All Questions
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When is the Infinite product of continuous functions a continuous function? Assume that the product is convergent.
Is there any theorem about the continuity of an infinite product of continuous real valued functions on compact Housdorff spaces, if the product is convergent?
I mean, for each natural number $n$, let ...
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Prove that the as Zn approached infinity, the Chordal Metric of Zn and zero approached zero.
We've been asked this problem to prove that as $z_n\to \infty$, the chordal metric $\rho(z_n,\infty)\to 0$. Where $\rho(z_1,z_2)= d(z_1',z_2')$ where $z_1'$ and $z_2'$ are points on the Riemann Sphere,...
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Why Does a Function Extends Holomorphically when The Related Sum Converges?
I have seen some cases where to prove a function is holomorphic, it is proven that a sum derived from that function is convergent.
For example, in Newman's Short Proof of the Prime Number Theorem by ...
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3 Exercises on uniform convergence of complex sequences and relation to Weierstrass M-Test
A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 7.19(a),7.20(a),7.21
What are the errors, if any, in the following proofs? I put $\color{...
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$ g\in C^{2}(\Omega)$ are dense in $f\in C(\partial\Omega)$?
Let $\Omega$ a compact set in the complex plane.
How I can show that the space of $ g\in C^{2}(\Omega)$ with a compact support are dense in $f\in C(\partial\Omega)$?
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Show that $\lim_{r \to 0}\int_{0}^{2\pi}f(r e^{i \varphi})d\varphi = 2 \pi f(0) $
I need to show that $\lim_{r \to 0} \int_{0}^{2 \pi}f(r e^{i \varphi})d \varphi = 2 \pi f(0)$ if $f$ is continuous on a neighborhood of $z = 0$.
I was given the following hint: Write $\int_{0}^{2 \...
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