Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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8
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2answers
982 views

Total ordering on complex numbers

Show that there doesn't exist a relation $\succ$ between complex numbers such that (i) For any two complex numbers $z,w$, one and only one of the following is true: $z\succ w,w\succ z,$ or ...
8
votes
2answers
281 views

The roots of the derivative $P'(z)$ of the polynomial $P(z)\in\mathbb C[x]$ lie in the convex hull of the set of roots of $P(z)$.

Assume $S=\{z_1,z_2,...,z_k\}, z_i\in \mathbb C$$, C(S)$ and define $$C(S):=\{z=a_1z_1+a_2z_2+...+a_kz_k | a_i\ge0 ,a_1+a_2+...+a_k=1\}$$ where $$A:=\{z\in \mathbb C:f(z)=0 ...
8
votes
3answers
148 views

What is the right treatment for $0^i$?

I need to calculate a limit of a complex expression (had it in a physics research) that contains a term $(r-b)^p$ for $r\rightarrow b+$ where $r,b$ are reals, and $p$ is complex, let's suppose for ...
8
votes
1answer
681 views

How to choose a proper contour for a contour integral?

When analyzing real integrals with contour integrals, how does one choose a proper contour integral? Many cases can be solved by integrating around the top half of a circle with radius of infinity ...
8
votes
2answers
193 views

Expressing a complex function in terms of z

Use the Cauchy-Riemann equations to determine all differentiable functions that satisfy $Re(f(z))=xy$ I think I know how to do this problem. If we let $z=x+iy$, then $f(z)=u(x,y)+iv(x,y)$. We ...
8
votes
2answers
123 views

In a complex vector space, $\langle Tx,x \rangle=0 \implies T = 0$

Suppose $T$ is a linear operator on a complex inner product space. Is it a theorem that if $\langle Tx,x\rangle=0$ for all $x$ in the space then $T=0$. The theorem fails in the real case, as seen for ...
8
votes
2answers
344 views

If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear

If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear. I really can't seem to get anywhere on this problem, ...
8
votes
3answers
194 views

Proving that the limit of a sequence is $> 0$

Let $u$ be the complex sequence defined as follows : $u_0=i$ and $ \forall n \in \mathbb N, u_{n+1}=u_n + \frac {n+1-u_n}{|n+1-u_n|} $ . Consider $w_n$ defined by $\forall n \in \mathbb ...
8
votes
0answers
54 views

An Inequality with complex numbers and $1/\pi$ [duplicate]

Let $\displaystyle \{z_1,z_2, \ldots, z_n\}$ be $n$ complex numbers such that: $\displaystyle \sum\limits_{k=1}^n|z_k| = 1$ Then we have to show that, there is a subset $S$ of $\{1,2,\ldots,n\}$, ...
8
votes
1answer
240 views

What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?

I'm starting to come around to an understanding of hypercomplex numbers, and I'm particularly fascinated by the fact that certain algebraic properties are broken as we move through each of the $2^n$ ...
7
votes
8answers
1k views

What's the thing with $\sqrt{-1} = i$

What's the thing with $\sqrt{-1} = i$? Do they really teach this in the US? It makes very little sense, because $-i$ is also a square root of $-1$, and the choice of which root to label as $i$ is ...
7
votes
4answers
761 views

Adding powers of $i$

I've been struggling with figuring out how to add powers of $i$. For example, the result of $i^3 + i^4 + i^5$ is $1$. But how do I get the result of $i^3 + i^4 + ... + i^{50}$? Writing it all down ...
7
votes
7answers
1k views

How to solve $e^{ix} = i$

I am taking an on-line course and the following homework problem was posed: $$e^{ix} = i$$ I have no idea how to solve this problem. I have never dealt with solving equations that have imaginary ...
7
votes
3answers
226 views

$ \exists a, b \in \mathbb{Z} $ such that $ a^2 + b^2 = 5^k $

I saw this problem recently and found an elegant solution to it, and was curious to see if anybody would think of something else. Nice solutions to nice problems are fun to see! Problem: Prove ...
7
votes
9answers
1k views

Positive and negative complex numbers?

Can there be such a thing as positive and negative complex numbers? Why or why not? What about positive or negative imaginary numbers? It seems very tempting to say $+5i$ is a positive number ...
7
votes
2answers
3k views

What does $\mathrm{Re}(x)$ mean?

I see this all the time in Mathematica output as well as in text, such as near the top of the Wikipedia Beta function page.
7
votes
3answers
644 views

What is the motivation for complex conjugation?

I have been dealing with complex numbers for few years now. But when I've tried to think about the motivation behind complex conjugation, I was not sure. Let me write what I am working with. For a ...
7
votes
8answers
738 views

what is$ \sqrt{8i}$

Very simple question with an answer that I cannot understand: I have $\sqrt{8i}$, which, I suppose, is the same as $\sqrt{\sqrt{-64}}$. How come that $2+2i$ is the same as $\sqrt{8i}$? My ...
7
votes
2answers
405 views

How does one find $z\in \mathbb{C}$ such that $\sin z=100?$

I am self-studying Complex Analysis and I am suppose to find $z\in \mathbb{C}$ such that $\sin z=100.$ I know that $$\sin z=\sin x \cosh y+i\cos x\sinh y$$ So I must have $\sin x \cosh y=100.$ I ...
7
votes
6answers
366 views

Prove if $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} $ is a real number

If $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} \in \Bbb R $ i found one link that had a similar problem. Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} ...
7
votes
3answers
435 views

Why are complex numbers considered to be numbers?

I've had Dave's Short Course on Complex Numbers on the web since 1999, and I'd like to add a page on why complex numbers are (or should be) considered to be numbers. I'm frequently asked that ...
7
votes
6answers
233 views

What is $(-1)^{\frac{2}{3}}$?

Following from this question, I came up with another interesting question: What is $(-1)^{\frac{2}{3}}$? Wolfram alpha says it equals to some weird complex number (-0.5 +0.866... i), but when I try ...
7
votes
2answers
1k views

Limit of complex function

Im trying to find the limit of: $$ \frac{\operatorname{Re}(z) \operatorname{Im}(z)}{z^2}$$ as z tends to zero.
7
votes
2answers
351 views

Why is $\left(e^{2\pi i}\right)^i \neq e^{-2 \pi}$?

Here's my (obviously flawed) proof that $1=e^{-2 \pi}$: $$ 1^i=1\\ e^{2 \pi i} = 1\\ \left(e^{2\pi i}\right)^i = 1^i\\ e^{-2 \pi} = 1 $$ What's the issue? I understand that exponentiation is not ...
7
votes
4answers
2k views

Non-integer powers of negative numbers

Roots behave strangely over complex numbers. Given this, how do non-integer powers behave over negative numbers? More specifically: Can we define fractional powers such as $(-2)^{-1.5}$? Can we ...
7
votes
4answers
116 views

Deriving an expression for $\cos^4 x + \sin^4 x$

Derive the identity $\cos^4 x + \sin^4 x=\frac{1}{4} \cos (4x) +\frac{3}{4}$ I know $e^{i4x}=\cos (4x) + i \sin (4x)=(\cos x +i \sin x)^4$. Then I use the binomial theorem to expand this fourth ...
7
votes
3answers
506 views

A difficult inequality involving complex numbers

Suppose that $z_1,\ldots,z_n$ are complex numbers with the property that there is some constant $C$ such that $$\big|z_1^r+\cdots+z_n^r\big|\leqslant C$$ for all integers $r\geqslant0$. Show that ...
7
votes
2answers
159 views

Invariant under transformation $i\mapsto -i$ implies real?

When one has an expression in terms of $i$, one can send $i$ to $-i$ and, if the expression remains unchanged, one can conclude that the expression is, in fact, real. Analogous statements hold for ...
7
votes
3answers
159 views

How to visualize $f(x) = (-2)^x$

Background I teach Algebra and second year Algebra to middle school students. We are currently studying Exponential, Power, and Logarithmic functions. We study exponential functions (of the form ...
7
votes
3answers
629 views

How do I completely solve the equation $z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$ where there is a root with the real part of $1$.

I would please like some help with solving the following equation: $$z^4 - 2z^3 + 9z^2 - 14z + 14 = 0$$ All I know about the equation is that there is a root with the real part of $1$. My approach ...
7
votes
5answers
223 views

How to find the center of the circle that contains three given complex numbers?

Suppose $\alpha_1, \alpha_2, \alpha_3 $ are complex numbers which are not collinear. Is it possible to use some geometry to find the center of the circle that contains $\alpha_1, \alpha_2, \alpha_3 $ ...
7
votes
3answers
74 views

Set Theoretic Definition of Complex Numbers: How to Distinguish $\mathbb{C}$ from $\mathbb{R}^2$?

I have spent some time looking for a rigorous, set-theoretic definition of the complex numbers. I have read the book Elements of Set Theory by Herbert Enderton (1977) which does an excellent job of ...
7
votes
2answers
302 views

How to calculate $i^i$ [duplicate]

I've been struggling with this problem, actually I was doing a program in python and did 1j ** 1j(complex numbers) (In python ...
7
votes
2answers
303 views

How to show that $\overline{zw}=\overline{z}\,\overline{w}$?

I thought about first multiplying two complex which aren't in the conjugate form: $$zw=a c+i a d+i b c-b d$$ Then multiply two complex conjugates: $$\overline{z}\,\overline{w}=a c\color{red}{-}i a ...
7
votes
1answer
407 views

Is every complex number the root of a polynomial? (Converse to fundamental theorem of algebra.)

For every polynomial with complex coefficients, the fundamental theorem of algebra guarantees the existence of complex numbers which happen to be roots of it. But is this everything? i.e. is the ...
7
votes
2answers
842 views

Why isn't $\log(-1)=i\pi$?

Reading http://people.math.gatech.edu/~cain/winter99/ch3.pdf, $\log(z)$ is defined as $=\ln|z|+i\arg(z)$. Looking on the Wessel plane, isn't $\arg(-1)=\pi$ (more generally $\pi \pm 2 \pi n$)? And ...
7
votes
2answers
351 views

Why is $x^2$ +$ y^2$ = 1, where $x$ and $ y$ are complex numbers, a sphere?

I've heard $x^2 + y^2$ = 1, where $x$, $y$ are complex numbers, is supposed to be a sphere with two points removed, or also a cylinder. The problem is I've been trying to wrap my head around this for ...
7
votes
3answers
1k views

How does this equality on vertices in the complex plane imply they are vertices of an equilateral triangle?

I've read that if the complex numbers $a_1$, $a_2$ and $a_3$ are the vertices of a triangle in the complex plane such that $$ a_1^2+a_2^2+a_3^2=a_1a_2+a_2a_3+a_1a_3 $$ then the vertices are actually ...
7
votes
2answers
284 views

Is it ever $i$ time?

I am asking this question as a response to reading two different questions: Is it ever Pi time? and Are complex number real? So I ask, is it ever $i$ time? Could we arbitrarily define time as ...
7
votes
3answers
186 views

What type of numbers are roots of $x^{2} = -1$ themselves?

Are the roots $i, -i$ themselves irrational numbers or complex numbers or left auxiliary so that undefined?
7
votes
1answer
505 views

To calculate residue of the function $f(z) = \frac{z^2 + \sin z}{\cos z - 1}$.

I was trying to find the residue of the function $$f(z) = \frac{z^2 + \sin z}{\cos z - 1}.$$ Here is the my attempt: The given function has a pole of order two at $z = 2n\pi$. So, we use the ...
7
votes
2answers
2k views

Prove that the zeros of an analytic function are finite and isolated

Let us assume that the zeros of $f = \{Z_1,\ldots,Z_n,a\}$ are infinite and converge towards $a$. The book which I am reading says that any neighborhood of $a$ will contain infinite zeros. Since $f$ ...
7
votes
3answers
132 views

Connectedness of $\lbrace z\in\mathbb{C} : |z^2+az+b|<r\rbrace$

What are the values of $r$ for which the set $$\lbrace z\in\mathbb{C} : |z^2+az+b|<r\rbrace$$ is connected ? Here $a,b\in\mathbb{C}$ and $r\in\mathbb{R}$.
7
votes
3answers
2k views

Drawing $z^4 +16 = 0$

I need to draw $z^4 +16 = 0$ on the complex numbers plane. By solving $z^4 +16 = 0$ I get: $z = 2 (-1)^{3/4}$ or $z = -2 (-1)^{3/4}$ or $z = -2 (-1)^{1/4}$ or $z = 2 (-1)^{1/4}$ However, the ...
7
votes
2answers
110 views

Product of $ |z^k - 1| $

Problem: Prove the following identity about the product involving the nth roots of unity: $$ \prod_{k=1}^{N-1}|z^k-1| = N $$ where $ z^k $ is the primitive nth root of unity. Attempt: $$ ...
7
votes
2answers
92 views

How to visualize $\mathbb{C}^2$?

In a homework question I had to do, the rotational matrix $A = \begin{pmatrix} 0&-1\\1&0 \end{pmatrix}$ was given. Its eigenvalues in $\mathbb{C}$ are $i$ and $-i$. The set of all eigenvectors ...
7
votes
1answer
91 views

Does $\exp(2ir\pi)$ equal $1$? What's wrong?

Since $e^{ix}=\cos x+i\sin x$, thus $e^{2\pi i}=\cos2\pi+i\sin2\pi=1$ Now I take arbitrary real number $r$ then $e^{i2\pi r}=(e^{i2\pi})^r=1^r=1?$ But this cannot be true since $\cos2\pi ...
7
votes
1answer
94 views

Inequality relating coefficients and roots of a complex polynomial

While going through some olympiad handouts I stumbled upon a problem related to an upper bound for the Mahler measure, which stated that Given a polynomial $f(x) = x^n + a_{n-1}x^{n-1} + \dots + a_0 ...
7
votes
3answers
129 views

Notation For Complex Numbers

I have seen many different notations used for complex numbers. Does it make a difference which notation is used, or is any one notation more standard than another? I see a+bi at ...
7
votes
3answers
212 views

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 ...