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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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}...
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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 ...
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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
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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}^...
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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) &:= \...
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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 $z_0$ ...
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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 ...
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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 ...
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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 ...
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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 $...
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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}{\...
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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 ...
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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 ...
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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 $$f(z)=c(1-i)$$ ...
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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+...
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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 ...
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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 $F(x_1,x_2):=(\text{Re}(f(x_1+ix_2)),\text{Im}(f(x_1+ix_2)))$ ...
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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,...
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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 \...
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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 ...
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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\...
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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 ...
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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-...
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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(...
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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 ...
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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 ...
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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 $u$...
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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 ...
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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 ...
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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)}...
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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 ...
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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. (...
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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 ...
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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{\...
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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{...
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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.
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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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 ...
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1answer
26 views

The integer function $f(z) = u(x,y) + iv(x, y)$ is such that $u(x, y) = e^{−x}\sin(y -1) + y$. What's the value of $f'(i)$?

The integer function $f(z) = u(x,y) + iv(x, y)$ is such that $u(x, y) = e^{−x}\sin(y -1) + y$. What's the value of $f'(i)$? The solution is $-2i$. I did: \begin{align}\frac{\partial u}{\partial x} &...
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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 ...
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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{\...
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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. $f=\frac{1}{1+z^2}=\frac{1}{1+(x+iy)(x+iy)}=\frac{1}{1+x^2-y^2+2ixy}=\frac{1+x^2-y^2-2ixy}{(1+x^2-y^2)^2-4x^2y^2}$ ...
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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}{\...
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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 ...
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42 views

$f(z)=exp(\frac{−1}{z^4})$ . 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?