Tagged Questions

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Complex differentiability equivalent to linear approximation

Let $G \subset \mathbb C$ be an open set and $f: G \to \mathbb C$ a complex function on $G$. Prove that the function $f$ is complex differentiable at a point $z \in G$ if and only if there exists a ...
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Where on the border of convergence circle series converges and where diverges?

I have power series of $\sum\limits_{k=2}^{\infty} (\ln k)^{\alpha} z^k$. Alpha is a parameter. I've found the radius of convergence. R = 1. If $alpha \geq 0$ then series diverges for z from boundary ...
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Fraction exponential limit in complex plane

Let $n$ be an integer. I need to compute the limit $$\lim_{z\rightarrow 2n\pi i}\dfrac{e^z-1}{z-2n\pi i}$$ for complex number $z$. I think I can't use L'Hospital here since $z$ is complex. How can I ...
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Identify the singularity in this ..

$\frac {\sin^2z}{z^2}$ What kind of singularity is present in this ? My take on this is that, the limit at $z = \infty$ , is $0\,\,$.So, limit is finite.Thus, there should be no essential ...
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Evaluating limit $\lim_{m\to{\infty}}\frac{\sum_{k=1}^m\cot^{2n+1}(\frac{k\pi}{2m+1})}{m^{2n+1}}$

How can I prove the following equality? $$\lim_{m\to{\infty}}\frac{\displaystyle\sum_{k=1}^m\cot^{2n+1}\left(\frac{k\pi}{2m+1}\right)}{m^{2n+1}}=\frac{2^{2n+1}\zeta(2n+1)}{\pi^{2n+1}}$$
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prove this complex function $f$ is complex differentiable

I have a complex function $f(z)=u(z)+iv(z)$ and I know both functions $u$ and $v$ are differentiable at some $z=z_0$. Also I know the following limit exists: ...
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Prove that $\lim_{n \to \infty}n\left(\frac{1+i}{2}\right)^n = 0$

I understand for this proof I must use the principle that I must find $N$, such that for $|u_n - 0| < \epsilon$ I have $n > N$ for $N$ that depends on $\epsilon$. However, I can't seem to get ...
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Let $f(z) = \frac{2z-1}{3z+2}$. Prove that $\lim_{h \to 0}\frac{f(z_0+h)-f(z_0)}{h} = \frac{7}{(3z_0+2)^2}$

I'm having a hard time with the problem stated. I understand this is an epsilon-delta proof. However, When I get to simplifying the numerator, I get $\frac{7h}{9z(z+h)+6(2z+h)+4}$ (this is through ...
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Uniformly bounded sequence of holomorphic functions converges uniformly

Consider an open connected set $\Omega\subset \mathbb{C}$, and $f_n\subset H(\Omega)$. Suppose $f(z)=\lim_{n\to\infty}f_n(z)$ exists and $|f_n(z)|\leq M$ for all $z\in \Omega$. Show that ...
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Show that if $\lim_{z\to\\z_0}f(z)$=0 and there is a positive M such that |g(z)|$\leq$M for all z in some neighborhood of $z_0$, then

$\lim_{z\to\\z_0} f(z)g(z)$=0. This is for complex analysis, and I know how to do this argument for regular epsilon calculus arguments. We want to say since $\lim_{z\to\\z_0} f(z)$=0 then we can take ...
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Understanding the definition of the order of an entire function

Let $f: \mathbb C \to \mathbb C$ be an entire function. The order of $f$ is defined by $$\lambda=\limsup_{r \to \infty} \frac{\log \log M(r)}{\log r},$$ where $$M(r)=\max_{|z|=r} |f(z)| .$$ The ...
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Liminf of a sequence obtained from Poisson Kernel

Fix $\theta_0\in[0,2\pi),$ and let $\{z_n\}\subset\mathbb{D}$ be a sequence converging to $\exp(\iota\theta_0)\in\mathbb{T}.$ Find: ...
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Why does $\lim_{n \to \infty} \sqrt[n]{(-1)^n \cdot n^2 + 1} = 1$?

As the title suggests, I want to know as to why the following function converges to 1 for $n \to \infty$: $$\lim_{n \to \infty} \sqrt[n]{(-1)^n \cdot n^2 + 1} = 1$$ For even $n$'s only $n^2+1$ has ...
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Find $\lim_{z\to 0}|\sin(1/z)/\sin(z)|, z\in \mathbb{C}?$

Find $$\lim_{z\to 0} \ \left|\frac{\sin(1/z)}{\sin(z)} \right|, z\in \mathbb{C}$$ I'm pretty sure that this limit doesn't exist, but i am not able to prove it.
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Asymptotic formula for complex gamma function at $+\infty+i \times y$

I am currently looking for the behaviour of the complex gamma function at real infinity: $\lim_{x \to \infty}\Gamma\left(x+i\times y\right)$ and more particularly for asymptotic formulas for the ...
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What is the limit of erf in $\infty+i\times\infty$?

What is the following limit: $\lim_{x \to \infty, y \to \infty} \mbox{erf}\left(x+iy\right)$ Wolfram alpha seems to give $1$, but here the unique answer seems to tell that $\mbox{erf}$ diverges. ...
Does it make sense to take the following limit? $$\lim_{\phi\to\infty}e^{i \phi}=?$$ And if yes, what does it yield? EDIT: I vaguely remember someone mentioning that this limit gives zero in a ...