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

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1answer
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Complex var. integral: $\oint_{|z|=1} \frac{z^2\ dz}{\sin^3{z}\cos{z}}$

Integrate $\displaystyle\oint_C \dfrac{z^2\ dz}{\sin^3{z}\cos{z}}$; $C \rightarrow |z|=1$ I already know that $|z|=1$ is a circumference with $r=1$ and center at $(0,0)$. I also know there are ...
2
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1answer
17 views

Length of a complex vector

From the definition of inner product in $\mathbb{F}^n$ $$\textbf{a}\cdot\textbf{a}=\sum\limits_{k=1}^na_{k}\overline{a_{k}}$$ Say $a_{k}=x_{k}+iy_{k}$, then ...
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0answers
31 views

proof of a vector identity

In an exercise I am asked to prove the following vector identity: $$\textbf{a}\cdot\textbf{b}=\frac{1}{4}\big(|\textbf{a}+\textbf{b}|^{2}-|\textbf{a}-\textbf{b}|^{2}\big)$$ Both the dimension of the ...
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1answer
50 views

Complex transformation $w=\sqrt \frac{1-iz}{z-i}$ the region $D=\{z\in \mathbb C:|z|<1\}$ [on hold]

Under the transformation $w=\sqrt \frac{1-iz}{z-i}$ the region $D=\{z\in \mathbb C:|z|<1\}$ is transformed to (a) $\{z\in \mathbb C:0<\operatorname{arg}(z)<\pi\}$ (b) $\{z\in \mathbb ...
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1answer
38 views

If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$.

Statement: If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$. I just wanted to verify my proof ...
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2answers
31 views

Modulus of a complex expression [on hold]

Show that for a complex number $a$, $$\left\lvert\frac{z-a}{1-\bar{a}z}\right\rvert = 1$$ for $|z| =1$ and $\bar{a}z ≠ 1$. I've tried to show that if $|z-a| = |1-\bar{a}z|$ then it's true but to no ...
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2answers
151 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 ...
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1answer
39 views

stuck with a cubic equation

I picked an equation $0= x^3 +x^2 -2x -1$ I plotted it with geogebra, to see if it had more than $1$ real root. It definitely cuts the $x$-axis $3$ times. But when I checked wolfram alpha, to see ...
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0answers
42 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
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1answer
21 views

Seperating points in the complex plane

Given a finite set of points say $p_1,p_2, \ldots, p_n$ in the complex plane, how do I find another point $q$ such that ray $R_i$ joining $q$ to $p_i$ are all distinct. I would be happy with any kind ...
3
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0answers
63 views

Limits of sequences connected with real and complex exponential

Let us denote $S_{n}(x)=1+\frac{x}{1 !}+\frac{x^{2}}{2!}+ ... + \frac{x^{n}}{n!}$. How could be calculated the limit $$L(x)=\lim_{n\to \infty}\frac{S_{n}(n x)}{e^{n x}}=\lim_{n\to ...
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1answer
44 views

Physical Proof of Euler's Formula

I would like to construct a geometrical or physical proof of Euler's Formula $e^{ix}=\cos x +i\sin x $. If anyone has constructed such a proof before I would love to see it, if not, I would like some ...
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0answers
40 views

Show that if $\lvert a \rvert \neq 1$, then the equation $\overline{z}^2 = az^2+bz+c$ has only a discrete number of solutions.

I knew the proof for this at some point, but I'm having trouble piecing it back together. At least, I think the proof I'm thinking of was for this result, or a result which implied this result. The ...
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1answer
65 views

you know root square of -1, what is the larger of the square? [on hold]

there is a square ABDC, $BD = \sqrt{-1}$ what is the value of AB=BC=DC=AD?
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3answers
231 views

Does $\sin(x+iy) = x+iy$ have infinitely many solutions?

How to prove that $\sin(x+iy) = x+iy$ has infinitely many solutions? I know how to prove that $\sin(x) = x$ has only one solution, but I do not know how to extend this to complex analysis.
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2answers
34 views

Polar form equations on the unit circle

If $l \in [0, 2 π)$, $k, n \in N$, proof the following equations: $$\mid{e^{i k l/n} - e^{i (k-1) l/n}}\mid = \mid e^{i l/n} - 1\mid$$ and: $$\lim_{n \to \infty} \sum_{k = 1}^n \mid e^{i k l/n} - ...
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1answer
17 views

Prove that if $C$ is anti hermitian matrix then $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $.

Suppose $C \in M_{n\times n}(\mathbb C)$ satisfies $C+C^* = 0$. Prove that $\forall v\in \mathbb C^n \ : \ Re(\langle Cv, v \rangle)=0 $. Here is what I was able to show so far: We know that $C$ ...
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19 views

Example on Complex Number using De moivrs theorem [on hold]

Prove That ( √3+i)^14 + ( √3-i )^14 = 2^14
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1answer
42 views

Inverting complex cosine

I have been working out problem 3a in chapter 1 section 3 in Basic Complex Analysis by Marsden. He asks to solve $$ \cos z=\frac{3}{4}+\frac{i}{4} $$ After putting cosine in its exponential form and ...
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1answer
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Let $w=1+3i$. Investigate whether $|iw+w|=|iw|+|w|$. [on hold]

Let $w=1+3i$. Investigate whether $|iw+w|=|iw|+|w|$. This is a question on complex numbers.
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0answers
61 views

Can an ordered field contain complex numbers?

I read a question about ordering of complex numbers, and saw an answer showing that there cannot exist an ordering of the complex numbers because regardless of how $i$ is placed in that order, it ...
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6answers
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Can a complex number ever be considered 'bigger' or 'smaller' than a real number, or vice versa?

I've always had this doubt. It's perfectly reasonable to say that, for example, 9 is bigger than 2. But does it ever make sense to compare a real number and a complex/imaginary one? For example, ...
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1answer
15 views

Sixth root of -64 using Euler's formula and De Moivre's theorem

I am attempting to solve: $$(-64)^{\frac{1}{6}}$$ Using the relation: $$a+bi=re^{i(\tan^{-1}(\frac{b}{a})+2\pi n)}$$ And then applying De Moivre's theorem: ...
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0answers
51 views

Complex Analysis (Complex Mapping) stuck on professor's method of simplification in math notes

I'm having an exam this university semester and need some help with my math notes. I've come accross some problems with the section "Complex Mapping." Link to Image of my Notes: Click Me (see first ...
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1answer
62 views

Find roots of $ω^x+(ω^x)^2+1=x$ [on hold]

We have to solve this equation at complex numbers group $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right
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1answer
49 views

When the imaginary part of a function is zero?

Let $z_k=x_k+ i y_k, x_i,y_i \in \mathbb{R}$ are the complex variables. Consider a polynomial of $z_k$ and its conjugates $f(z_1,\ldots,z_n, \bar{z}_1, \ldots,\bar{z}_n).$ Question:Is there any ...
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1answer
11 views

Distribution of magnitude squared for complex Gaussian

$\def\Re{\operatorname{Re}}\def\Im{\operatorname{Im}}$ If we have a random complex variable $h_l$, with $\Re[h_l]\sim \mathcal{N}(0,\sigma_l^2/2)$ and $\Im[h_l]\sim \mathcal{N}(0,\sigma_l^2/2)$ ...
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3answers
38 views

Show that $1+z=2\cos\frac 12\theta(\cos\frac 12 \theta + i\sin \frac 12 \theta)$

Let $z=\cos\theta+i\sin\theta$. Show that $1+z=2\cos\frac 12\theta(\cos\frac 12 \theta + i\sin \frac 12 \theta)$ Can anyone show me how to show the equation? I can't think of how to get $\frac 12 ...
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0answers
33 views

Question about galois imaginary and modular arithmetic

Let $p$ be a prime of type $3\space mod \space 4$. Then there is no solution $x^2 = -1 \space mod \space p$. Therefore we can define the so-called Galois imaginary $i$. ( $i^2 = -1 \space mod \space ...
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1answer
30 views

Complex Analysis (Limits)

Let $a, b$ be complex numbers. Use the definition of a limit directly (not just the properties of limits) to prove that $$ \lim_{z \to z_0}az + b = az_0 + b. $$ Sorry for the wrong notation, I do ...
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1answer
35 views

Representation theory of $\mathbb{Z}_k$ and complex roots of unity

Is there a natural relationship between the (characters?) of irreducible representations of $\mathbb{Z}_k$ and the $k$ complex-roots of unity? Can they be like thought of as characters of its ...
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0answers
53 views

Complex Analysis ( Open and Closed Sets) [on hold]

I am supposed to show that if T is a closed set of complex numbers and S is contained in T, then the modulus of S is contained in T. I know a closed set means it does not extend to infinity, S=x+iy, ...
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1answer
28 views

Complex Plane ( $\arg(z)$)

Sketch the following regions of the complex plane. For each, say whether it is open, closed, or neither, and whether it is connected. No proofs necessary. $$\left\{z \in \mathbb{C}\mid -\dfrac{\pi}{2} ...
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If two solutions of arg$(z)$ are in interval $−\pi<$arg$(z)≤\pi$ are both correct?

For example there is complex number $z=\sqrt3-i$ Are the answers $\frac{5}{6}\pi$ and $-\frac{\pi}{6}$ correct as $\text{arg}(z)$?
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1answer
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Use the Newton-Raphson algorithm to find all roots accurate to $10$ decimal places of the polynomials

Use the Newton-Raphson algorithm to find all roots accurate to $10$ decimal places of the two polynomials $p(x)=5ix^4-(9+2i)x^3+7x+6-i$ and $q(x)=9x^5-x^3+7x+6$. The roots, with accurate to $10$ ...
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1answer
17 views

Express $\sin(z)$ and $\cos(z)$ in Rectangular Form

"Express $\sin(z)$ and $\cos(z)$ in rectangular form." For $z \in \mathbb{C}$ (complex numbers), we have defined \begin{equation} \sin (z)=\frac{e^{iz}-e^{-iz}}{2i} \end{equation} and ...
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1answer
50 views

Show that if $\lim_{n\rightarrow\infty}z_n=0$ then $\lim_{n\rightarrow\infty}(1+\frac{z_n}{n})^n$ [on hold]

Can you help me with the following problem. I dont have any idea how to start. Prove that if $\lim_{n\rightarrow\infty}z_n=0$ then $\lim_{n\rightarrow\infty}(1+\frac{z_n}{n})^n=1$.
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2answers
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Plotting on a complex plane

I'm very confused how you would plot the relationship $|z-4| \leq |z|$. I tried to change it in form which could become $-|z|\leq|z-4|\leq|z|$ and I guess the same can be done for z-4. But I don't ...
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1answer
14 views

Computing Principal Logarithm on Different Intervals

Compute the principal logarithm of a complex number $z=\sqrt{3}+i$ using $\mathrm{Arg}(z) \in [0,2\pi)$ and $\mathrm{Arg}(z) \in (-\pi,\pi]$. Wikipedia shows how the answer can be different for the ...
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1answer
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Question about determining accumulation points

So far the way I have determined accumulation points of given sequences or relations has been by drawing them out. However I would like some clarification to see if my thinking is correct or not. a) ...
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2answers
16 views

Defining domain in complex plane

I am asked to define the domain for the following given that $z=x+iy$: $a) \quad f(z) = \dfrac 1 {z^2 + 1}$ $b) \quad f(z) = \dfrac 1 {1 - |z|^2}$ How would this be different from a normal domain ...
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1answer
33 views

Conditions To Make Complex Numbers $z_1, z_2, z_3, z_4$ Vertices of a Square

Let $z_1,z_2,z_3,z_4\in\mathbb C$ be distinct. State conditions in terms of computation of complex numbers, which make $z_1,z_2,z_3,z_4$ vertices of a square (in the counterclockwise direction). ...
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1answer
14 views

Complex number and orthogonal axis

What are the properties of complex numbers which allow us to plot the real and complex part on orthogonal axis? One thing I understand is that complex portion cannot be represented as scalar multiple ...
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1answer
27 views

Algabreic manipulation with complex numbers

How does $(iwl + \frac{1}{iwc})^2$ equal to $(wl - \frac{1}{wc})^2$? Let me clarify. In physics there is the impedance which is a complex number Z = R + iwl + 1/iwc R, w, l, and c, are ...
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2answers
69 views

Proof of $\cos(y)$ and $\sin(y)$ using $e^{iy}$

I need to use that $e^{iy} = \cos y + i \sin y$ (for $y \in \mathbb{R}$) to prove that $$\cos y = \frac{e^{iy}+e^{-iy}}{2}$$ and $$\sin y = \frac{e^{iy}-e^{-iy}}{2i}$$ I'm really clueless, any ...
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3answers
57 views

A complex Analysis proof

Let $a \in \mathbb{C}$ and $\phi \in \mathbb{R}$. Prove that if $|a+1|=|1+ae^{i \phi}|$ then $ae^{i \phi} = a$ or $ae^{i \phi} = \bar{a}$. I need an idea of how to approach here please anyone.
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1answer
39 views

Check if $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ is compact

I want to check, if this set is compact: $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ Thoughts: $z:= a +bi$ real part $a$ is ...
0
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1answer
19 views

Deriving definition of the complex logarithm

Given that: $$z = Re^{i\theta} = R(a + bi) = R\left( \cos(\theta) + i\sin(\theta) \right)$$ In its polar form. $$\log(z) = \log(R) + i\theta$$ $$|z| = \sqrt{(Ra)^2 + (Rb)^2} = R\sqrt{a^2 + b^2} ...
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3answers
41 views

Eigenvalues of skew-symmetric matrix

Prove that all of the eigenvalues of skew-symmetric matrix are complex numbers with the real part equal to 0. Has anyone got a clue how to do it?
0
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1answer
34 views

Complex analysis basics

If I z = x + yi and w = f(z), describe the image R of D in the w-plane when $$0<x<\pi/2, 0<y<\infty;w = e^{iz}$$ Wouldn't this mean that in the w-plane the argument arg(w) = $\infty$ ...