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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0
votes
1answer
41 views

Inequality of complex numbers involving modules

Let $z \in \Bbb C$ such that $|z| \ge 1$. Show that $$\sqrt[6] \frac {|2z-1|^2} {7} \ge \sqrt[7] \frac {|z-1|^2} {3}.$$ My try: I wrote $|z|^2$ as $z\times \bar z$, but I didn't get to any result. Can ...
0
votes
2answers
19 views

Finding Possible Meromorphic functions on $\mathbb{C}$

I am trying to find all meromorphic functions on $\mathbb{C}$ such that: $$ \mid f(z) \mid \leq (\frac{3 \mid z \mid}{\mid z + 1 \mid})^{3/2}$$ Can I express the functions as: $$f(z) = ...
2
votes
0answers
39 views

Multiplying two radicals with negatives, simple algebra? [duplicate]

Evaluate $$ \sqrt{-9}\sqrt{-4} $$ Now, I am told that $\sqrt{a}\sqrt{b}=\sqrt{ab}$, so I should be able to simply write $$ \sqrt{-9}\sqrt{-4} = \sqrt{(-9)(-4)}=\sqrt{36} = 6 $$ However, I am also told ...
0
votes
0answers
15 views

Proof of an inequality in C ,(2)

Let $n\ge 2$is a integer,$z_{1},z_{2},\cdots,z_{n}$ are $n$ complex numbers Prove that $$\sum_{k=1}^{n}|1+z_{k}|+\dfrac{1}{n-1}\sum_{1\le i<j\le n}|1+z_{i}z_{j}|\ge\sum_{k=1}^{n}|z_{k}|$$ for ...
2
votes
1answer
44 views

Show that $\cot \frac{\pi}{2m}\cot \frac{2\pi}{2m}\cot \frac{3\pi}{2m}…\cot \frac{(m-1)\pi}{2m}=1$

Prove: $$\cot \frac{\pi}{2m}\cot \frac{2\pi}{2m}\cot \frac{3\pi}{2m}...\cot \frac{(m-1)\pi}{2m}=1$$ This is a roots of unity problem. I managed to show a similar example for $\cos$ by the ...
0
votes
0answers
18 views

Prove that the point will go 3 times around ellipse

I'd like to prove that if a point $z$ goes once around ellipse with focus $2,-2$ then point $z^3-3z$ goes 3 times around some ellipse with the same focus. I was thinking (since ellipse is a set of ...
0
votes
0answers
10 views

How can I separate the real and imaginary parts of this Ikeda mapping?

How might I separate the real and imaginary parts of this mapping? So I can plot and compare real curves. $E_{n+1} = A+BE_ne^{i\left|E_n\right|^2}$ where $E_n = x_n+iy_n$.
0
votes
3answers
45 views

Show that any conjugate pair of complex numbers (with non-zero imaginary part) cannot be the spectrum of any 2x2 matrix with real, nonnegative entries [duplicate]

My professor showed me this in her office today but I didn't like her method and wanted to use another method. So, I computed the characteristic polynomial of some arbitrary $2 \times 2$ matrix ...
2
votes
2answers
156 views

Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $z_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the ...
2
votes
1answer
24 views

Quick Question - Complex roots of polynomials?

I was asked to find solutions to $z^3 = 1$ and give my answer in Cartesian form. I got $1, -1/2 \pm i\sqrt{3}/2$ (b) Hence solve the equation $(z+i)^3 = (2z-i)^3$ Little help on this one? Any help ...
0
votes
0answers
12 views

Complex Number Powers of Coprime Rational Powers

I'm trying to figure out $z^{p/q}$ where $p,q$ are coprime. Suppose I want to find $z^{2/7}$ where $z=128$. I can rewrite $z=128e^{0}$ Now I know that the $z^{1/7}$ roots are $2e^{k2\pi i/7}$ for ...
-1
votes
1answer
40 views

Algebra Roots (Cubic/Complex)

Show that the equation $3z^3+(2-3ai)z^2+(6+2bi)z+4=0$ (where both $a$ and $b$ are real numbers) has exactly one real root, and find this root. I've dealt with quadratics in this form but never with ...
1
vote
1answer
3k views

Straight Line Equation in Complex Plane

I'm confused about the straight line equation in complex plane: how does $0 = Re((m+i)z + b)$ come from $y = mx + b$? I mean when I see $y = mx + b$, I can draw a graph in my mind, but when I see $0 ...
0
votes
1answer
63 views

How to express $\sin \sqrt{a-ib} \sin \sqrt{a+ib}$ without imaginary unit?

I got this kind of expression as a value of an infinite product: $$\prod_{k=1}^{\infty} \left(1-\frac{A}{k^2}+\frac{B}{k^4} \right)$$ It's easy to see how it can be factored into a product of two ...
0
votes
0answers
15 views

limits and convergence of sequences complex

For the following sequence discuss its limits and whether the convergence is uniform, in the region $\alpha \leq \left | z \right |\leq \beta $, for finite $\alpha$,$\beta >0$. $$\left \{ ...
0
votes
0answers
11 views

little agebra help, complex numbers

Can someone please explain this to me, I dont understand how to go from $ [ \psi-1+r( e^{2i\omega} - 4e^{-\omega}+6-4e^{-i\omega} + e^{-2i\omega})] A\psi6{n}e^{i\omega j} $ to this line here. $ ...
0
votes
0answers
34 views

Inequality with complex numbers involving 6-th and 7-th root [on hold]

Let $z \in \Bbb C$ such that $|z| \ge 1$. Show that $$\sqrt[6] \frac {|2z-1|^2} {7} \ge \sqrt[7] \frac {|z-1|^2} {3}.$$
1
vote
1answer
37 views

Singular points of $\displaystyle \sin \left( \frac{1}{\cos\frac{1}{z}}\right)$

Specifically, $\displaystyle f(z) = \sin \left( \frac{1}{\displaystyle \cos \frac{1}{z}} \right)$ has singular points at $z = \displaystyle \frac{2}{\pi + 2\pi k}$, among others. Now, I am trying to ...
0
votes
2answers
51 views

Complex numbers ; $(1+i)^n=(1-i)^n$, Find $n$. [on hold]

Find $n$, if $$(1+i)^n=(1-i)^n.$$ I don't really remember complex numbers, but the problem is very easy I think.
0
votes
1answer
16 views

Discrepancy between text's answer and mine: singular points of $\cot\left(\frac{1}{z}\right) - \frac{1}{z}$

The points $\frac{1}{k\pi}$, where $k \in \mathbb{Z}$ are all singularities of the function $f(z) = \cot\left(\frac{1}{z} \right) - \frac{1}{z}$. My textbook seems to think that they are simple ...
0
votes
1answer
27 views
-4
votes
2answers
81 views

Trying to derive a contradiction with this simple inequality, [on hold]

I am stuck at $$(a+d)^2 - 4(ad-bc) < 0$$ $$\implies (a+d)^2<4(ad-bc)$$ $$\implies (a+d)<2\sqrt{ad-bc}$$ where $a,b,c,d \ge 0$. Is there a contradiction to derive here? Also, the square ...
3
votes
1answer
54 views

Singular point of $f(z)$ also a singular point of $1/f(z)$ and $f^{2}(z)$

Suppose $z_{0} \in \mathbb{C}$ is an isolated singular point of the function $f$ of a given type (removable, pole of order $N$, essential). I need to show that $z_{0}$ is an isolated singular point of ...
4
votes
0answers
194 views
+500

Bounding a sum involving a $\Re((z\zeta)^N)$ term

This is a follow up to this question. Any help would be very much appreciated. Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$. Let ...
-3
votes
0answers
55 views

Complex Analysis Questions Compilation [on hold]

All of my questions are in relation to Gamelin's Complex Analysis. How was the parametric form of the line from the North Pole on the unit sphere through a point P come to be? It is $$ N + t (P-N) ...
-1
votes
0answers
17 views

bessel function integration [on hold]

$$ I(u,v_p) = \int_{0}^{{\it v_p}}\! \left| \int_{0}^{1}\!{{\rm e}^{iu{\rho}^ {2}}}{{\rm J}_{0}\left(v\rho\right)}\rho\,{\rm d}\rho \right| ^{2}v\,{\rm d}v $$ Suppose: P=1 and J is the ...
1
vote
0answers
19 views

Isolated Singular Points

I would like to check and see if my reasoning for this question is correct: Find the singular points of the function, and classify them if they are isolated singular points. Also, evaluate if ...
1
vote
1answer
11 views

How do the coefficients in the linear combination of cosines impact the number of local minima of the sum?

Consider the following function: $$f(\theta) = r_0 + r_1 \cos(\theta + \phi_1) + r_2 \cos(2\theta + \phi_2)$$ where $\theta$ is an angle between 0 and $2\pi$. For all $0\leq k\leq 2$ we have $r_k\geq ...
4
votes
4answers
147 views

How can a complex exponential represent a real world quantity?

Equations containing complex exponentials are mysterious. The complex exponential merely embodies a complex number but in a more compact form where doing maths is easier. Right? If this complex ...
0
votes
1answer
37 views

Complex exponential with 2 pi

I wonder why is it wrong to do the following: $e^{i2\pi x}=(e^{i2\pi})^x=1^x=1$ for a real $x$ but not for an integer $x$
6
votes
1answer
155 views
+50

$m+ni+k\lambda,\,\Re(\lambda),\Im(\lambda)\notin \mathbb{Q}$ is dense in $\mathbb{C}$!

As said in the comments below, it's needed to suppose $\{1,\Re(\lambda),\Im(\lambda)\}$ linearly independent over $\mathbb{Q}$, otherwise the result is false, according to Christian's example. ...
3
votes
0answers
26 views

Find and classify singular points of $\cot\left(\frac{1}{z}\right)$

I need to find and classify singular points (i.e., decide whether the point is removable, a pole of order $N$, essential, or not an isolated singular point), including infinity, of ...
4
votes
0answers
78 views

Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
1
vote
1answer
383 views

Limit and complex numbers to infinity

The following sentence $$\lim_{z \rightarrow 0} f(z) = 100i\quad \Leftrightarrow \quad\lim_{z\rightarrow \infty} f(1/z) = 100i.$$ Is this true?
0
votes
0answers
15 views

Prove that the eigenvalues of a skew-symmetric matrix are purely imaginary [duplicate]

Proof idea: $A$ is a skew symmetric matrix. $A$ is similar to $A^t$ because every matrix is similar to it's transpose. $$A^t = -A $$ $A$ is similar to $-A$. Let $P_{(\lambda)}$ be the characteristic ...
0
votes
0answers
9 views

how to map sequel arcs forming a curve to a unit circle

I would like to map a sequel of arcs with different radii and origins that form a closed curve on to the unit circle. Each contact point of two sequel arcs as well as their origins lie on the same ...
0
votes
0answers
22 views

Linear factors of minimal polynomial dividing $x^r$ - 1

I have a monic minimal polynomial $m(x)$ that divides $x^r - 1$. Apparently $m(x)$ has distinct linear factors over the complex numbers $\mathbb C[x]$. I understand this part, since $\mathbb C[x]$ is ...
0
votes
0answers
13 views

Find $\frac{d}{dt}[\bar{f(\gamma(t))}]$ in the context of of finding $\frac{d}{dt}[|f(\gamma(t)|^2]$

I am trying to prove this exact problem, but more rigorously and without referencing analyticity: Ahlfors complex integration. I think the way to proceed is to try and find ...
38
votes
8answers
88k views

How do I get the square root of a complex number?

If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?
2
votes
0answers
34 views

$f$ has pole of order $m$ and $g$ has a pole of order $n$ at $z_{0}$, show $f+g$ has isolated singular point there

I am faced with the following problem: Suppose $f(z)$ and $g(z)$ have poles of order $m$ and $n$ respectively, at a point $z_{0} \in \mathbb{C}$ with $m \neq n.$ Show that $z_{0}$ is an isolated ...
-2
votes
1answer
32 views

Help with complex numbers question [on hold]

Could someone please solve and explain this question about complex numbers? I'm having a lot of difficult understanding parts (b)(iii) and (b)(iv) especially. I understand what they are asking but ...
27
votes
7answers
3k views

Can a complex number be prime?

I've been pondering over this question since a very long time. If a complex number can be prime then which parts of the complex number needs to be prime for the whole complex number to be prime.
6
votes
0answers
220 views

Complex prime numbers

I was just learning about how, if you multiply any two complex numbers together, you get a new complex number. With that in mind, is there such a thing as complex numbers that cannot be factored, or ...
2
votes
1answer
52 views

$n$ complex numbers with modulus $1$

The problem: Let $z_1$,$z_2$,...$z_n$ $(n \geq 3)$ be complex numbers such that $\left| z_1 \right|=\left| z_2 \right|=\ldots=\left| z_n \right|=1$. Then show that the following statements are ...
1
vote
2answers
52 views

Complex Analysis: How isolated singular points behave

I am working on the following question: Suppose $z_0 \in \mathbb{C}$ is an isolated singular point of the function f of a given type (removable, pole of order N, essential). Show that $z_0$ is an ...
1
vote
2answers
31 views

Prove a doubly periodic entire analytic function in complex plane is a constant [duplicate]

I got stuck on this problem. So I really appreciate if anyone can give me some hint to move on. Thanks a lot. Prove that an entire analytic function $f:\mathbb{C} \rightarrow \mathbb{C}$ is a ...
0
votes
1answer
44 views

How to calculate a definite integral with complex numbers involved?

I'm trying to calculate this integral, and I find it difficult when coping with complex numbers. $$ f(k) = \int_{lnK}^{\infty} e^{ikx} (e^{x}-K) dx ...
2
votes
2answers
52 views

prove $\sum_{k=1}^n(-1)^{k+1}\cos^{2n+1}\left(\dfrac{k\pi}{2n+1}\right)=\frac{1}{2}-\frac{2n+1}{2^{2n+1}}$

Today I found the identity : $$\sum_{k=1}^n(-1)^{k+1}\cos^{2n+1}\left(\dfrac{k\pi}{2n+1}\right)=\frac{1}{2}-\frac{2n+1}{2^{2n+1}}$$. How to prove or disprove this? Thank you.
0
votes
2answers
38 views

Fiding imaginary part of a complex number [on hold]

What is the imaginary part of $i^i$ ? I've tried multiple approaches, including using log. I can't seem to understand how to work with complex numbers as logarithmic functions. Also, it would ...
0
votes
1answer
29 views

the closure of the set $\{ e^{in\theta}:n\;\text{non-negative integer numbers} \}$

May I ask a question about the closure of the set $\{ e^{in\theta}:n\;\text{non-negative integer numbers} \}$, where $\theta\in\mathbb R$. Many thanks.