Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

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3answers
63 views

By using De Moivre's Theorem, show that $\cos5\theta = 16\cos^5 \theta - 20 \cos^3 \theta + 5 \cos \theta$ [closed]

First step is $$\cos5\theta + i \sin 5\theta = (\cos \theta + i \sin \theta)^5$$ thanks
0
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3answers
55 views

How $\sqrt{\cos (106.3) + i \sin (106.3)} = \cos 53.15 + i \sin 53.15$ [closed]

The question is find sqrt of $-7 +24i$ solution: $$\sqrt{-7+24i} = z$$ $$-7+24i = z^2$$ $r=25$, $106.3^\circ$ $$\sqrt{\cos (106.3) + i \sin (106.3)} = \cos 53.15 + i \sin 53.15 /*HOW?*/$$ thanks
1
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4answers
58 views

$\text{If } |z_1| = |z_2|, \text{ show that } \frac{z_1 + z_2}{z_1-z_2} \text{is imaginary.} $

$\text{If } |z_1| = |z_2|, \text{ show that } \frac{z_1 + z_2}{z_1-z_2} \text{is imaginary.} $ The first thing I tried to do was to multiply both top and bottom by the conjugate of the denominator... ...
1
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0answers
35 views

Proof Strategy: For all nonzero complex numbers $z$ and all nonzero rational numbers $a$ and $b, \mathbb Q (az+b)=\mathbb Q(z)$

I am having trouble proving (or finding a counterexample but I believe it to be true) the following. Prior to this I did some problems such as: Show that $\mathbb Q (-3+i\sqrt{2},2-\sqrt{8})=\mathbb Q ...
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1answer
36 views

Is product of two square roots of two integers square root of their products? [duplicate]

Suppose $a,b\in \mathbb{Z}$. Is it true $\sqrt{a}\sqrt{b}=\sqrt{ab}$. If so, then $\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1$ But we know $\sqrt{-1}=i$ and so $i^2=-1.$ ...
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1answer
45 views

Use complex number to solve this equation $\int e ^{3x} cos x dx$?

I can solve it another way, but am not sure how to use complex numbers to solve it. Thanks for your help
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1answer
20 views

Radius of Convergence (Non-Series)

I am confronted with the following exercise: Compute the radius of convergence for the expansion at the point $z=4+4i$ for \begin{equation} f(z)=\frac{z^{5}e^{z}}{(2-z)(3i-z)} \end{equation} I ...
0
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1answer
37 views

The set of points in complex plane that satisfy a strict linear inequality is open

Let $S = \{(x,y)\in \mathbb C: y > 3x+2\}$. Show that $S$ is an open set. I can imagine what it looks like; a shaded region above a line. I also imagine that we must choose $ε$ (the radius ...
2
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1answer
34 views

Finding the cube root of a complex number $z$

$\text{Let }z = -2-2i \text{ where }i \text{ is imaginary. Find in Modulus-Argument form the cube roots of }z$ So far I've done this $$r = \sqrt8 = 2 \sqrt2 \\ \alpha = \frac{-\pi + \frac{\pi}{4}}{3} ...
2
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1answer
44 views

If $z_i+z_jz_k$ are real, then $z_1z_2z_3=1$

Let $z_j=r(cosφ_j+isinφ_j), r\in R$ for $j=1,2,3$ be different complex numbers. If the numbers $w_1=z_1+z_2z_3$, $w_2=z_2+z_1z_3$, $w_3=z_3+z_1z_2$ are real, prove that $z_1z_2z_3=1$ I know one ...
0
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0answers
16 views

How to find z of cosh(z) = -2 & choosing value

$\cosh(z) = -2$ $z = \cosh^{-1}(-2)$ $z = \ln(-2 \pm i\sqrt{4-1})$ $z = \ln(-2 \pm \sqrt{3}) $ -> I wolfram this and it choose only $-2 - \sqrt{3}$ not $-2 + \sqrt{3}$ I would like to know what is ...
0
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1answer
26 views

Simple identity involving complex numbers

We have to prove the following identity: $$z_1 \bar{z_2} = \frac{1}{4}(|z_1 + z_2|^2 + i|z_1 + iz_2|^2 - |z_1 - z_2|^2 - i|z_1 - iz_2|^2)$$ It says to use the identity we just proved, which is ...
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2answers
43 views

How do I show $|\frac{i\overline{z}}{2} - \frac{i}{2}|=|z - 1|?$

I was looking over an example from our book concerning limits, and I'm having trouble seeing how this equality holds.
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2answers
82 views

Prove that $\prod_{k=0}^{n-1}(z-\mathrm{e}^{2k\pi i/n})=z^n-1$

Prove that $$ \prod_{k=0}^{n-1}(z-\mathrm{e}^{2k\pi i/n})=z^n-1. $$ In my some problem I have used $$ \prod_{k=0}^{7}(z-\mathrm{e}^{2k\pi i/8})=z^8-1. $$ I have verified this. So I think in general ...
0
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1answer
46 views

A problem on Complex differentiability

I following problem was given as a homework, I have explained how I approached it I need to know if it was correct and even then if it there wasn't any easier way, because that way only had tedious ...
2
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2answers
67 views

General formula for $\sin\left(k\arcsin (x)\right)$

I'm wondering if there's a simple way to rewrite this in terms of $k$ and $x$, especially as a polynomial. It seems to me to crop up every so often, especially for $k=2$, when I integrate with trig ...
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2answers
37 views

Prove $\left(z^n\right)' = nz^{n-1}$

I'm trying to solve this complex-variable problem: Prove, using direct Calculus, that $\left(z^n\right)' = nz^{n-1}$ ($n \in \mathbb{N}$). I tried the following steps to solve that: I saw that ...
1
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1answer
24 views

Square root and distribution

I see two ways of definining the imaginary number $i$, e.g. $i^2 = -1$ or $i = \sqrt -1$. I always thought the second one was right, yet I saw lots of websites saying that : $1=\sqrt ...
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2answers
44 views

Complex power of a real number

What is the meaning of $(-1)^{i}$, where $i^{2}=-1$ and what is its value?
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1answer
33 views

Express complex function in the form $u+iv$

One of the parts of the question I'm working on goes something like this: Express $z^i = \exp(i \log_I(z))$ in the form $u+iv$, where $u,v$ are real-valued functions, and the log is defined on the ...
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8answers
159 views

Why is $\frac 25$ the real part of $\frac{1}{2+i}$?

According to Wolfram Alpha, Re(1/(2+i))=2/5. How did it calculate that?
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8answers
86 views

How ? $| z + 1 | = | \overline z - 1 |$ [closed]

Can someone talk me through this? Determine all complex numbers that satisfy the equation $| z + 1 | = | \overline z - 1 |$
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1answer
34 views

Which complex vector multiplied by its conjugate returns the identity matrix

I am trying to find (in case there is any) which complex vector $n$ of 2 dimensions, multiplied by its conjugate transpose, returns a diagonal matrix. $n = [a, b]^T = [a_1+ja_2, b_1+jb_2]^T$ ...
1
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1answer
25 views

Integral on complex plane of a gaussian times power

I can't solve the integral $$ I = \int_\mathbb{R} \int_\mathbb{R} \ (x + i y)^{2k} \ e^{\displaystyle - \frac{(x + i y)^2 R^2}{1+R^2} - y ^2} d x d y $$ which can be rewritten as $$ I= \int_\mathbb{R} ...
0
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0answers
17 views

Complex Plane - Analytic Function

I am trying to understand the definition of an analytic function and how to solve for it's domain. I understand that for $f(z) = {1\over z}$ the function is analytic on the complex plane except for 0. ...
0
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0answers
58 views

Analyze branch cuts and discontinuities of function $f(z)=\sqrt{1-z^2}$

Analyze the function $f(z)=\sqrt{1-z^2}$, where the square root is defined by the principal branch of the log function. Where does it have discontinuities? Here's what I did: We have $I = ...
0
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1answer
37 views

Finding the set of analytic functions whose image is a subset of a given set

Let $A=${$z\in\mathbb{C}||z|=1$} and $B=${$z\in\mathbb{C}||z|<2$}. I want to find the the set of analytic functions such that $f(B)\subset A$. Is there a way to solve this? Hope someone could help ...
0
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0answers
44 views

Calculating the real and imaginary parts of a holomorphic function

Calculate the real and imaginary parts of the holomorphic function $f(z)=z^2\cos(z)-e^{z^3-z}$ and verify directly that each of these functions is harmonic. I believe I know how to the question, ...
0
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1answer
18 views

Can I extend these ODE formulas to complex numbers?

In my calculus class, we recently covered first-order, linear ODEs. Specifically, we discussed the formula for the solution of one (and its derivation): $$y=\frac{1}{u(x)}\int Q(x)u(x)dx$$ where ...
0
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1answer
37 views

The points $z_1, z_2, z_3$ are three complex numbers

The points $z_1, z_2, z_3$ are three complex numbers lying on a the circumference of a circle passing through the origin in the Argand diagram. Show that $\frac 1{z_1} , \frac 1{z_2}$ and $\frac ...
1
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0answers
30 views

The differentiability of the complex valued function $(Rez)(Imz)z\over|z|^2$

$$ f(z) = \left\{ \begin{array}{ll} \Re(z)\Im(z)z\over|z|^2 & \quad z \neq 0 \\ 0 & \quad z = 0 \end{array} \right. $$ I want to prove that this ...
0
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0answers
19 views

Complex number theory - limits

Can anyone please help me with those two limits? I have missed the first two weeks of a new semester, so really not experienced with this. $$a) \lim\limits_{z \to \infty} \frac{z^2-\overline z^2 + ...
0
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1answer
10 views

Norm2 of a vector of complex numbers

I am migrating a matlab code into C++ and I need to know how does matlab calculate the norm of below matrix. For two numbers, A=a+ib , B=c+id, I know I should do [(a-c)^2+(b-d)^2]^1/2. But how is it ...
0
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2answers
49 views

Question about a step in the proof of the Cauchy-Schwartz inequality in $\mathbb{C}$

I'm studying the proof of the Cauchy-Schwartz inequality, which states that for complex numbers $z_1,\ldots. z_n,w_1,\ldots, w_n$ we have $$ \Big\vert\sum_{j=1}^nz_jw_j \Big\vert^2\le ...
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2answers
68 views

How do you represent $\Im$ and $\Re$ on paper?

Do you draw $\Im$ and $\Re$ just like they are or you write $\mathtt {Im}$ and $\mathtt{Re}$?
1
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2answers
62 views

The image of the curve with equation $z\bar z = 2\operatorname{Im} z $ under the map $w=1/z$

Let $z \in \mathbb{C}$ satisfy $z \overline{z} = 2\Im (z)$ Let $w=\frac{1}{z}$ Find the equation describing the curve $w$ forms on the complex plane and the $z$ that has the minimun distance from ...
1
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0answers
20 views

Find similar estimate for complex numbers

According to Borwein, page 356 Prop. 2, $\left|\ln\left(\frac{4}{k}\right)-\operatorname{I}(1,k)\right|\leq 4k^2\left(8+\left|\ln k\right|\right)$ holds for $k\in\left(0,1\right]$. ...
2
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1answer
38 views

Does $\lim_{x \to 0}({z^2\over \overline z})$ exist? $(z\in \mathbb{C})$

I am trying to figure out if $\lim_{x \to 0}({z^2\over \overline z})$ exists or not. This is a way I though to show that this does not exist but I am not entirely sure. Let $a_n={1\over n}$ and ...
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0answers
34 views

$f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$

I want to prove that if $f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$. This is my rather incomplete approach to ...
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votes
1answer
50 views

When equality holds in an inequality

I am working on a class project, the passage I quoted in here is from a book Complex Numbers & Geometry by Hahn. For any four complex numbers $a$, $b$, $c$, $d$, the following identity is easy ...
0
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2answers
30 views

Function with exponent imaginary power

If we have $u=\frac{4c(e^{-is}-e^{is})}{(e^{-is}+e^{is})^2} \tag 1$ where c is a constant and s is a variable. Can we write $e^{is}$ in terms of u ? Means Can we write $e^{is}$ as $\psi(u)$ , a ...
0
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1answer
20 views

Is this log identity true?

I'm wondering if the exponent property carries forward to the complex log. In other words, for some complex numbers $z$ and $w$ does $\ln(z^w) = w\ln(z)$?
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1answer
26 views

Principal branch of the complex logarithm does not always satisfy the product formula

My book asks to prove: $\text{Ln}[i \cdot (-1+i)]$ does not equal to $\text{Ln}(i) + \text{Ln}(-1+i)$ where $\text{Ln}$ gives the principal log of the complex number. I don't see why this is true ...
5
votes
3answers
96 views

Can anyone tell me how $\frac{\pi}{\sqrt 2} = \frac{\pi + i\pi}{2\sqrt i}$

I was working out a problem last night and got the result $\frac{\pi + i\pi}{2\sqrt i}$ However, WolframAlpha gave the result $\frac{\pi}{\sqrt 2}$ Upon closer inspection I found out that ...
5
votes
5answers
61 views

Is $\{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\}$ open or closed?

I am trying to figure out if the set $\{z\in\mathbb C\mid|\text{Re }z|+|\text{Im }z|\le1\}$ is open or closed or maybe none of that. I hope someone could provide a hint to solve this. Can this set be ...
0
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0answers
17 views

Is a complex function really just an infinite dimensional matrix?

I have recently sort of come to the understanding that integrating two functions multiplied together is a sort of infinite dimensional dot product, and I only know this from taking an undergraduate ...
2
votes
1answer
34 views

Proving $\left(\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}\right)^n=\cos n\left(\frac{\pi }{2}-x\right)+i\sin n\left(\frac{\pi }{2-x}\right)$

How to solve the following question? If $n$ is an integer, show that \begin{eqnarray} \left(\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}\right)^n=\cos n\left(\frac{\pi }{2}-x\right)+i\sin ...
0
votes
3answers
29 views

All Values of a Complex expression

I am asked to find all values to $$\left(\frac{1-i}{\sqrt2}\right)^{1+i}$$ I do not know how to approach a power with complex part. Any help would be appreciated.
0
votes
4answers
38 views

Find all complex and real roots of higher degree polynomials, given one root

$2+3i$ is a zero of $f(x)=x^4-4x^3+17x^2-16x+52$, find all of the zeros of $f(x)$ thanks!
0
votes
1answer
56 views

Why does $2x_1x_2y_1y_2 \leq x_1^2y_2^2+x_2^2y_1^2$?

When I tried to prove the triangle inequality $|z_1+z_2| \leq |z_1| + |z_2|$ algebraically for complex variables $z_1$ and $z_2$, I came across this inequality and found that this is always true no ...