2
votes
1answer
37 views

Fake proof using mean value property

Let $f : \Delta \rightarrow \mathbb{C}$ be a holomorphic function on the unit disk. Let $u = |f|$. Assume that $u(0)=f(0)=0$. Question : since $u$ is harmonic, the mean value property should imply ...
1
vote
1answer
22 views

Showing that a function is harmonic using a specific method

I'm trying to show that if $f$ is a harmonic function, then so is $\log|f|$. Moreover, I'm trying to do this using the following operator: $$ \Delta = 4\frac{\partial}{\partial z} ...
1
vote
1answer
28 views

When is a harmonic map biholomorphic?

Let $D_1$ and $D_2$ be simply-connected bounded open domains on $\mathbb{C}$. Riemann mappping theorem tells us that there exist biholomorphisms between them. On the other hand, let $\gamma : ...
3
votes
0answers
22 views

Harmonic except at a point [duplicate]

If $u$ is harmonic and bounded in $0<\lvert z \rvert< \rho$, show that $u$ becomes harmonic in $\lvert z \rvert< \rho$ when $u(0)$ is properly defined. What I was thinking so far is that ...
2
votes
0answers
23 views

Proof that function is real part of $\sec(z)$ [duplicate]

I'm working on the following problem: I've deduced that the key is to show that $u$ is the real part of $\sec(z)$. But, I'm getting stuck in the algebra and am hoping someone can point me in the ...
0
votes
0answers
16 views

Property about positive harmonic functions

Let $U$ be a region and $K$ a compact subset of $U$. Fix a $z_{0} \in U$. Why does there exist positive real numbers $\alpha$ and $\beta$ such that $$\alpha u(z_{0}) \leq u(z) \leq \beta u(z_{0})$$ ...
1
vote
1answer
38 views

Find an harmonic function in $\mathbb R^n$ which is a polynomial of degree 4 and is 1 at the origin

Find an harmonic function in $R^n$ which It is a polynomial of degree 4 and is =1 at the origin. It is a polynomial of degree 5 and its partial derivatives are both =0 at the origin. Important ...
1
vote
1answer
33 views

Showing that two given functions are harmonic

I'm preparing for my complex analysis midterm on Thursday and our professor gave us the following as a practice problem: I'm a bit confused on how to approach part (a). Here's my train of thought: ...
0
votes
1answer
39 views

Showing that $u(x, \, y) = \ln(x^2 + y^2)$ is harmonic without computing partial derivatives

I'm trying to show that $u(x, \, y) = \ln(x^2 + y^2)$ is a harmonic function, without explicitly computing the partial derivatives and showing that $u_{xx} + u_{yy} = 0$. I believe that it would ...
0
votes
1answer
29 views

Harmonic function and Poincaré metric

Let $u$ be a harmonic function on the unit disk $\Delta$, taking values in $[0,1]$. Is it true that this implies that $u$ is Lipschitz for the Poincaré metric ? If not, what can be said about a ...
3
votes
2answers
31 views

Harmonic non-surjective functions are constant

Let $u:\mathbb R^2 \to \mathbb R$ be a non-surjective harmonic function. $(i)$ Show that $u$ is bounded from below or from above. $(ii)$ Prove that $u$ is constant (and therefore any harmonic ...
2
votes
3answers
51 views

Harmonic Maximum modulus

So, i am starting to solve some exercises of complex analysis, and i am a little rusty, so if anyone could help me with this exercise. I think that if i just can prove the mean value theorem for ...
0
votes
1answer
22 views

Function $u(x,y) \in C^2$ that admits harmonic conjugate

Problem 1) Prove that if the real and imaginary part of a holomorphic function are of class $C^2$, then they are harmonic. 2)Deduce from 1) that if $u(x,y) \in C^2$ is a function that admits a ...
0
votes
1answer
70 views

Dirichlet problem on a disk with polynomial boundary values

Suppose that $\phi$ is a real valued harmonic function on the unit disc that is continuous up to the boundary such that $\phi$ agree with a real valued polynomial on the unit circle. Then $\phi$ ...
2
votes
1answer
47 views

Uniqueness of harmonic function

Let $u\in C(2\overline{\mathbb{D}})$ be harmonic in $\mathbb{D}$, and also harmonic inside the annulus $\{1<z<2\}$. Suppose $v\in C(2\overline{\mathbb{D}})$ is another function that is harmonic ...
0
votes
1answer
27 views

The limit function of decreasing sequence of subharmonic is also subharmonic

Let $u(z)$ be a continuous function on a domain $D \subset \mathbb{C}$ to $[−\infty, \infty)$. Suppose $u_n(z)$ is a decreasing sequence of subharmonic functions on $D$ such that $u_n(z) \to u(z)$ for ...
3
votes
3answers
105 views

Positive harmonic function with harmonic reciprocal must be constant

Let $f(z)$ be a positive harmonic function on the unit disk such that $\frac{1}{f(z)}$ is also harmonic. Show $f(z)$ must be constant.
2
votes
1answer
43 views

Estimates for harmonic functions

Assume $u$ is harmonic in $U$. Then $$ |D^{\alpha}u(x_{0})|\leq \frac{C_{k}}{r^{n+k}}\|u\|_{L^{1}(B(x_{0},r))} $$ for each ball $B(x_{0},r)\subset U$ and each multi-index $\alpha$ of order $|\alpha| ...
0
votes
0answers
34 views

Vanishing Partial Derivatives of a Harmonic Function

If $u$ is a harmonic function such that all of its partial derivatives vanish at some point $z$, show that $u$ is constant.
0
votes
1answer
31 views

If $f$ and $g$ are holomorphic, then $\log(|f|+|g|)$ is subharmonic

Let $f$ and $g$ be two holomorphic functions on a plane domain, and let $u(z)=\log(|f(z)|+|g(z)|)$. Is it true in general that $u$ is subharmonic? I know it is true if $g=0$, but here I have some ...
3
votes
0answers
79 views

Subharmonic function and holomorphically parametrized integrals

Let $f_\lambda$ be a family of $L^1$ functions (say on $\mathbb{C}$) such that for all $z$ the map $\lambda \mapsto f_\lambda(z)$ is holomorphic. Consider the map $N(\lambda)=\log \int |f_\lambda(z)| ...
1
vote
0answers
26 views

Complex Variables Conformal Mapping in Complex Plane of harmonic Functions

Consider the harmonic function $u(x,y) = 1 - y + x/(x^2+y^2)$ on the upper half plane $y > 0$. What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
0
votes
0answers
28 views

example of harmonic function on sphere

Can anyone give me an example of a harmonic function on the sphere $S^{2}=\{(x,y,z):x^2+y^2+z^2=1,x,y,z\in{\mathbb{R}}\}$, which equals $1$ on the northern hemisphere and $-1$ on the southeren ...
2
votes
1answer
47 views

Harmonic conjugates on annulus slit

Let $D$ be an annulus slit with $$D= \{a<|z|< b \}$$ excluding $(-b,-a)$. Show that any harmonic function on $D$ has a harmonic conjugate on $D$. The hint says to fix $c$ between $a$ and ...
0
votes
1answer
44 views

What is a harmonic conjugate of $u=Arg(z)$?

Let $u=Arg(z)$ be a function maps $\mathbb{C}\setminus \{0\}$ to $ (-\pi,\pi]$. How do i find a harmonic conjugate of $u$ when $Arg(z)\in (-\pi,\pi)$?
0
votes
1answer
37 views

Extending bounded functions on $\mathbb{R}$ to $\mathbb{H}$ with the Poisson kernel.

Let $h(\phi):\mathbb{R}\to\mathbb{R}$ be a bounded piecewise continuous function on the real line. Define a function on the upper half-plane by the formula $\tilde{h}(s+it):=\int_{-\infty}^{\infty} ...
1
vote
1answer
44 views

Need help about harmonic functions!

I have trouble on solving the following problem: Show that there doesn't exist a non-constant function $u$ such that $u$ is harmonic on C and for $z=x+iy$ in C that $u(z)>4x^2+9y^2+1$.
1
vote
2answers
43 views

Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic?

Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic? If yes, in which domain?
3
votes
1answer
114 views

The derivative of harmonic function at origin is a bounded linear functional

The following problem is the 5th problem in the qualifying exam of UCLA (spring 2013). Let $\mathbb{D}=\{(x,y):x^2+y^2<1\}$ and let us define a Hilbert space $$H:=\{u:\mathbb{D} \rightarrow ...
1
vote
2answers
64 views

Harmonic function vanishing on a set of positive measure.

I'm preparing for a qualifying exam, and came across a question I couldn't figure out: If $\Omega$ is a region and $h:\Omega\to \mathbb{R}$ is a harmonic function vanishing on a set of positive ...
3
votes
1answer
57 views

Show $\Omega$ is simply connected if every harmonic function has a conjugate

Prove: If every harmonic function on $\Omega$ has a harmonic conjugate on $\Omega$, then $\Omega$ is simply connected. Same question is asked here but no proof is presented: is the converse true: in ...
1
vote
1answer
89 views

harmonic conjugate of a natural log

Find the harmonic conjugate of $\ln \sqrt{x^2+y^2}$ on some open nonempty subset of the plane. Ok I got stuck. So I set the function as $u(x,y)=\ln \sqrt{x^2+y^2}$. So to show that $u_x(x,y)= ...
0
votes
1answer
37 views

Why do level curves of a function and its harmonic conjugate intersect each other orthogonally?

So I've had this assignment in which I had to proof that two level curves of a function and one of its harmonic conjugates intersect each other orthogonally. The proof itself wasn't that difficult, ...
1
vote
0answers
44 views

Does a complex polynomial can be approximated by a Euler-type exponential function?

Can one uniformly approximate a complex polynomial with a Euler-type exponential function? I mean: let $E\subset\Bbb C$ be a compact set with the connected complement $\Omega:= \bar {\Bbb C}\setminus ...
1
vote
1answer
54 views

Harmonic functions on $\mathbb C-\{0\}$

Find all real-valued $C^2$ differentiable functions $h$ defined on $(0,\infty)$ such that $u(x,y)=h(x^2+y^2)$ is harmonic on $\mathbb C-\{0\}$. This is one of my homework problem. As I understand I ...
0
votes
1answer
39 views

Conformal Mapping and Relating Solutions (of Laplace) of Domains (via the Mapping)

Find a conformal equivalence between the following domains: the strip $ S = \{ z \in \Bbb C \ | \ 0 < \Bbb Im(z) < 1 \} $ and the quadrant $ Q = \{z \in \Bbb C \ | \ \Bbb Re(z) > 0, \Bbb ...
3
votes
0answers
41 views

The Schwarz reflection principle and harmonic function (Big Rudin chapter 11)

In his book page 250 Exer 11: Suppose that $I=[a,b]$,$\Omega$ is a region ,$I\subset\Omega$,$f$ is continuous in $\Omega$,and $f\in H(\Omega-I)$,prove that actually $f\in H(\Omega)$. If I follow the ...
0
votes
1answer
62 views

Function must be constant comparison

The following 2 problems are past exit exam problems for my major. I see that they're worded differently but are asking me to do the same thing. Not sure how they differ much I'd appreciate if anyone ...
1
vote
2answers
49 views

Harmonic functions constant on circumferences

I want to find all the harmonic functions in $\mathbb{R}^{2}-\{(0,0)\}$ which are constant on circumferences with center in $(0,0)$. $\mathbb{R}^{2}-\{(0,0)\}$ isn't simply connected so we can't ...
3
votes
2answers
102 views

$uv$ is harmonic if and only if $u+icv$ is analytic for some real c

Let $u$ and $v$ be non constant harmonic functions on a complex domain. Prove that $uv$ is harmonic if and only if $u+icv$ is analytic for some real $c$. I can prove the "if" part. I am having some ...
8
votes
1answer
249 views

Error on Wikipedia: Nelson's proof of Liouville's theorem works only for bounded modulus?

On Wikipedia, it is stated: If $f$ is a harmonic function defined on all of $\mathbb{R}^n$ which is bounded above or bounded below, then $f$ is constant...Edward Nelson gave a particularly ...
0
votes
1answer
134 views

Dirichlet Problem on the unit disk

Find a C-harmonic function in the unit disk with boundary values $x^3-xy$. I know the answer is $u(x,y)=\frac{(x^3-3xy^2)}{4} + \frac{3x}{4} - xy$ but don't know how to solve it Any hint or help is ...
2
votes
1answer
86 views

Upper bound for coefficients of a power series

I am doing the following problem. Suppose $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is an analytic function on the unit disc $|z|<1$. Let $0<r<1$. Prove that $$|a_n|r^n\leq \max\{4A(r),0\}-2Ref(0),$$ ...
3
votes
1answer
160 views

Finite order function in the complex analysis.

Assume that an entire function $f$ be finite order with finitely many zeros. Please show that either $f(z)$ is a polynomial or $f(z) + z$ has infinitely many zeros. Thank you. And I know the ...
1
vote
0answers
40 views

Find a function $f$ such that $f$ is harmonic on $D$ and $f|_{\partial D}$.

I understand its solutions in general. But my question is how to decide whether I sould take $Im z^4$ or $Re z^4$? I have two similar examples. And in one example, the real part is taken, but in ...
1
vote
0answers
373 views

Poisson Integral Formula

I'm looking at the following problem. Prove that if $h$ is harmonic on an open neighborhood of the disc $B(w,\rho)$, then for $0 \leq r < \rho, 0 \leq t < 2\pi$, $$h ...
2
votes
1answer
59 views

A question on Harmonic functions.

$f$ and $g$ are two given continuous functions on $\overline {D(0,1)}$) and both $f$, $g:\overline {D(0,1)}\to\Bbb R$ are harmonic on unit disc $D(0,1)$. Now, If $f\equiv g$ on some $C=\{e^{i\theta} ...
2
votes
2answers
77 views

The conditions for Harmonics functions in complex analysis

Let $\sigma(u, v)$ and $\gamma(u, v)$ be harmonic functions on a region $D$ in $\Bbb C$. What are the conditions on $\sigma$ and $\gamma$ such that $\sigma \gamma$ is harmonic on $D$. And I want to ...
1
vote
1answer
91 views

Show whether $\log r$ has a conjugate harmonic function on $\mathbb{C} \setminus \{0\}$

Can someone help me understand this passage in a student-written wiki article? The question is whether $u(x+iy) = \log \sqrt{x^2+y^2}$ has a conjugate harmonic function on $\mathbb{C}\setminus \{0\}$. ...
4
votes
2answers
139 views

Problem of Harmonic function.

If H is a harmonic function on an unit disk; And $H=0$ on $R_1\cup R_2$, here $R_1, R_2$ are radius of $D(0,1)$. The angle between $R_1$ and $ R_2$ is $r\pi$; here $r\in (0,1]$. If $r$ is an ...