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

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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 ...
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0answers
13 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, ...
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1answer
15 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 ...
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1answer
34 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 ...
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26 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 ...
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0answers
16 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 + ...
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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 ...
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1answer
29 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
52 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 $\mathtt{Re}$?
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2answers
51 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 ...
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0answers
14 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 $k\in\left(0,1\right]$. ...
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1answer
33 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
24 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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0answers
26 views

How to solve this complex-valued exponential signal ? Help !!! [on hold]

Consider the complex-valued exponential signal $$x(t)=Ae^{αt+jωt}, α>0$$ Evaluate the real and imaginary components of $x(t)$.
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1answer
37 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 ...
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2answers
19 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 ...
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1answer
16 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
22 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 ...
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1answer
22 views

Integration of exponential with a complex [on hold]

i want to prove the left side of the equation to the right side, can some one please help me with this
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3answers
95 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 ...
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5answers
53 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 ...
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0answers
15 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 ...
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1answer
33 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 ...
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3answers
26 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.
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4answers
29 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!
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1answer
55 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 ...
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1answer
76 views

Imaginary Numbers [duplicate]

Can someone help me by explaining what imaginary numbers actually are, please? I know that it is defined as $i = \sqrt{-1}$ and I can find umpteen sources telling me how to calculate with them but I ...
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12 views

If $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$

I tried proving that if $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$. This is my attemt. Let $\epsilon>0$. Then there exists $N\in ...
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2answers
62 views

What is the generalized definition of '<' and '>' for complex numbers? [duplicate]

I'd expect this question to be asked here before, but I've not been able to find it. The generalized definition of the multiplication operator for complex numbers is simple: The product of the ...
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1answer
32 views

$\Im\left ((a+bi)^n\right )=0, a,b \in \mathbb{R}, n \in \mathbb{N^*}$

I am trying to connect $a,b,n$ such that $$\Im\left ((a+bi)^n \right )=0, a,b \in \mathbb{R}, n \in \mathbb{N^*} \mathrm{or } \; \mathbb{R^*}$$ What I tried was write $(a+bi)^n$ as $\sqrt{a^2+b^2} ...
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1answer
66 views

When is $x^{2^n}$ dense in $\mathbb{S}^1$, for $|x|=1$?

Motivation: So I just saw this question: Limit when $n\rightarrow\infty$ of $\text{sgn}(\sin(2^n \pi x))$ with $x\in(0,1)$ fixed., and the answer involves diadic numbers and things of the kind. Most ...
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1answer
13 views

Question on the Closure Properties of $\mathbb C$

I read here: https://proofwiki.org/wiki/Properties_of_Complex_Numbers, that the Complex Numbers are closed under addition and multiplication. I'm having trouble understanding why. I realize this is a ...
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1answer
15 views

Straight Line Equation in Complex Plane

Hi there, 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 ...
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1answer
32 views

Roots of a quintic function

I need some pointers in the right direction for this question: Three of the roots of the equation $ax^5+bx^4+cx^3+dx^2+ex+f=0$ are $-2$, $2i$ and $1+i$. Find $a$, $b$, $c$, $d$, $e$ and $f$. I ...
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3answers
76 views

Proof of $\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos n\theta=\frac{\sin\frac12n\theta}{\sin\frac12\theta}\cos\frac12(n+1)\theta$

State the sum of the series $z+z^2+z^3+\cdots+z^n$, for $z\neq1$. By letting $z=\cos\theta+i\sin\theta$, show that $$\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos ...
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1answer
76 views

About the identity $e^{i\pi}=-1$

I have a question about the famous identity of Euler $e^{i\pi}=-1$. I opened the other day this question about the number of roots of a complex number with irrational exponent. Under this light and ...
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1answer
27 views

Every compact set $S\in \mathbb{C}$ is bounded

This is my proof for every compact set $S \subseteq \mathbb{C}$ is bounded. Let $S \subseteq \mathbb{C}$ be compact and assume that it is not bounded. Then for each $z\in \mathbb{C}$ and for each ...
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1answer
60 views

Given $w=e^{2i\pi/7}$, perform algebraic manipulations with complex numbers like $w+w^2+w^4$

It is not my own homework and I forgot how to solve this kind of things. Anyway, the following are the statement of the homework and my attempts: The homework: Let $w=e^{2i\pi/7}$, $u=w+w^2+w^4$, ...
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3answers
46 views

Cross Ratio is positive real if four points on a circle

Given four points $a, b, c, d \in \mathbb{C}$ on a circle I want to show that the ratio $ \frac{(a-c)(b-d)}{(a-b)(c-d)}$ is a positive real number. I have shown that it is real by expanding $ ...
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1answer
23 views

Evaluating $|a^b|$ when $a,b$ are complex

Here, $a^b=e^{b\log a}$ for some suitable (but fixed in advance) branch of the $\log$ function. What is the most general formula for $|a^b|$ when both $a$ and $b$ are complex, and what are the ...
2
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1answer
26 views

Variance of amplitude and phase from sin and cos regressors in polar coordinates

On a data set, I estimated the sine and cosine weights at a specific frequency, $\beta_{\sin}$ and $\beta_{\cos}$. I can extract the amplitude and phase from these regressors as follows: ...
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1answer
38 views

Finding which base number given operations

$$ (35_a + 24_a) * 21_a = 1081_a $$ Which base is the above number? Any advice on how to solve questions like these? I tried making it in to a polynomial: $(3a+5 + 2a+4) * (2a+1) = 108a + 1$ ...
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2answers
22 views

Calculating the argument of a complex number… something tends towards infinity?

A simple question, but I like to be clinical with my choice of words: I have a complex number, $z=-i$. If I were to calculate the argument of this complex number, $arg(z) = tan^{-1}( \frac{-1}{0}) ...
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4answers
35 views

Complex integration by shifting the contour

In section 12.11 of Jackson's Classical Electrodynamics, he evaluates an integral involved in the Green function solution to the 4-potential wave equation. Here it is: $\int_{-\infty}^\infty dk_0 ...
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1answer
38 views

$D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact

This is the proof I wrote for $D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact. $$ \bigcup_{n=2}^{\infty}D_{1-(1/n)}(0) $$ is clearly a open covering of $D_1(0)$. Consider the finite ...
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0answers
25 views

Complex trigonometric equation

Find a solution to the equation $\tan(z)=7i$ which satisfies the condition $0<\Re(z)< \pi$} We use the $\sin(z)=\frac{e^{zi}-e^{-zi}}{2i}$ and $\cos(z)=\frac{e^{xi}+e^{-xi}}{2}$. Here is ...
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1answer
74 views

Prove that the sum of all distinct $n$-th roots of unity is zero? [closed]

Prove that the sum of all distinct $n$-th roots of unity is zero. Interpret the fact geometrically.
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0answers
49 views

Proving Ptolemy Theorem using complex number

I am working on an assignment proving Ptolemy Theorem using complex number, and I am looking at a textbook Complex Numbers and Geometry by Hahn. Here is what I am working at this moment: THE ...
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4answers
58 views

Why has my prof worked out the multiplicative inverse of complex numbers this way?

She explains how to obtain the multiplicative inverse: In what follows, z=a+bi and w=c+di are complex numbers with a,b,c,d∈R. Is there a multiplicative inverse of z? If so, what is it? Note ...
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1answer
52 views

Definition of $z^0, z=a+bi, a,b \in \mathbb{R}, z \neq 0$

Today I found in a True-False the question; Does the equality $$z^0=1, z=a+bi, a,b \in \mathbb{R}$$ hold $\forall z \in \mathbb{C^*}$? The thing is, this was never clearly defined in the book, and ...