# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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### Intuitive explanation of solution to Laplace's equation on unit disc with boundary conditions (i) $u(1,\theta)=c$, & (ii) $u(1,\theta)=\sin(\theta)$?

I know it can be explicitly shown that for (i) $u(r,\theta)=c$, and for (ii) $u(r,\theta)=r\sin(\theta)$ by separating variables and finding the Fourier coefficients, but is there any mathematical ...
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### $2\log |f(z)|-2\log |z|$ is harmonic function where $f$ is holomorphic in an annulus

Let $U$ denote the open annulus $\{z\in \Bbb C:1<|z|<2\}$ and suppose $f:U\to U$ is a holomorphic bijection. I want to show that $u(z)=2\log |f(z)|-2\log |z|$ is a harmonic function on $U$. ...
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### Critical points of harmonic functions

Let $(M^3,g)$ be a compact Riemannian manifold with boundary $\partial M \neq \emptyset$, and consider a non constant harmonic function $f : M \to \mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$ satisfying ...
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### Are all odd harmonic functions are half-wave symmteric?

I know that the Fourier series expansion of a half-wave symmetric function contains only odd harmonics. Is the reverse true?
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### Maximal operator inequality $P^*$

The maximal operator function $P^*$ is defined in this way: DEF: If $f\in L^{p}(\mathbb{T})$, $P^*f(x)=sup_{0<r<1}|P_{r}*f(x)|$.Where $P_{r}(t)=\sum_{I=-\infty}^{\infty}r^{|i|}\phi_{i}(t)$ is ...
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### Why is $\nabla u (r\cdot 0) = r\nabla u (0)$ true for a harmonic function $u$?

I've stumbled across an older post here trying to solve the same problem the asker of the post had. The solution that was provided stated that for a harmonic function $u$ on $\mathbb{R}^n$ we have ...
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### A continuous function for which Poisson's equation has no C^2 solutions

I am trying to solve exercise 4.9 of Gilbarg and Trudinger, and in particular need to show that for the function $f(x)=\sum_{k=0}^{\infty}\frac{1}{k}\Delta(\eta{P})(2^kx)$ the problem $\Delta{u}=f$ ...
### Why is $\int_{-\infty}^\infty |\text{sinc}(t)|\ dt=\infty$? [duplicate]
Can you help me with that: Why is $\int_{-\infty}^\infty |\text{sinc}(t)|\ dt=\infty$ ? I saw this in Signal Processing course and I can’t understand why this is true. Reference: https://dsp....