Tagged Questions

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0answers
21 views

Harmonic Function Transformation Help

Consider the harmonic function $$u(x,y)=1-y+\frac{x}{x^2+y^2}$$ on the upper half plane $y>0$. What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
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1answer
56 views

Showing particular harmonic function is constant

Suppose $u$ is a real valued continuous function on $\overline{\mathbb D}$, harmonic on ${\mathbb D}$\ $\{0\}$ and $u=0$ on $\partial\mathbb D$, show $\mathbb u$ is constant in $\mathbb D$. I'm going ...
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1answer
53 views

How to determine if the sums and products of harmonic functions is also harmonic?

Suppose I know that $u + iv$ is non-constant and analytic on a domain $D$, then I know that $u$ and $v$ are harmonic on $D$ and not both constant; but how do I then determine whether $3u^2v - v^3 + ...
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0answers
30 views

Angle between Harmonic functions

I have a question about a practice prelim problem: What is the angle between the curves $Re(z^3) = 1$ and $Re(z^3) = Im(z^3)$? Also, What is the angle between the curves $Re(z^3) = 0$ and $Re(z^3) ...
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0answers
33 views

Prove that the function is bounded and harmonic

Let $ f: \{ z: $ Re $z$ = $0$ $\}$ $\rightarrow$ $\mathbb {R}$ be a bounded continuous function and define $ u: \{ z: $ Re $z$ > $0$ $\}$ $\rightarrow$ $\mathbb {R}$ by $$ u(x+iy) = \frac{1}{\pi} ...
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1answer
31 views

Harmonic functions and real valued function related to it

Find all real-valued functions $h$, defined and of class $C^2$ on the positive real line, such that the function $u(x,y)=h(x^2+y^2)$ is harmonic.
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0answers
37 views

Bounded harmonic functions on the right half plane

Let $ f: \{ z: $ Re $z$ = $0$ $\}$ $\rightarrow$ $\mathbb {R}$ be a bounded continuous function and define $ u: \{ z: $ Re $z$ > $0$ $\}$ $\rightarrow$ $\mathbb {R}$ by $$ u(x+iy) = \frac{1}{\pi} ...
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0answers
35 views

Composition of a subharmonic function and a conformal mapping

this is q.4 of p.248 of Ahlfors book: Prove that a subharmonic function remains subharmonic after a composition with a conformal mapping. What I'v tried: Let $u:\Gamma \rightarrow \mathbb{R}$ and ...
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4answers
191 views

A question in Complex Analysis $\int_0^{2\pi}\log(1-2r\cos x +r^2)\,dx$

My problem is to integrate this expression: $$\int_0^{2\pi}\log(1-2r\cos x +r^2)dx.$$ where $r$ is any constant in $[0,1]$. I know the answer is zero. Can you explain you idea to me or just prove ...
3
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1answer
45 views

Bound for analytic function from unit disk into punctured unit disk

Suppose $f$ is analytic in the unit disk $D$ and satisfies $0<|f(z)|<1$. Show that $|f(z)|\leq|f(0)|^{\frac{1-|z|}{1+|z|}}$ for all $z\in D$. I tried to work with $\log|f|$. It seems that ...
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1answer
109 views

Proving that $f$ is analytic if $f$ and $z f(z)$ are harmonic

If $f$ is harmonic and $zf(z)$ is harmonic, then $f$ is analytic. Please help me prove this. Thanks.
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1answer
67 views

Poisson integral

Given a bounded harmonic function $u(z)$ on the open unit disk, and given radial limits $\lim_{r\rightarrow 1^{-1}}u(re^{i\theta})$ being some constant $a$ for $0<\theta<\pi$ and being some ...
0
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1answer
27 views

Conformal map projecting a line to a sine wave

I'm looking for an analytic complex function that will map a straight line on to a sine wave. Are there any known examples? To be more specific, let $f(x+iy) = u(x,y) + iv(x,y)$. I want to find a ...
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0answers
35 views

Geometric Condition for Harmonic Function

Under what geometric condition on real harmonic functions u and v on a region G is the function uv also harmonic?
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1answer
29 views

Harmonic conjugate of $u,v$ in $f=u+iv$

Do I understand correctly the definition of being harmonic conjugate if I understand it that: $v$ is the harmonic conjugate of $u$ but $u$ is not the harmonic conjugate of $v$, but rather $-u$ ?
2
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0answers
30 views

For compact $K \subset \mathbb C$, show that $u(z) = -\log(\mathrm{dist}(z,K))$ is subharmonic

Let $K \subset \mathbb C$ be compact, and let $u(z) = -\log(\mathrm{dist}(z,K))$ be defined on $\mathbb C \setminus K$. May I get a hint for proving that $u(z)$ is subharmonic? Subharmonic, here, is ...
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2answers
49 views

Harmonic function, existence of a constant

May i ask you for a little help about a problem with harmonic function? It seems to be not that difficult, in a way even intiutively obvious but i don't really know how to show this explicitly. We ...
4
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3answers
111 views

Is it a harmonic function or not?

I am trying to resolve a question whether a certain function is harmonic or not. If yes, I should find its harmonic conjugate. The function is $u = \frac{x}{x^2+y^2}$. I found that it is a harmonic ...
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1answer
139 views

Derive Poisson's integral formula for Im z>0

How to derive Poisson's integral formula for $\text{Im }{z}>0$ given that for $|z|<1 $ we have ...
6
votes
2answers
135 views

How do you prove that $\ln|f(z)|$ is harmonic?

Suppose that $f(z)$ is analytic and nonzero in a domain $D$. Prove that $\ln|f(z)|$ is harmonic in $D$. I know the laplacian equation but I'm not sure how to use it.
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0answers
52 views

Elementary question about differentiating harmonic complex functions

Consider the complex analytic function $\Omega(z) = \phi(x,y,) + \psi(x,y)$ in a domain D. If we transform from z to w using $w=f(z), w = u+ iv$, with f(z) analytic in D, with the co-domain in the w ...
4
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1answer
105 views

Harmonic functions with zeros on two lines

For which pairs of lines $L_1$, $L_2$ do there exist real functions, harmonic in the whole plane, that are $0$ at all points of $L_1 \cup L_2$ without vanishing identically? Note: This is ...
3
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1answer
107 views

Characterization of positive harmonic functions on unit disc with $0$ radial limits

Suppose $u$ is a positive harmonic function in $U$, and $u(re^{i\theta}) \to 0$ as $r \to 1$, for every $e^{i\theta} \ne 1$. Prove that there is a constant $c$ such that $$u(re^{i\theta}) = ...
4
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1answer
109 views

Simply connected domain and harmonic function

Let $\Omega$ be a simply connected domain that is properly contained in $\mathbb C$, and $u(x,y)$ is harmonic on the unit disk $\mathbb D $, then there is a funtion $f(z)$, that is one-one and ...
0
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1answer
87 views

Harmonic Extension

Let be $u$ a harmonic function defined on an open set $\Omega \setminus \{p\} \subset \mathbb{C}$ of the complex plane. Show that if $u$ is bounded in a neighborhood of $p$ then $u$ admits a harmonic ...
1
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1answer
82 views

Is this function a subharmonic function?

Does anyone know, is $h\left(z,w\right):=\frac{\left|zw\right|}{\left|z\right|+\left|w\right|}$ for $z$ and $w$ in the unit disk $\mathbb{D}$ of the complex plane a (pluri)subharmonic function? ...
2
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0answers
30 views

hyperbolic group ; showing the existence of a ration function with a certain condition

I'm currently working out of a book called differentialgeometry and minimal surfaces written by Jost-Hinrich Eschenburg und Jürgen Jost. Right now I'm looking at an exercise (12.5) under the ...
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2answers
264 views

Logarithm of absolute value of a holomorphic function harmonic?

Let $f:U\rightarrow\mathbb{C}$ be holomorphic on some open domain $U\subset\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ and $f(z)\not=0$ for $z\in U$. Is it true that $z\mapsto \log(|f(z)|)$ is ...
2
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1answer
148 views

Let $f$ be a harmonic function. Prove that $\overline{f}$ is harmonic.

Let $f$ be a harmonic function. Prove that $\overline{f}$ is harmonic. I need help to write a rigorous proof. Thank you
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1answer
80 views

is the converse true: in a simply connected domain every harmonic function has its conjugate

The question is. Is the converse true: In a simply connected domain every harmonic function has its conjugate? I am not able to get an example to disprove the statement.
1
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1answer
66 views

a special extension of a two variable function

We consider the function $f(x,y)=x^2+y^2$ in $\omega = (0,1)^2.$ I am wondering about the existence of a $C^2-$extension $F$ of $f$ in $\Omega = (0,2)^2$ such that $F$ is harmonic in ...
1
vote
2answers
322 views

Proving the maximum principle for harmonic functions

I am in the middle of the proof of the maximum principle for harmonic functions. Given a harmonic function $u$ on the complex plane and $M_0\in \mathbb{C}$. Take $r>0$ and suppose there is an open ...
2
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2answers
71 views

Boundary values of harmonic $u$ are $ u(e^{it}) = 5- 4 \cos t $; find $u(1/2)$ and $v(1/2)$.

My problem is the following: Let $u$ be a continuous real-valued function in the closure of the unit disk $\mathbb{D}$ that is harmonic in $\mathbb{D}$. Assume that the boundary values of $u$ are ...
5
votes
3answers
644 views

Composition of a harmonic function.

I noticed this question while reading several pdfs of lecture notes, and I'm having trouble figuring it out. Can anyone help? If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ ...