1
vote
1answer
25 views

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 ...
0
votes
2answers
61 views

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
0
votes
0answers
20 views

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 ...
2
votes
0answers
54 views

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 ...
0
votes
2answers
59 views

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?
2
votes
1answer
54 views

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 ...
0
votes
2answers
41 views

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 ...
5
votes
2answers
269 views

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 ...
0
votes
1answer
34 views

$-ia(1\pm \sqrt{1-1/a^2})$, $a>0$ 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?
1
vote
2answers
168 views

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 ...
1
vote
1answer
60 views

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 ...
0
votes
1answer
31 views

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 ...
1
vote
1answer
70 views

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 ...
1
vote
1answer
70 views

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 ...
1
vote
0answers
32 views

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 ...
0
votes
0answers
28 views

inequality for Weierstrass sine representation

probably it's another stupid question, but I want to prove that, if $|w|>n$ for some real number $w$, then $$ \left| \frac{\sin \pi w}{\pi w} \prod_{j=1}^n\left( ...
0
votes
0answers
149 views

Quaternion exponential map, rotations and interpolation

A code snippet I need to optimize is performing something peculiar. It seems that it's somehow related to transforming from a frame of reference to another. This is what it does, in mathematical ...
4
votes
2answers
394 views

condition for roots of quartic equation to be purely imaginary

(a) Show that the roots of equation $z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$ where $a_1, a_2, a_3, a_4$ are real constants different from zero, has a pure imaginary root if $a_3^2 + a_1^2 a_4 = ...
0
votes
1answer
367 views

the equation of circle in complex plane passing through three points

I need hints on this question Q.1 Show that the equation of circle passing through three points $z_1, z_2$ and $z_3$ is given by $$\displaystyle \frac{(z-z_1)/(z-z_2)}{(z_3-z_1)/(z_3-z_2)} = ...
5
votes
2answers
167 views

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 ...
1
vote
2answers
61 views

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?
3
votes
2answers
226 views

A question on the Fundamental Theorem of Algebra

I just read about a wikipedia page on Fundamental Theorem of Algebra, and it says "Some proofs of the theorem only prove that any non-constant polynomial with real coefficients has some complex ...
1
vote
0answers
293 views

Unclear on an algebra step in Laurent series of cotangent.

I'm reading through a calculation for $\displaystyle z\pi\cot(\pi z)=\pi iz+\frac{2\pi iz}{e^{2\pi iz}-1}$ which confuses me. It states $$ \begin{align*} z\pi\cot(\pi z) &= \pi iz+\frac{2\pi ...
2
votes
2answers
75 views

Simplifying this exponential equation

I am wondering how does $$\frac{{{e^{zk}}}} {{{z^2} + 1}} = \frac{1} {{2i}}\left( {\frac{{{e^{zk}}}} {{z - i}} - \frac{{{e^{zk}}}} {{z + i}}} \right)?$$ I can see that $z^2 + 1 = (z + i)(z − ...
0
votes
1answer
132 views

Inequality question (with complex numbers)

I was wondering how would you derive/get $|q(z)|\ge R^2-|a|R-|b|>R^2/2 $? Thanks.
2
votes
1answer
115 views

Bounding a Complex Polynomial

Given the complex polynomial $P(z) = z^2 + a_1z + a_0$ and the constraint that $|z| > 1$, I'm trying to show that $|P(z)| \geq |z|^2 - |a_1||z| - |a_0|$. The obvious thing to do here of course is ...
3
votes
2answers
83 views

What was used to establish this equivalence? [Textbook]

To solve for the $\sin^{-1} z$ ($z$ element of $\mathbb{C}$), the book reads that $w = \sin^{-1} z$ when $z = \sin w$ implies: $w = \sin^{-1} z$ when $z = (e^{iw} - e^{-iw})/2i$ implies: ...