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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Prove that $|a|+|b|\le \sqrt{2}|z|$

I was solving maths and got struk on this question.might you help me with this one. If z=a+ib Then, prove that $|a|+|b|\le \sqrt{2}|z|$ I don't know how to start it. Help me.
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1answer
22 views

$|g(t_{1}) e^{-(t_{1}-x)^{2}}- g(t_{2})e^{-(t_{2}-x)^{2}}|\leq |f(t_{1}) e^{-(t_{1}-x)^{2}}- f(t_{2})e^{-(t_{2}-x)^{2}}| $?

Suppose $f, g: \mathbb R \to \mathbb C$ such that $|g(t_{1}) -g(t_{2})| \leq |f(t_{1})- f(t_{2})| $ for every $t_{1}, t_{2} \in \mathbb R.$ Take any $x\in \mathbb R$ and fix it. Edit: We also assume ...
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1answer
26 views

Find the number of distinct elements.

Let $\omega$ denote a non-real cube root of unity. Then find the number of distinct elements in the set $\{ (1+\omega + \omega^2 + \cdots + \omega^n)^m | m,n \in \Bbb Z_+ \}$
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43 views

what are contraction(Lipschitz) maps on $\mathbb C$?

We say a map $f:\mathbb C \to \mathbb C$ is contraction(Lipschitz) if $|f(z_{1})- f(z_{2})| \leq C |z_{1}- z_{2}|$ for every $z_{1}, z_{2} \in \mathbb C$ and $C$ is some constant. Trivial Examples: ...
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2answers
35 views

sum of complex number of different magnitude

Is there a systematic way to express the sum of two complex numbers of different magnitude (given in the exponential form), i.e find its magnitude and its argument expressed in terms of those of the ...
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1answer
44 views

Complex number calculations

Can someone explain to me, step by step, how to calculate this, $$x=(-1-i)^{15}+(-1+i)^{11}$$ Method 1 (Transform the numbers (−1−i) and (−1+i) to polar coordinates ) by Mr 5xum ...
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1answer
121 views

Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described ...
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0answers
136 views

A uniformly convergent series

How does one show that the series $$\sum_{k = 1}^\infty \left\{\frac{s}{k} - \log\left(1 + \frac{s}{k}\right)\right\}, \quad s \in \mathbb{C} \setminus \{0, -1, -2, \ldots\}$$ is uniformly convergent? ...
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48 views
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2answers
73 views

prove the following equation about inverse of tan in logarithmic for

$$\arctan(z)=\frac1{2i}\log\left(\frac{1+iz}{1-iz}\right)$$ i have tried but my answer doesn't matches to the equation .the componendo dividendo property might have been used. where ...
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2answers
159 views

Evaluating the real and imaginary parts of a nasty complex number

This seems like an elementary question, but I was unable to find an clear answer to it. Generally, the real and imaginary parts of a complex number comprised of radicals are not expressible by ...
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1answer
77 views

Linear Algebra,Conjugate Transpose

Let $ M_n(\mathbb C) $ be the space of all $ n\times n $ matrices with complex entries. Prove that function $ \langle, \rangle : M_n(\mathbb C) \times M_n(\mathbb C) \to \mathbb C $ defined by $ ...
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1answer
185 views

Orthogonality with complex numbers

I have these 2 equations: $$\begin{align} (68-4i,44+12i,-38-2i)\mathbf x=0 \\ (-66i-18,-52i-10,42i+12)\mathbf x= 0 \end{align}$$ I need to find the span of vector $\mathbf x \in \Bbb C^3$. I'm ...
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1answer
47 views

Complex numbers exercise - homework

We have to prove that $z_1^{24n}+z_2^{24n}=2^{12n+1}$ if we know that a)$z_1z_2=2$ b)$z_1^3+z_2^3=-4$ I have tried many things but nothing worked so far
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1answer
70 views

How to $ (3-2i)(2+3i)/(1+2i)(2-i)$ [closed]

I got my admit card today noon and my exam (Lateral Entry for B.Tech) is tomorrow 11:15A.M. LoL...that means I have to go through my guide of > 500 pages in less than 18 hours (I know its not ...
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1answer
88 views

Locus of Complex Number

Would be great to get your help in finding the locus of this complex number $z$: $|z-z_1|+\sin \alpha|z-z_2|=\sin \theta$ From this question I proceed to a refined one- What would ...
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1answer
37 views

Complex argument and nyquist plot

I'm trying to sketch the nyquist plot of $$\frac{j\omega-1}{j\omega+1}$$ but can't seem to calculate the argument correctly. I think it should be $$\arctan(-\omega) - \arctan(\omega) = ...
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1answer
86 views

Complex modulus? No, not the absolute value.

I was trying to make a class for complex numbers (VB.NET) but then I stumbled upon a problem. How do I define the $mod$ operator for Complex numbers? First I asked Wolfram Alpha. It didn't help much. ...
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1answer
62 views

Laurent Expansion partial fractions

I have a function: $\frac{1+2z}{z^3 + z^2}$ for $0 < |z| < 1$ (about $z=0$) I need to find the Laurent expansion of this function. However, I'm a bit confused how to find the partial fractions ...
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2answers
23 views

Solving two varibles system equation above $\mathbb{C}$

A bit emmbarrassed to ask this newbie question: Let: $$(1+i)x + y = 2$$ $$(1-i)x + iy = 0$$ Multiplying the first equation by $(-i)$ and summing the two equations, we have: $$(2-2i)x + 2i = 0$$ ...
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1answer
91 views

Clarification on a step in the proof of Lagrange's identity for complex numbers.

I wrote this proof of the following identity and I want to verify that a certain step is correct. $\newcommand{\conj}[1]{\overline{\vphantom{b}#1}}$ $\newcommand{\on}[1]{\operatorname{#1}}$ ...
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1answer
30 views

Exponential of complex variable

What is the equivalent to " $(e i)^z$ " , where i is the imaginary "i" and z is a variable (maybe a complex one) ? (I'm thinking in a possible symmetry with $e^{iz}$)
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1answer
33 views

What is the special thing about |z|=2, will this point lie in the mandelbrot set?

For the quadratic iteration $z \to z^2+4$, if you perform a few iterations letting $z_0 =0.5+1.936491673i$, the modulus of the points will be 2 ( or closer 2 two because of the inaccuracy of the ...
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1answer
78 views

Geometric intuition behind subspaces in $\mathbb C^n$

While learning elementary linear algebra one develops a great deal of geometric intuition in $\mathbb R^n$. It helps to see the forest for the trees and leads through proofs. After meeting ...
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1answer
21 views

How to show that $f(A_1)=\{z: z(t)=x+i, x \in \mathbb{R} \}$?

Let $A_1=\{z: z(t)=\frac{1}{2}e^{it}- \frac{1}{2}i, t\in [\frac{\pi}{2}, \frac{3\pi}{2}] \}$ and $f(z)=\frac{1}{z}$. Show that $f(A_1)=\{z: z(t)=x+i, x \in \mathbb{R} \}$. My attempt: ...
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2answers
31 views

How to expand out negative powers for complex numbers

I have the following expansions but I don't know how my teacher gets them. Apparently there is a formula for it (though the guy who told me didn't know it well), but I cannot find it in my notes. For ...
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3answers
60 views

Proof using complex numbers

Prove that $\left|\dfrac{z-w}{1-\bar{z}w}\right| = 1$ where $\bar{z}$ is conjugate of $z$ and $\bar{z}w\ne 1$ if either $|z| = 1$ or $|w| = 1$. I used $|c_1/c_2| = |c_1|/|c_2|$ and multiply out with ...
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1answer
67 views

Is there infinitely many “complex units”

As we know, $i$ = $\sqrt{-1}$, a simple complex unit. In complex space of two dimensions, you graph an axis of $a+bi$ where $i$ is your second dimension axis. Now, you also know, in three and four ...
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1answer
38 views

Convert complex logarithms to inverse tangents

I'm doing an exercise from the book "Algorithms for Computer Algebra" by Keith O. Geddes. I'm asked to show that if $u$ and $v$ are two relatively prime polynomials in $\mathbb{Q}[x]$ and $s$ and $t$ ...
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1answer
149 views

How to show $\displaystyle\lim_{n\to\infty}\sqrt[n]{|z^n|}=|z|$

For $z\in\mathbb{R}$ it's very easy to show that it holds $$\displaystyle\lim_{n\to\infty}\sqrt[n]{|z^n|}=|z|$$ But how do we show the same thing for $z\in\mathbb{C}$
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1answer
58 views

Laurent Series of $f(z) = \frac{1}{e^z - 1}$

The Laurent Series of $f(z)$ centred at $0$ can be written as, $$f(z) = \frac{1}{z} - \frac{1}{2} + \frac{z}{12} - \frac{z^3}{720} + \cdots$$ So we see that $f(z)$ has a simple pole at $0$. Can we ...
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5answers
59 views

Explain why there are two complex numbers z such that $|z| = 1$ and that satisfy the equation $|z| = |z-1|.$

I must find both such complex solutions and express them in Euler form and usual form. So it's been a while since I've touched the imaginary/real plane. However, from what I remember, $z = a + bi$. ...
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1answer
167 views

locus of a complex number in the form |z-a|=k|z-b|

how does one find the locus of a set of complex numbers defined in the form |z-a|=k|z-b| for example in the question (CIE ALEVELS MATHS/9709/May-June 2013/Paper 33/Question 7) below we have to find ...
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38 views

About gaussian integers and orders.

I noticed $(1+i)^{16}= 256$ so $(1+i)^{16} - 1$ is a multiple of $17$. So $(1+i)^{16} - 1$ is a multiple of $(1+4i)$ or $(1-4i)$. $(1+i)^{|1+4i|}$ is congruent to $1$ or $i$ mod $(1+4i)$. I think . ...
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124 views

Why does the Mandelbrot shape show up in other fractals?

In the pictures below, the Collatz map fractal includes parts resembling the Mandelbrot set. Why? Do other fractals do so? The Mandelbrot set From Wikimedia Commons Part of the Collatz map fractal ...
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2answers
664 views

Finding the least value for points in a loci

The question goes like this: On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z+i|=1$ and the locus representing the complex numbers $w$ satisfying ...
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0answers
38 views

Complex Analysis Questions Compilation

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) ...
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3answers
249 views

Find all complex numbers z such that $\Im(-z+i) = (z+i)^2$

$$\Im(-z+i) = (z+i)^2$$ So I tried solving this by replacing $z=a+bi$, thus getting $$\Im(-a+(1-b)i) = (a+(b+1)i)^2$$ From here I split it into the imaginery and real part: \begin{equation*} ...
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2answers
70 views

Proving a simple equation with complex numbers

Fix $A \in ℂ$ and $B \in ℝ$ Let $z \in ℂ$. Show that the equation $|z^2| + Re(Az) + B = 0$ has solutions iff $|A^2| ≥ 4B$ I have no trouble proving the forward implication, but its the "only if" ...
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2answers
52 views

Show that $(x-\alpha)(x-\overline{\alpha})$ is a also a factor of $p(t)$ over the complex numbers

Here is the full question. Lots of struggles: Let $p(t)$ belong to $P(R)$. a) If $(x − \alpha)$ is a factor of $p(t)$ over the complex numbers (i.e. $p(t) = (x − \alpha)\cdot q(t)$, for ...
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3answers
56 views

Suppose $z \neq -1$ is a complex number of norm 1. Prove that $(\frac{1+z}{|1+z|})^2 =z $

Suppose $z\neq -1$ is a complex number of norm 1m $(|z| =1)$. I know that $z= n + mi$, what is the most straightforward way of solving this problem? I was also given the following sketch for a ...
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2answers
79 views

Complex Analysis Polar Circle Question

My book (Gamelin ' s Complex Analysis) says that the roots of a complex number are distributed in equal arcs on the circle centered at $0$ with radius $|w|^{1/n} $. Why is the radius centered at zero ...
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2answers
79 views

A solution to the equation $\frac{1}{x}=0$ [duplicate]

The number $i$ is defined as a solution to the equation $x^2+1=0$. How come no one has yet defined a number $j$ as a solution to the equation $\frac{1}{x}=0$? The purpose of course is to be able to ...
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2answers
41 views

Factorization of a Complex Polynomial

My book (Complex Analysis by Gamelin) states that a complex polynomial $$p(z) = a_nz^n + a_{n-1}+\cdots +a_1z + a_0$$, where $z$ is an element of the complex numbers, can be factored as a product of ...
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1answer
128 views

Conformal map entire domain to a strip with specific branchcuts

I am looking for a conformal mapping function that maps the entire z-plane to an infinite strip. (e.g. T=f(z) & -b < Real(T) > b ) I hope to find a function that cuts open to original domain ...
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1answer
82 views

Why is $\int |e^{ix}|^2 dx = x + C$?

Quick question: Wolfram Alpha tells me that $$\int |e^{ix}|^2 dx = x + C$$ Why is that?
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1answer
82 views

Regarding Gaussian integers and primitive roots.

Can modular arithmetic be set up using gaussian integers instead of (non-complex) integers? If so is there an analogue of 'primitive roots' with Gaussian integers?
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2answers
25 views

Prove that function transforms $S$ into $S$

I have to prove, that function $\Phi(z):=\frac{1-\overline a z}{z-a}$ transforms $S$ into $S$ where $S:=\{|z|=1\}$ and $|a|>1$ I don't know where can I start. Any hint?
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0answers
39 views

Why is $\Re(e^{(\lambda e^{i\theta}-\lambda)})\neq e^{(\lambda(\Re (e^{i\theta}-1)))}$?

Why is $\Re(e^{(\lambda e^{i\theta}-\lambda)})\neq e^{(\lambda(\Re (e^{i\theta}-1)))}$ ? I always thought, that $\Re$ is linear, but if I compute LHS; $\Re(e^{(\lambda ...
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1answer
85 views

Real and Imaginary [duplicate]

$$Re\Big(({\frac{1+i\sqrt{3}}{1-i})^4\Big)} = 2$$ $$Im\Big(({\frac{1+i}{1-i})^5\Big)} = 1$$ I got that $Re\Big(({\frac{1+i\sqrt{3}}{1-i})^4\Big)} = 1 \ne 2$ And, that ...