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.
111
questions
-5
votes
0answers
31 views
Previous year queation complex numbers [closed]
my attempt
I solved the derivation part but unable to prove that the Ux=Vy AT f(0)...kindly direct me how to solve it..... I got Ux=(X^4 +3X^2Y^2 + 2XY^3)/(X^2 + Y^2)^2
Uy= /fract{-Y^4 -2X^3Y -3X^2Y^2}...
-5
votes
0answers
80 views
How to show a given function satisfies the Cauchy Riemann equation at a point, but is not differentiable at that point.
This is my question:
Prove that the function $f(z) = u+iv$, where
$$ f(z) = \begin{cases} \frac{x^3(1+i) - y^3(1-i)}{x^2+y^2}, &\text{ if }z\neq 0,\\ 0 &\text{if }z=0.\end{cases}$$
satisfies ...
-6
votes
0answers
16 views
to check differentiability in complex analysis using c- r equations [closed]
Prove that the function f(z)=x^2*y^2 is differentiable at all point in each of the coordinate axis but not at any other points in the complex plane
1
vote
0answers
39 views
Integral of a f(x,y)=xy over the circle with radius 2 oriented positively (Complex Analysis)
I have to integrate $f(x,y) = xy$ over $|z| = 2$. I've worked out the integral using the fact that $x = \frac{z+\bar{z}}{2}$ and $y = \frac{z-\bar{z}}{2i}$ ending up with $f(z) = \frac{-i(z^2-\bar{z}^...
1
vote
1answer
48 views
Harmonic function problem
I'm attempting to show that the following function (which is defined everywhere on $\mathbb{C} $ with the exception of a discrete set) is harmonic,
\begin{align*}
\psi(z) &:= \...
1
vote
1answer
48 views
Limit of modulus of the difference quotient exists, implying complex differentiability.
A question from a past qualifying exam at my university reads:
Suppose that $f(z)=u(x,y)+iv(x,y)$ is a function on a domain $D$ and $z_0\in D$. Show that if: a) $u$ and $v$ are differentiable at $z_0$ ...
1
vote
1answer
69 views
showing that an function $f$ is constant
I am trying to solve the following problem.
$f(z)=u(x,y)+iv(x,y)$ is an analytic function in $D$ ($D$ is connected and open).
If $u, v$ fulfill the relation $G(u(x,y), v(x,y))= 0 $ in $D$ for some ...
1
vote
0answers
19 views
Relative version of identity theorem for pseudoholomorphic curves
Let $(M,J)$ be an almost complex manifold and $N\subset M$ a closed almost complex submanifold, i.e. the tangent space $T_xN$ at every point $x\in N$ is invariant under $J$.
Let $\Sigma$ be a ...
1
vote
1answer
42 views
Proof of the partial converse of Cauchy Riemann equations: how does the author conclude the limit is $0$?
I was reading through the book "Complex Analysis" by John M. Howie. On page 54, he goes through a proof of the partial converse of the Cauchy Riemann equations - that is, given an open ...
1
vote
1answer
32 views
Question Cauchy Riemann equations
Let $f(z) =zRe(z)$. Determine all points $z_0$ for which the complex derivative $f'(z_0)$ existst.
I wrote $f(z)$ as $f(z)=f(x+iy)=(x+iy)Re(x+iy)=x^2+(xy)i:=u(x)+v(x,y)i$.
So we get the partials $...
0
votes
2answers
66 views
Prove that $u(x,y) = \frac{x}{x^2+y^2}$ is harmonic in $\mathbb{R}^2\setminus\{(0,0)\}$
I want to check if I did this right. I reached the conclusion that $u$ is not harmonic. We know that a function is harmonic if $$\displaystyle\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\...
3
votes
3answers
92 views
How to remember the Cauchy-Riemann equations?
A complex function $f$ with real and imaginary parts $u$ and $v$ respectively is holomorphic in some domain $\Omega$ iff $u$ and $v$ satisfy the Cauchy-Riemann equations in $\Omega$:
$$\frac {\partial ...
1
vote
2answers
44 views
Find an example of an entire function such that $f(z)=-f(iz)$ for all $z$
Is there a systematic way to find an example of an entire function $f$ such that $f(z)=-f(iz)$ for all $z$? By testing at monomials, we find that $f(z)=z^6$ is a solution. But how can we find other ...
1
vote
3answers
45 views
How should I structure this complex analysis proof?
Suppose that $f$ is an entire function (i.e. analytic over the entire complex plane). Show that if
$$f(z)=\overline{i f(-z)}$$
then $f$ must be a constant function of the form
$$f(z)=c(1-i)$$
...
0
votes
1answer
22 views
For $f(z)=2x^3+3iy^2$, $f'(x+ix^2)=?$
For $f(z)=2x^3+3iy^2$, f'(x+ix^2)=?
I said this must be analytic and provide Cauchy-Riemann equations
And my answer ==> $u_x=6x^2$, $v_y=6y$, $u_y=0$, $v_x=0$
So $6x^2=6y$
==> $x^2=y$
==>$z=x+...
5
votes
1answer
55 views
Why is $\int_\Gamma \frac{\log(b-az)}{z} dz=2\pi i\log(b)$?
Let $a\leq b$ and $\Gamma$ be the unit circle (in the complex plane). I found that $\int_\Gamma \frac{\log(b-az)}{z} dz=2\pi i\log(b)$. It seems like Cauchy's integral formula has been used here.
I ...
0
votes
0answers
25 views
Not complex differentiable function
Define $f: \mathbb{C} \rightarrow \mathbb{C}$ as
$f(z):= \begin{cases}\exp(-z^{-4}), &x \ne 0\\ 0, &x=0.\end{cases}$
Show that $F(x_1,x_2):=(\text{Re}(f(x_1+ix_2)),\text{Im}(f(x_1+ix_2)))$ ...
0
votes
2answers
29 views
Finding analytic function with given condition
I have this task from complex analysis: Find analytic function $ f(z) $ such that $ |f(z)|=e^{{\rho}^2\cos(2\theta)} $ where $ z=\rho e^{i\theta}. $ I'm guessing I should use Cauchy-Riemann conditions,...
0
votes
2answers
30 views
Find all values of a complex constant c where a function is analytic in $\mathbb{C}$
Let $f:\mathbb{C} \to \mathbb{C}$ be a function defined by $f(x+iy) = x^2-y^2+2cxy$ , where $c$ is a complex constant.
In other words, $c=c_1+ic_2 \in \mathbb{C}$.
We also define $x,y,c_1,c_2 \in \...
0
votes
1answer
53 views
If $f(z)$ is analytic in a domain D then show that $f^2(z)$ is analytic there.
If $f(z)$ is analytic in a domain D then show that $f^2(z)$ is analytic there.
Let $f(z)=u+iv$, then $f^2=u^2-v^2+2iuv=X+iY$, say
then by CR equation due to analyticitiy, it can be shown that CR ...
0
votes
0answers
16 views
Cauchy-Riemann equations and definition of analytic function at point.
investigate the analicity of $f(z)=r^2cos^2\theta+ir^2sin^2\theta$
I formed the Cauchy Riemann equation and it turns out , the C-R eqs hold in two cases below:
1-$r=0$ ( origin )
2-$sin\theta=cos\...
-1
votes
2answers
53 views
If Cauchy-Riemann equations are satisfied everywhere, then does it mean the function is differentiable for all $z$ [duplicate]
$e^z=e^x cos\,{y}+ie^x sin\,{y}=u+iv$, then $u_x=v_y, u_y=-v_x$ for all $z\in C$.
So $e^z$ is differentiable everywhere in $C$.
True?
But we know that if
$f(z)$ satisfies Cauchy Riemann equations ...
0
votes
1answer
24 views
Study the differentiability of these functions
I have the following functions: $f:\mathbb{C} \to \mathbb{C}$ and $g:\mathbb{C} \to \mathbb{C}$, defined as
$$f(z)=z^2e^{\bar{z}}, \quad g(z)=\sin(z)f(z)$$
I am trying to solve this using Cauchy-...
0
votes
1answer
79 views
Apply Cauchy-Riemann equation on $\sqrt{z}$ to show it is analytic
I want to show that the function $f(z) = \sqrt{z}$ is analytic on $D = \{z\in\mathbb{C}:Re(z)>0\}$ by the Cauchy-Riemann equation.
But here is the thing. I fail to rewrite $f(z)$ into $u(x,y) + iv(...
0
votes
1answer
28 views
Holomorphism of an equation
I am confused about the property of holomorphism of complex functions.
Take the function $\frac{1}{z}$ as an example. This function satisfies the Cauchy-Riemann equations and is thus said to be ...
0
votes
1answer
32 views
analyticity of f(x+iy)=x^3+ax^2y+bxy^2+cy^3
please have a look at the question,
Necessary condition for analyticity of $f(x+iy)=x^3+ax^2y+bxy^2+cy^3$
to solve the question i started from
f(x+iy)=u(x,y)+iv(x,y) since it is analytic it will hold ...
1
vote
1answer
43 views
Pointwise Cauchy-Riemann equations suffice?
Another question just now reminded me of something I realized a while ago I didn't know how to do:
Say $V\subset\Bbb C$ is open, $f:V\to\Bbb C$, $f=u+iv$, and at every point of $V$ the partials of $u$...
0
votes
0answers
46 views
find all functions f, for which g is analytic
let f(z)=u+iv be analytic in open disk $D=\{z\in C s.t |Z|<1\}$.
assume $u\neq v$ for all (x,y) in D.
Find all such f, for which the funstion $g(z)=u^2+iv^2$ in analytic in D.
My answer is: g is ...
1
vote
2answers
52 views
Cauchy Riemann equations, application
For $u: \Bbb C \to \Bbb R$ with
$$u(x + iy) = 2x^3 - 6xy^2 + x^2 - y^2$$
find a function $v: \Bbb C \to \Bbb R$ s.t. $f = u + iv$ is holomorphic.
I see that I have to verify the Cauchy Riemann ...
0
votes
1answer
36 views
How to solve this system of equations:
The original question is: Where is the function below differentiable:
$$\begin{cases}
f(z) = e^{x^2-y^2}*[\cos(2xy) - i\sin(2xy)]
\end{cases}$$
Using Cauchy-Riemann:
$$\begin{cases}
\frac{\delta(u)}...
0
votes
1answer
48 views
holomorphic function, Cauchy Riemann equation
How can I show that the following functions are holomorphic on their domain of definition and that they satisfy the Cauchy Riemann equation?
1) $\displaystyle f(z) = \frac{1}{z+1}$
2) $\displaystyle ...
0
votes
1answer
69 views
Showing that a function is constant on a disc D
Recall that a function $f(z)$ is called radial if it is constant along the circles of center $0$. Let $f$ be a radial holomorphic function defined on the unit disc $D$.
Show that $f$ is constant. (...
2
votes
0answers
25 views
Bounded real component, Cauchy-Riemann implications
I was wondering if there is a way to show that an entire function $f(z)=u(x,y)+iv(x,y)$ is bounded, given its real component is bounded, from the Cauchy-Riemann equations. My intuition is that, since ...
0
votes
2answers
62 views
Cauchy-Riemann Equations + Locally Invertible implies non-zero derivative
Suppose $C^1$ $f:\mathbb{R}^2 \to \mathbb{R}^2$ satisfies the Cuachy-Riemann equations $\frac{\partial f_1}{\partial x} = \frac{\partial f_2}{\partial y}, \frac{\partial f_1}{\partial y} = -\frac{\...
0
votes
1answer
42 views
Show that a complex function is not complex differentiable.
Show that a complex function $f(x,y)=u(x,y)+iv(x,y)$ with components $u(x,y)=\left\{\begin{matrix}
\frac{x^{3}-y^{3}}{x^{2}+y^{2}}& &(x,y)\neq (0,0) \\
0& &(x,y)=(0,0)
\end{...
1
vote
0answers
38 views
Proving Cauchy-Riemann equations for PDE
Need a little help guys, can someone help me on this? I am unable to think about how to prove this. Any help will be greatly appreciated, thanks in advance.
1
vote
1answer
49 views
Let $f: \mathbb{C} \to \mathbb{C} $ such that $f(x+iy)=x^3+3xy^2+i(y^3+3x^2y)$
Let $f: \mathbb{C} \to \mathbb{C} $ such that
$$f(x+iy)=x^3+3xy^2+i(y^3+3x^2y)$$
let $f'(z)$ be the derivative of $f(z)$ ,Then which of the following is true
1)$f'(1+i)$ exists and $|f'(1+i)=3\...
0
votes
1answer
63 views
Is $f(x)=x$ analytic?
The real function $f(x)=x$ has no imaginary part. In particular, it does not satisfy the Cauchy-Riemann conditions (the necessary and sufficient condition for a function to be analytic). So from this ...
2
votes
2answers
80 views
About the notation of derivative used in Cauchy-Riemann equations
This is going to be a very basic ( and dumb ) question but please bear with me, I have a lot of problems in notation whenever I start off with new math and this is the first time that I am being ...
2
votes
0answers
66 views
What do the Cauchy-Riemann equations tell us about being differentiable on a non-open subset?
For an open set $U$ we have that a complex function $f: U \to \mathbb{C} $ is analytic iff it satisfies the Cauchy-Riemann equations and it's partials are continuous. So for open sets the situation is ...
0
votes
2answers
66 views
How to prove a complex function is constant if it is given that complex function is entire function (Holomorphic function).
Let $f$ be an entire function such that for all z with $\lvert z\rvert>M$, for $M\in \Bbb R$ there holds $$\lvert \text{Re}(f(z))\rvert \geq \lvert \text{Im}(f(z))\rvert$$ Prove that $f$ is ...
2
votes
0answers
35 views
Holomorphic complex parameter integrals
Maybe it is a basic question, but I can not find an answer in standard textbooks or at StackExchange.
Let $D\subseteq \mathbb C$ be an open set and $\Omega\subseteq \mathbb C^n$. Further, let $f:D\...
1
vote
0answers
70 views
Shortcut to find imaginary part of analytic function given real part, without standard form of Cauchy-Riemann equations?
I am supposed to find the imaginary part of the analytic complex function, which has real part:
$$e^{y^2-x^2}\cos(2xy)$$
and $z=x+iy$. I know that the answer is $e^{-z^2}+ic$. This can be verified ...
1
vote
1answer
26 views
The integer function $f(z) = u(x,y) + iv(x, y)$ is such that $u(x, y) = e^{āx}\sin(y -1) + y$. What's the value of $f'(i)$?
The integer function $f(z) = u(x,y) + iv(x, y)$ is such that $u(x, y) = e^{āx}\sin(y -1) + y$. What's the value of $f'(i)$?
The solution is $-2i$.
I did:
\begin{align}\frac{\partial u}{\partial x} &...
0
votes
1answer
35 views
Complex Differentiability of $f(x+iy)=\frac{(1+i)x^3-(1-i)y^3}{x^2+y^2}$
I am tasked with showing that $$f(x+iy)=\frac{(1+i)x^3-(1-i)y^3}{x^2+y^2}\text{ if }x+iy\neq0;\quad f(0)=0$$
satisfies the Cauchy-Riemann equations at $0$, despite not being differentiable at $0$. We ...
0
votes
0answers
26 views
About Partial Derivative with respect to Complex Conjugate
I understand the definition below
$$\frac{\partial}{\partial z} = \frac12(\frac{\partial}{\partial x} + \frac1i\frac{\partial}{\partial y})$$
Since if $f$ is homolorphic,
$$f'(z) = \frac{\...
0
votes
1answer
23 views
Two different forms of Cauchy-Riemann equation:
Let $f=\frac{1}{1+z^2}$.
I am trying to determine whether f is differentiable.
$f=\frac{1}{1+z^2}=\frac{1}{1+(x+iy)(x+iy)}=\frac{1}{1+x^2-y^2+2ixy}=\frac{1+x^2-y^2-2ixy}{(1+x^2-y^2)^2-4x^2y^2}$
...
0
votes
1answer
53 views
Let $z = x + iy$, then $f(x) = x^2 - y^2$ is this analytic?
I do not know if I did this fully correctly this is what I have:
$$f(z) = (x + iy)^2 - y^2 = (x^2 + 2xyi - y^2) - y^2$$
so $u(x,y) = x^2 - 2y^2$, $v(x,y) = 2xy$
so by C-R
$$\frac{\partial u}{\...
0
votes
1answer
33 views
Find an expression for $f(z)$ using Cauchy-Riemann Equations:
Suppose that $f: \mathbb{C} \to \mathbb{C}$ with $f(z)=u(x,y) + iv(x,y)$ is differentiable and that
$3u(x,y) + v(x,y) = 7$, $\forall z=x+iy \in \mathbb{C}$.
Find an expression for $f(z)$.
I've been ...
0
votes
0answers
42 views
$f(z)=exp(\frac{ā1}{z^4})$ . Prove Cauchy-Riemann's equations are true for z=0. [duplicate]
why if we split f=u+iv, it is true that u and v satisfy Cauchy-Riemann's equations? what are the steps?
