# Tagged Questions

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### Sequence of alternating $0$'s and $1$'s in terms of $i$?

How to redefine the function $f(n) = \begin{cases} 1, & \text{if$n$is even} \\ 0, & \text{if$n$is odd} \end{cases}$ in terms of arithmetic operations using ⅈ?
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### Laurent series for $(\sin 2z)/z^3$

I have to find the Laurent series for $(\sin2z)/z^3$ in $|z|>0$, but I really don't know how to start. And I thought that in this area it's a Taylor serie because the singularity isn't in the area, ...
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### Laurent Series Difficulty

Hello all at StackExchange, I'm having some trouble understanding computing the Laurent series for different domains. Here's my approach to finding the Laurent series for $\dfrac{3}{(z+1)(z-2)}$ for ...
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### Convergence of $\sum_{n=2}^{\infty}n^2\left(\frac{1-i}{2+i}\right)^n$

Does this sequence converge/diverge and if so, does it in a (not)absolute way? $$\sum_{n=2}^{\infty}n^2\left(\frac{1-i}{2+i}\right)^n$$
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### Convergence of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ If $z=re^{i\theta}=x+iy$, $$F(z)=\sum_{n=1}^\infty r^na_ne^{in\theta}$$ is a harmonic ...
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### Can't match boundary conditions on a perturbation series solution to a non-linear ODE?

I'm trying to generate a naive perturbation series solution (with all associated secular terms included) to the Rayleigh equation: \frac{d^2y}{dt^2} + y = \epsilon ...
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### How to evaluate a zero of the Riemann zeta function?

Here is a super naive question from a physicist: Given the zeros of the Riemann zeta function, $\zeta(s) = \sum_{n=1}^\infty n^{-s}$, how do I actually evaluate them? On this web page I ...
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### Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$

I'm going to teach a preparation course for the complex analysis qualifying exam from my university (which basically consists of me doing some problems from past exams) and I'm trying to solve some ...
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### Prove a Trigonometric Series

Question: $$\cot^{2}\frac{\pi }{2m+1}+\cot^{2}\frac{2\pi }{2m+1}+\cdots+\cot^{2}\frac{m\pi }{2m+1}=\frac{m(2m-1)}{3}$$ $m$ is a positive integer. Attempt: I started by showing that ...
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### Why does $\sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )+{1\over2}-{1\over3} = \gamma$?

How could one prove that $$x = \sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )$$ is such that $x+{1\over2}-{1\over3} = \gamma$ ? I am having problems just calculating ...
Let $w = e^{i\frac{2\pi}5}$. I would like to evaluate $$w^0 + w^1 + w^2 + w^3 +...+ w^{49}$$ Can anyone please give me an idea how to evaluate the expression? Thanks in advance
### When are we (not) allowed to replace $x$ by $ix$?
It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...