# Tagged Questions

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### Making $-{{\pi i}\over n} e^{\alpha i}({{1 - e^{2 n \alpha i}\over{1-e^{2 \alpha i}}}})={\pi \over {n sin(\alpha)}}$; $\alpha={{2m+1}\over{2n}} \pi$

As part of a (much) longer problem in complex analysis, I need to show that the equality mentioned in the title makes sense, but I can't seem to find the right algebra tricks to get from point A to ...
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### How could I prove this trigonometric identity?

Show that: $$\left(\frac{1+\tan \theta}{1 - \tan \theta}\right)^n = \frac{1+i\tan n\theta}{1-i\tan n\theta}$$ Original image: http://i.stack.imgur.com/q8Yxj.jpg
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### Show that if R$_{m,n}$ and r$_{m,n}$ agree at m+n+1 distinct points, then R$_{m,n}$ = r$_{m,n}$

Let R$_{m,n}$ = P(z)/Q(z) and $r_{m,n} = p(z)/q(z)$ be two rational functions each with numerator degree m and denominator degree n. Show that if $R_{m,n}$ and $r_{m,n}$ agree at $m+n+1$ distinct ...
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### Proof that $p(z)^2=a^2$ always has a nonreal solution.

Let $p(z)$ be a nonconstant integer polynomial of degree $n$ such that $p(0)=0$ and let $a$ be a nonzero real number. It seems that $$p(z)^2=a^2$$ Always has a nonreal solution (in $z$) if ...
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### On finding the zeros of a polynomial

What is the zero (real) of the polynomial $$x^{k+1}-2x^{k}+1=0$$ If there is such, how can I find it or what method can I use?
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### Simple expression for this sum?

Is there any simple expression for the sum: $$S = \sum_{n = 0}^{N-1} \frac{1}{a + e^{2 \pi i n / N}}$$ where $N$ is a positive integer and $a$ is some real number. It feels to me like there ...
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### Fractional exponents and when they commute.

In any elementary algebra class students are taught that if $a$ and $b$ are coprime, $x^{a/b}=(\sqrt[b]{x})^{a}$ or $\sqrt[b]{x^{a}}$. But only after teaching this lesson I realized this isn't always ...
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### Definition of a logarithm

My question is as follows: Is the below a useful elementary way of dealing with negative arguments? If not, what is a better (elementary or not) way of dealing with negative arguments of the ...
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### $-ia(1\pm \sqrt{1-1/a^2})$, $a>0$ inside unit circle?

Given $a>0$ I would like to know whether: $\alpha=-ia(1+ \sqrt{1-1/a^2})$ and $\beta =-ia(1- \sqrt{1-1/a^2})$ are inside the unit circle. How can I check that?
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### Find all the values of $(1+i)^{(1-i)}$

The question says to find all the values of $(1+i)^{(1-i)}$ I have trouble figuring out firstly, exactly what values are being looked for. I can toy around with the equation a bit to try to make it ...
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### Complex polynomial identity with norm condition

In this question, the following was shown: If $R(z)=\dfrac{P(z)}{Q(z)}$, where $P,Q$ are polynomials in a complex variable $z$, satisfies the condition that $|R(z)|=1$ whenever $|z|=1$, then the ...
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### Use of polynomial with reciprocals

Let $P(z), Q(z)$ be polynomials, and define $R(z)=\dfrac{P(z)}{Q(z)}$, where $P(z)$ and $Q(z)$ have no common factors. Greater unity is achieved if we let the variable $z$ as well as the values ...
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### Definition of complex number

In many situations (problems as well as solutions) I encounter the complex number $i$ which many times is defined as $i^2=-1$ instead of $i=\sqrt{-1}$, since it is "more preferred". Does anyone know ...
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### Showing $\max\limits_{|z|=r}|p(z)| \ge |a_n|r^n$, without Cauchy integral formula.

Let $p(z) = a_n z^n + a_{n-1}z^{n-1} + \cdots + a_0$. My question is: Is there an elementary way to show that for all $r > 0$ $$\max \limits _{|z| = r} |p(z)| \ge |a_n|r^n$$ without using ...
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### using Paley-Wiener to get support and then estimate inf sup

I try to make a complete question out of my previous ones. Define the function $$\tilde{f}_n(\omega)=\frac1{\sqrt{2\pi}} \frac{\sin R\omega/2}{R\omega/2} s_n(R\omega/2\pi),$$ where (using Weierstrass ...
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### how to show that $\lim_{z \to 0}z^z$ does not exist?

What makes $0^0$ indeterminate. Here is a video by numberphile that claims that $z^z$ does not exist as $z \to 0$ where $z \in \mathbb C$. I tried tried $\lim_{x \to 0}(x+ix)^{(x+ix)}$ and replaced ...
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### Is there a similar function in complex number system corresponding to logarithim in real number system?

i notice that there are $e^{i\theta}$ in math,so is there a similar function in complex number system corresponding to logarithim in real number system?