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# Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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49 views

### Bound on gradient positive harmonic function

My question is essentially the same as An inequality concerning an harmonic function , however I did not find the answer given satisfactory. To restate it, I would like to solve the following: Let $h$...
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### Mapping a curve-sided quadrilateral to a rectangle

I am currently investigating different ways of solving the Laplace equation $$\frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial z^2} = 0$$ numerically on the domain $\Omega$ shown as ...
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### Analytic continuation of harmonic series

Is there an accepted analytic continuation of $\sum_{n=1}^m \frac{1}{n}$? Even a continuation to positive reals would be of interested, though negative and complex arguments would also be interesting. ...
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### Harmonic functions in the half-plane

Denote by $\mathbb{H}$ the upper half-plane $$\mathbb{H} := \left\{ x \in \mathbb{R}^n : x_n > 0\right\}.$$ Suppose that $u \in C^2(\mathbb{H}) \cap C(\bar{\mathbb{H}})$ is a bounded harmonic ...
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### For which subsets does the harmonic to analytic connection hold?

I'm a bit confused on the choice of sets that authors choose and why. For example : "Any harmonic function $u$ on an open subset $\Omega$ of $R^2$ is locally the real part of a holomorphic function." ...
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### Analytic functions having harmonic real and imaginary parts.

I've bee set the following question in a homework assignment for my complex analysis class, but have literally no idea what it means by sufficiently regular. Let $f : \mathbb{C} \to \mathbb{C}$ be an ...
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### Why is it that for a hamonic $u$, $\int_{\gamma}*du = 0$ for any cycle $\gamma$ then $u$ has a harmonic function there?

Let $u$ be a harmonic function on a connected open set. If $\int_{\gamma}*du = 0$ for any cycle $\gamma$ then $u$ has a harmonic function. This question arises from an answer to this post Please do ...