# Tagged Questions

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### 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 ...
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### 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 ...
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### $\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$ right? Fundamental solution to Laplace in $\mathbb{R}^3$

OK, I can't figure out why I can't get this right: $$\nabla_y \left( \frac{1}{|x-y|} \right)= \frac{x-y}{|x-y|^3},$$ right? I've checked the calculation several times, although this student-written ...
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### Prove that $u \circ f$ is plurisubharmonic on $\Omega_1$

I'm trying to show that the theorem in my book: Let $f: \Omega_1 \to \Omega_2$ be a holomorphic map between open sub - sets $\Omega_1, \Omega_2$ of $\Bbb C^n$. If $u$ is plurisubharmonic on ...
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### Show that $\widetilde{u} \in PSH(\Omega)$

EDITED I'm reading the book: (Oxford science publications._ London Mathematical Society monographs, new ser., no. 6) Maciej Klimek -Pluripotential theory -OUP (1992). I don't understand ...
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There are 3 theorems in my textbook (My textbook doesn't have a solution): Why do we have ${\color{Red} (I)}$. $1/$ Let $\Omega$ be an open subset of $\mathbb{C}$. If $f : \Omega \to ... 2answers 85 views ### Prove that$\log(\max(\left | z_1 \right |,\left | z_2 \right |,\ldots,\left | z_n \right |)) \in MPSH(\Omega)$This's an example: For$u(z_1,z_1,\ldots,z_n)=\log(\max(\left | z_1 \right |,\left | z_2 \right |,\ldots,\left | z_n \right |))$, where$z=(z_1,z_1,\ldots,z_n) \in \Omega=\mathbb{C}^n \setminus\{0\} ...
A simple-layer potential is defined as $$\Psi(M)=\iint_{S}\dfrac{\sigma(N)}{R(M,N)}dS(N)$$ where $S$ denotes a flat region in the plane $z=0$; the coordinates of $M$ and $N$ are $(\rho,\phi,z)$ and ...