$ where $Df$ is the ...
0
votes
0answers
44 views
Inverse Function Theorem for Complex Functions
I want to prove the inverse function theorem for complex functions using a planar mapping approach. All that's left for me to do is to show that:
$$ g(w) = \xi(u,v)+i\ \eta(u,v) $$
satisfies the C-R ...
-1
votes
2answers
74 views
Complex function $\operatorname{Log}(\frac{z-a}{z-b})$ and Cauchy-Riemann-equation [closed]
I need a proof that the following complex function satisfies the Cauchy-Riemann equations
$$\operatorname{Log}\left(\frac{z-a}{z-b}\right),$$
where $z\in\mathbb{C}$ and $a,b\in\mathbb{R}$ such that $a&...
1
vote
3answers
85 views
Is it analytic ? $f(z)=z^2\bar{z}$
Please find the question and its solution.
My answer gone wrong dont know why, I think the answer marked in the question is wrong.
My try:
I solved for CR equations and replaced x=y=0 and found CR ...
0
votes
1answer
36 views
Proving an equation relating to the Cauchy-Riemann Equations.
Here is what the question asks:
Let $\Omega \subset \mathbb{C}$ be a domain (meaning that it's a connected and open), and let $f: \Omega \rightarrow \mathbb{C}$ be holomorphic. Prove that if $f(z) = ...
0
votes
0answers
40 views
Holomorphy and polynomials: Cauchy-Riemann equations
I'm having some trouble proving this assertion:
Let $\mathcal{P}\in \mathbb{C}[x,y]$ (where $x$ and $y$ are real variables) and $z=x+iy$.
Prove that $\mathcal{P}(x,y)$ satisfies the Cauchy-Riemann ...
1
vote
1answer
25 views
Checking holomorphic domain
Let $z=x+iy$ and $f(z) = 3 x y^2 + i y^3$ let $\frac{\partial}{\overline{\partial z}} = \frac{1}{2} (\frac{\partial}{\partial x} + i \frac{\partial}{\partial y})$
Computing the Cauchy–Riemann ...
1
vote
1answer
38 views
The solution of this Cauchy–Riemann equations.
・$f(z)=f(x+iy)=e^xu(y)+ie^xv(y)$.
・$f(z)$ is regular.
・$f(x)=e^x.$ (When $y=0$, $f(z)=e^x$.)
Under this condition, determine $u(y), v(y).$
The answer is following.
From Cauchy–Riemann equations, $u(y)=...
2
votes
1answer
46 views
If $v$ is an incompressible, irrotational fluid satisfying Euler's equation, show $\overline{v}$ is holomorphic
This feels like it'll be simple once I see it, but I've been stuck for a while now. The setup is from Sijue Wu's 1997 paper Well-posedness in Sobolev spaces of the full water wave problem in 2D. $v=(...
3
votes
1answer
141 views
Cauchy-Riemann Equation for Surface Equations
Suppose we are given smooth functions $W,V:\mathbb{R}^2\times\mathbb{C}\to\mathbb{C}^2$ and are asked to determine if the function $Q(W(x,y,z(x,y)),V(x,y,z(x,y))$ solving
$$
Q(W(x,y,z(x,y)),V(x,y,z(x,...
1
vote
1answer
181 views
Showing the Polar form of the Cauchy-Riemann Equations imply the Cartesian Form
Often times, when the Cauchy-Riemann equations $$u_x = v_y; \ u_y = -v_x$$ are introduced, it's an exercise to show that one can arrive at the "polar form" of those equations. That is, the ...
1
vote
0answers
25 views
Complex differentiation, Cauchy Riemann
By using the definition of complex differentiation and the complex derivative, how can we show that $f'(z_0)= \frac{1}{2} (\frac{\partial f}{\partial x}(z_0)-i\frac{\partial f}{\partial y}(z_0)). $
I ...
1
vote
1answer
34 views
Why $\dfrac{\partial v}{\partial y}=0$ which also gives us $\dfrac{\partial v}{\partial x}=0$?
I passed through this answer here and I don't understand the following. We are at the moment:
If $u^2+v^2 \neq 0$, then $\dfrac{\partial v}{\partial y}=0$ which also gives us $\dfrac{\partial v}{\...
-1
votes
1answer
38 views
For which $z \in \mathbb{C}$ does $f(z) = |z|^2 +2z$ satisfy Cauchy-Riemann equations?
The task is as follows:
$$
\text{Check, for which } z\in\mathbb{C} \text{ function } f(z) = |z|^2 +2z \text{ satisfies the Cauchy-Riemann equations.}
$$
I figured I need to transform $f(z)$ to $f(x,y) ...
0
votes
2answers
84 views
show $f$ complex analytic is constant
$U\subseteq \mathbb{C}$ open and connected, $f : U \rightarrow \mathbb{C}$ complex analytic function.
Suppose $f(z) = u(x, y) + iv(x, y)$, $z = x + iy$, and $u$ and $v$ are
real-valued functions.
...
0
votes
1answer
20 views
Find a real-valued function which satisfied Cauchy-Riemann equations
Question: Find a real-valued function $v(x,y)$ such that $v(0,0)=1$ and together, $u$ and $v$ satisfy the Cauchy-Riemann equations in the entire complex plane.
Let $v(x,y)=M(x,y)+i.0$ which is a real-...
2
votes
1answer
40 views
Cauchy-Riemann conditions implied by harmonicity and orthogonal gradients?
If I have two functions $\eta:\mathbb{R}^2\to\mathbb{R}$ and $\xi:\mathbb{R}^2\to\mathbb{R}$ for which the following relations hold:
$$
\begin{align}
\left( \frac{\partial\xi}{\partial x} \right)^2 + \...
2
votes
1answer
90 views
Cauchy-Riemann Equations
I have a question.
If I have a complex function and I know that for $r=1$, $f(z=e^{iθ})=\cos(θ)\sin(θ)$
And I know that $f(z)$ is entire.
Now, I need to calculate $f'(z)$ for every $z$ in $r=1$, ...
1
vote
0answers
24 views
Show that the inverse function also satisfies C-R equation.
Suppose that a smooth function $f=\left(f_{1}, f_{2}\right): \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ satisfies the following property:
$$
\frac{\partial f_{1}}{\partial x}=\frac{\partial f_{2}}{\...
0
votes
1answer
59 views
Prove $\int_1^{\infty}\psi(x)(x^\frac{s}{2}+ x^\frac{1-s}{2})\frac{dx}{x} $ is Entire
Define $$f(s)= \int_1^{\infty}\psi(x)(x^\frac{s}{2}+ x^\frac{1-s}{2})\frac{dx}{x} $$ where $ \psi(x)= \sum_{n=1}^{\infty}e^{-n^2\pi x}$ is the Jacobi Theta function.
Claim- $f(s)$ is Entire
My Try :-
...
0
votes
1answer
84 views
Question regarding analyticity, differentiability and the Cauchy-Riemann Equations ( Complex Analysis )
For non-analyticity, we can show that the Cauchy-Riemann equations don't hold anywhere. But here we can see that the Cauchy-Riemann equations hold when $y=2x$. I understand what the book did here but ...
0
votes
1answer
30 views
How can I take a partial derivative of an equation like $2u+v=5$?
I know that $\frac{du}{dx} = 0$ and $\frac{du}{dy} = 0$. But I don't undestand how? how can you take a derivative of this when you don't know the value of either u or v?
0
votes
3answers
67 views
Let $f(z) = u(x,y) + iv(x,y)$ be an holomorphic function and $2u(x,y) + v(x,y) = 5$. Show that $f$ is constant
I understand that it suffices to show that $f'(z)=0$
What I don't understand though, is how to take a partial derivative from $2u(x,y) + v(x,y) = 5$ ...
Another solution I thought of is,
If $2u(x,y) + ...
0
votes
0answers
22 views
Are conjugate complex functions independent of each other when differentiating if the Cauchy Riemann conditions are satisfied?
Let $f(z)$ be a function of a complex variable that satisfies the Cauchy-Riemann conditions (i.e. is analytic). Let $f^*(z)$ be its complex conjugate (such that for any value of z, the complex ...
0
votes
0answers
75 views
Does this piecewise function satisfy the Cauchy Riemann equations?
We were asked if the function:
$$
f(z) = \left\{
\begin{array}{ll}
e^{-1/z^4} & \quad z \neq 0 \\
0 & \quad z = 0
\end{array}
\right.
$$
satisfies ...
0
votes
0answers
62 views
Why does an analytic function $z^2$, which satisfies the Cauchy-Riemann equations, not satisfy the condition of being harmonic?
Why does an analytic function $z^2$, which satisfies the Cauchy-Riemann equations, not satisfy the condition of being harmonic?
We have $u = x^2-y^2$ and $v = 2xy$
\begin{align}
du/dx &= 2x, &...
0
votes
1answer
15 views
Proof that harmonic functions $u,v$ satisfy this condition.
Question: Suppose $u,v$ are harmonic and satisfy the Cauchy-Riemann equations in $\mathbb{R}^2$. Show that $f = u + iv$ satisfies $f'(x) = u_x(x,0) - iu_y(x,0)$ for real $x$.
My attempts: I considered ...
0
votes
1answer
41 views
Where is $e^z(z-\bar{z})^2$ differentiable?
Question: Where is $f(z) = e^z(z-\bar{z})^2$ differentiable?
My attempts: By letting $z = x+iy$ and by using the Cauchy Riemann Equations, I was able to get to a point where $\sin y = 0$ and $\cos y = ...
1
vote
1answer
185 views
Derivative of a Complex Function as a $\mathbb{C}$-linear Transformation
I'm trying to show that the derivative of a differentiable complex function is a $\mathbb{C}$-linear mapping of $\mathbb{C}$ to itself, and since every $\mathbb{C}$-linear map is of the form
$$
\...
1
vote
1answer
65 views
Determine where the Cauchy Riemann conditions are satisfied for $w=\overline{z}$ (the complex conjugate of $z$)
I have a question in my complex analysis worksheet.
Determine where the Cauchy Riemann conditions are satisfied for $w=\overline{z}$ (the complex conjugate of $z$).
As far as I know the function has ...
-1
votes
1answer
253 views
Prove if a function is holomorphic, then its complex conjugate is holomorphic, by Cauchy-Riemann Equations
Given a function $f(z)$ is holomorphic on a disc $D(0,R)$, we want to prove that $g(z)= \overline {f(\bar z)}$ is holomorphic.
Below is my proof.
$\lim_{h\to 0} \frac{\overline {f(\overline{z+h})}-\...
1
vote
2answers
113 views
Wirtinger derivative form of Cauchy–Riemann equations
I'm trying to understand the Cauchy-Riemann equations using the traditional $u, v$ form and the Wirtinger derivative form.
Taking $\ln|z|$ as an example function, for the normal $u, v$ form I have:
$$\...
