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

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Which $k$ value from $2\pi k$ do you pick as the $n$th root of the solution?

$$W = \frac{1+i}{\sqrt{2}}$$ I need to find 5th root of $W$ where $Z^5 = W$ $\theta$ is: $$\frac{\pi}{20} + \frac{2\pi k}{5}$$ I always thought You need to plug in $K = 0, \pm 1, \pm 2,\ldots$ to ...
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0answers
20 views

How to use the Cauchy Integral for a certain problem [duplicate]

I'm having a lot of trouble solving a problem for my Fourier Analysis class. I have searched the web for hours looking for answers, but I can't find anything that can break this down for me, with my ...
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2answers
24 views

Symbolic expansion of complex arithmetic expressions

I have a lengthy complex polynomial-type equation. There is a software that can solve this type of equations provided I pass it as a coupled system of two real-valued equations. Hence, I am wondering ...
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0answers
22 views

Find a value for a number to the power of a complex number

Find a value for $2^{-4i}$? I have no idea what to do or how to find the value. My thoughts are that I should use logarithm. Can someone please show me how to solve this?
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0answers
17 views

How to use three complex vector components to calculate resultant complex vector

This is a practical problem related to complex vectors. Imagine you want to find the resultant electric field of multiple electromagnetic waves that have parallel and perpendicular components ...
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6answers
44 views

Compute: $|z + 2| = |z − 3i|$

Find all complex numbers that solve this equation: $|z + 2| = |z − 3i|$ How would I go on about solving this one? 4 times? Like this? $I. z+2=z-3i$ $II. z+2 = -(z-3i)$ $III. -(z+2) = z-3i$ $IV. ...
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2answers
26 views

Compute $2i\Re(z)\Im(z) = \bar{z} + 3 + i$

Compute: $2i\Re(z)\Im(z) = \bar{z} + 3 + i$ How do you solve this? What do you change $\Re(z), \Im(z)$ to? I tried it like this, but I'm unsure if I've done it right; $2i·a·bi = a-bi+3+i$ $i(b-1) - ...
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3answers
80 views

Compute $\bar z - iz^2 = 0$

$\bar z - iz^2 = 0, i = $ complex unit. I've found 2 solutions to this, like this: $x - iy - i(x+iy)^2 = 0$ $i(-x^2+y^2-y)+2xy+x=0$ $2xy + x = 0$ --> $y=-\frac{1}{2}$ $x^2 - y^2 + y = 0$ --> ...
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4answers
70 views

Compute $z^3 = -26 -18i$

$z^3 = -26 -18i, i =$ imaginary unit. How do I solve this? So far I've calculated the length of it; $|z| = \sqrt{(-26)^2+(-18)^2}=10\sqrt10$ I think I'm supposed to use the polar form of it, but ...
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0answers
20 views

Method's name/Theory: Equivalence of complex and real matrices of double dimension

I remember reading a document where it was explained, how complex matrices are equivalent to real matrices of double size, according (as far as I remember): Let $C$ be a complex matrix, then $D = ...
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3answers
100 views

Is integrating over complex numbers like this valid?

I had to evaluate the integral $$I=\int_{0}^{\pi} x^4 \sin{x} \ \mathrm{d}x$$ I thought that integrating by parts would be to long, and so, planning to use the property $\displaystyle\int e^{x} ...
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2answers
33 views

If $u(z) = \frac{az+b}{cz+d}$ is bijective on the upper half complex plane, show that $a,b,c,d$ are real.

The following question is from Greene-Krantz, Function Theory of One Complex Variable (Q 1.10) Let $U = \{ z \in \mathbb{C} \colon \textrm{Im} \, z > 0 \}$. Prove that if $ u(z) =\displaystyle ...
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1answer
357 views

De Moivre's theorem:

Could someone help me to expand and express: $$ \sum_{k=0}^N \cos(k\theta) $$ And: $$ \sum_{k=0}^N \sin(k\theta) $$ In terms of $$\cos\theta/2$$ and $$\sin\theta/2$$ Using De Moivre's ...
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3answers
95 views

Proving that $x=\arccos(\sqrt{\sin\theta})$ is $\sin(x+iy)=\cos\theta+i\sin\theta$

Given: $$\sin(x+iy)=\cos\theta+i\sin\theta$$ To prove: $$x=\arccos (\sqrt{\sin\theta})$$ How I tried: $$\begin{align*} \sin x \cosh y &= \cos\theta \\ \cos x \sinh y &= \sin\theta ...
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3answers
61 views

Finding the value of $|z|$

Let $z$ is a non-real number such that, $$\frac{1+z+z^2}{1-z+z^2}$$ is purely real. Find the value of $|z|$. Hello everyone! For this question, when I set ...
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1answer
15 views

Complex numbers property proof. [duplicate]

I eas given this quesstion in one of my Linear Algebra course with the excercises regarding minimal polynomialsm eigenvalues and diagonalizable matrix: Show that for any two numbers $a,b \in ...
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1answer
23 views

Inequality including complex numbers

I'm trying to prove the following inequality. Let $ s = x+iy \in \mathbb{C} $. Prove that $$ \left\lvert\frac{1}{n^s} - \frac{1}{(n+1)^s}\right\rvert \leq \frac{\lvert s\rvert}{n^{x+1}} $$ The ...
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1answer
20 views

Absolute value of the sume of two complex number

I have a question about the following. $|A+B|^2$, where $A, B $ is complex number. The question is , when can $|A+B|^2$ be equal to $|A|^2 + |B|^2$?
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1answer
43 views

Absolute Value Trig Sum

I have been trying to solve $$y(x)=\sum_{k=1}^{\infty} \frac{|\cos(kx)|}{k}$$ however, this is proving to be more difficult than I had hoped, and cannot seem to figure this out. What I have figured ...
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2answers
61 views

Let $z_1,z_2$ be complex numbers such that $Im(z_1z_2)=1$ Find the minimum value of $|z_1|^2+|z_2|^2+Re(z_1z_2)$

Question : Let $z_1,z_2$ be complex numbers such that $Im(z_1z_2)=1$ Find the minimum value of $|z_1|^2+|z_2|^2+Re(z_1z_2)$ I know that $|z_1+z_1| \leq |z_1|+|z_2|$ Also if I consider two ...
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3answers
89 views

Showing $\left(\frac{-1+\sqrt{3}i}{2}\right)^3=1$

In my textbook they asked me to show that $$\left(\frac{-1+\sqrt{3}i}{2}\right)^3=1$$ but this is not true, I think. I put down $$\begin{align} ...
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1answer
60 views

What is the error in this fake-proof of the complex number i? [duplicate]

The error is from the 3rd step to the 4th step. But why is this an error? Can't $i$ be interchangeable with $\sqrt{-1}$? $-1 = i\cdot i = \sqrt{-1}\cdot \sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1$
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3answers
202 views

Why do I get two different results for the reciprocal of $i$?

I am aware that the correct answer is $$\frac{1}{i}=\frac{1}{i}\frac{i}{i}=\frac{i}{i^2}=\frac{i}{-1}=-i$$ But equally, I find no error here: $$\frac{1}{i}=\frac{1}{\sqrt{-1}}= ...
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1answer
28 views

Prove that $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$

Prove that: $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$ We just learned about the characteristic/minimal polynomial and diagonalization but I am not sure if it has ...
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1answer
49 views

Unable to solve any Euler questions. Fundamental error I cannot find

Good day, I have been trying to solve Euler based questions for a day now. And i realize I still cannot solve a single one, and am getting errors for all my questions. I feel like I am getting ...
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1answer
21 views

Solving a system of complex equations

$$u = (1+i, i), v = (1-i, 2i), w = (2,3+i)$$ I'm asked to find is there's $z$ such that: $$v = zu$$ So if I suppose $z = a+bi$ I have the system: $$(1-i, 2i) = (a+bi)(1+i, i)\implies\\(1-i, 2i) = ...
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2answers
28 views

How to find root of the complex number

When finding n-th root of the complex number z=cos(x)+sin(x)i, we can use known De Moivre's formula: My task is to find without using De Moivre's formula, and my question is how to do that? Thank ...
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1answer
61 views

For which positive integers $n$ does $P(n)$ fail to hold?

Let $n$ be a natural number and let $z$ be a complex number. Consider the following proposition: $P(n)$: If $\cos (nz)$ is bounded above by one in absolute value, then $\cos z$ ...
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1answer
26 views

Conformal mappings to polygons: why is my integral conformal?

I'm learning about conformal mappings into polygons in a class,(undergrad complex analysis) and am having trouble understanding one of the examples given in my book. (Stein & Shakarchi) Here it ...
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1answer
18 views

Why is $z^{1/k}=|z|^{1/k}e^{iarg(z)/k}$ for $k\in \mathbb{Z}$ true?

I'm reading a short paper here Which attempts to prove/discuss all the properties of the complex logarithm, exponential, and power functions. In the paper, the following statement is made on pg 9, ...
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1answer
71 views

Evaluate $\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos (\theta )\right)^2}$

I am trying to evaluate the integral $$\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos \left(\theta\right)\right)^2}$$ via change of variables and applying Cauchy's Residue Theorem. Here is how I'm ...
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1answer
13 views

Limit of complex exponential

The following is the characteristic function of a random variable $X_n$:$$\phi_{n}(t)=\frac{1-e^{it}e^{\frac{it}{n}}}{(n+1)\left(1-e^{\frac{it}{n}}\right)}$$ for $t \in \mathbb R$. I am trying to ...
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2answers
44 views

Faster way to for $z^3 = -2 (1+i \sqrt 3) \bar z$ than complex algebra

What is the fastest way to solve for $z^3 = -2 (1+i \sqrt 3) \bar z$? I know how to do this using complex algebra. but that takes a long time. Can someone show me a faster way?
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0answers
67 views

What is a complex number that can't be written in polar form?

What is the cartesian form of a complex number that can't be written in polar form? Why can't it be written in polar form?
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1answer
25 views

Prove of an addition theorem for the general binomial coefficients

Prove that: $\sum_{k=0}^n \binom{s}{k} \binom{t}{n - k} = \binom{s + t}{n}$ for all $s, t \in\Bbb C $, $n \in N\cup {0}$. That's pretty much all I'm given, and therefore, I haven't come quite far ...
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0answers
18 views

How to determine the limit of a complex function

It is easy to show that a complex function doesn't have a limit as it approaches a certain point, but is there any way to know for sure whether any given complex function has a limit as it approaches ...
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1answer
27 views

Circle in the plane of complex numbers

Let $K = \{z \in \mathbb{C}: |z−a|=r \}$ be a circle in $ℂ$. Show that, for the case that $|a|$ is not equal to r, the image of $K$ under the transformation $z$ $\to$ $\frac {1}{z}$ is a circle too. ...
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1answer
99 views

Relating convergence theorem for Newton-Raphson method to Newton fractal

I have created a Newton fractal (below) using the Newton-Raphson method to find the five solutions of f = (z^5-1) The convergence theorem of Newtons method say ...
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3answers
49 views

Finding the roots of a polynomial on a complex plane [duplicate]

I use an online calculator in order to calculate $x^5-1=0$ I get the results x1=1 x2=0.30902+0.95106∗i x3=0.30902−0.95106∗i x4=−0.80902+0.58779∗i x5=−0.80902−0.58779∗i I know that this is the ...
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1answer
50 views

complex plane questions

Find where the points of the complex plane are if, a) $|\pi - \arg z| < \pi/4$ b) $|\Re z| < 1$ c) $\Im \left(\frac{z+1}{z+i}\right) = 0$ d) $z = z_1 + t(\cos x + i\sin x), 0\leq ...
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3answers
55 views

Factoring a polynomial of fourth degree with false roots: $x^4+4$

I want to write this polynomial in factored form: $$x^4+4$$ The reason I want to do this is to be able to make partial-fraction decomposition on it to make an integrand easier to integrate. What's ...
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1answer
18 views

A linear fractional transformation and mapping of concentric circles

Q: A fractional linear transformation maps the annulus $r < \|z \| <1$ (where $r > 0$) onto the domain bounded by the two circles $\|z- \frac{1}{4} \|=\frac{1}{4}$ and $\|z \|=1$. Find $r$. ...
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1answer
28 views

Where the power series is convergent

Where $f(z)=\sum_{n=1}^{\infty}\frac{(2i)^n}{n}z^n$ is convergent? I checked that the radius of convergence is equal to $\frac{1}{2}$. Now, since we know that the series ...
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0answers
29 views

A function on the punctured complex plane which turns out to be constant

Let $f: \mathbb{C}- \{0\} \rightarrow \mathbb{C}$ be a holomorphic function on the punctured complex plane, and suppose that $f(2z)=f(x)$ for all $z \neq 0$. Prove that $f$ is constant. Proof: ...
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1answer
35 views

Laurent Series Expansion for $f(z)=\dfrac{z+2}{(z+1)(z-2)}$ in $\{1<|z|<2\}$ and $\{2<|z|<\infty\}$

I'm trying to get the Laurent Series expansion of the function stated in the title in the stated regions. My approach is as follows: We can first break up $f(z)$ using partial fractions ...
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1answer
30 views

Zeros of $z^2 \text{cos}z^2$

Is there an easy way to find the zeros of the function $z^2 \text{cos}z^2$, $z\in \mathbb{C}$ and the respective orders (multiplicities)? All I can think of is to find $f^{(1)},f^{(2)},...$ but then ...
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1answer
20 views

Find the basis of set given by matrices

In linear space of matrix $2\times 3$ over $C$ we have subspace generated by: $ A= \{{\left[\begin{array}{ccc}i&i&i\\i&0&1\end{array}\right]}$ ...
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1answer
14 views

Phase of relative coordinate in the complex plane

If we have two points $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ in the complex plane and define the relative coordinate $z=z_2-z_1$, we have that the length of $z$ is the Euclidian distance between the ...
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1answer
44 views

Solve $z^{1+i}=4$

$\def\Log{\operatorname{Log}}$ I have to solve $z^{1+i}=4$. Is there any easy way? I'm starting like this: $$e^{(1+i)\Log z}=e^{2\Log2}$$ Then I solve $$(1+i)\Log z=2\Log2$$ But I really doubt I ...
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3answers
65 views

Find all $z \in \mathbb{C}$ that satisfy $z^3 = −2(1 + i\sqrt{3})\bar{z}$

Find all $z \in \mathbb{C}$ that satisfy $z^3 = −2(1 + i\sqrt{3})\bar{z}$. You must express your answers in the standard form. So far, I'm thinking of writing $z = a + bi$, but then I have to ...