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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64 views

$z= \frac{u-\overline{u}v}{1-v}$ is real is equivalent to $|v|=1$.

Let $u,v$ be complex numbers such as $u,v\notin \mathbb{R} $, and : $$z= \frac{u-\overline{u}v}{1-v}$$ Prove that : $z\in\mathbb{R} \Longleftrightarrow |v|=1$.
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236 views

Prove $i\notin \mathbb R $

Prove that $\imath$ (defined by $\imath^2=-1$) does not have a position on the Real number line. That is, show that there does not exist two real numbers $a$ and $b$ such that $a<\imath<b$. (I'...
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119 views

A question on complex numbers

We are given If $\cos(a+ib)$=$r (\cos\theta +i\sin\theta)$ then prove that $e^{2b} = \sin(a-\theta)/­\sin(a+\theta)$ I just tried and got $b = 0$ such that $\cos(a) = ra$. Will there be other ...
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224 views

Non-existence of a certain holomorphic function on the unit disk

I am trying to prove the following: Let $n\in \mathbb{N}$. Prove that $\not \exists$ a holomorphic function $f$ on the open unit disk satisfying: $f\left(\displaystyle \frac{1}{n}\right) = 2^{-n}$ ...
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175 views

simplify $ (-2 + 2\sqrt3i)^{\frac{3}{2}} $?

How can I simplify $ (-2 + 2\sqrt3i)^{\frac{3}{2}} $ to rectangular form $z = a+bi$? (Note: Wolfram Alpha says the answer is $z=-8$. My professor says the answer is $z=\pm8$.) I've tried to figure ...
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276 views

Euler's formula and $i^x = \cos(x \cdot \frac{\pi}{2})$

While playing around with a plotting software, i just found out that $$f(x) = i^x = \cos(x·\frac{\pi}{2})$$ How does this connect to Euler's formula? Obviously, here, the alternating sign change ...
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1answer
73 views

Which region is defined by $\frac{z}{i}-\bar{z}=0$

Which region on complex plane is defined by the geometric images of $z$ that satisfied this condition: $\frac{z}{i}-\bar{z}=0$ One of my trials was: $\frac{z}{i}-\bar{z}=0\Leftrightarrow z- \...
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167 views

How to analyze the modulus of $\lambda = (1-2\mu)+2\mu\cos\theta+i\nu\sin \theta$?

Consider the complex number $$ \lambda = (1-2\mu)+2\mu\cos\theta+i\nu\sin \theta $$ where $i=\sqrt{-1}$, $\mu,\nu$ are constants, and $\mu>0$. Question: How can I get that $|\lambda|\leq ...
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144 views

the power sum can completely determine the complex numbers self

If $a_1,a_2,\dots,a_n,b_1,b_2,\dots,b_n$ are complex numbers in $\mathbb C$, and for every $j\in\mathbb N$, we have the power sums $$\sum_{i=1}^n a_i^j=\sum_{i=1}^n b_i^j$$ I want, without applying ...
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1k views

A square root of i with negative imaginary part

In an ODE class, one assignment question says find the “rectangular” expression z = a + bi (with a and b real) and the “polar” expression |z|, Arg(z) where z is "a square root of i with negative ...
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1k views

Raising a square matrix to the k'th power: From real through complex to real again - how does the last step work?

I am reading Applied linear algebra: the decoupling principle by Lorenzo Adlai Sadun (btw very recommendable!) On page 69 it gives an example where a real, square matrix $A=[(a,-b),(b,a)]$ is raised ...
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1answer
80 views

Is it true that $ \sqrt{-z} = i \sqrt z $?

Is it correct to write $ \sqrt{-z} = i \sqrt z $ , for every complex $z$? I think it's not true but I have seen it in some books . The reason I think it's not correct is for example if $z=i$ then $\...
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1answer
40 views

A fallacy in the imaginary numbers. [duplicate]

$$\sqrt{-5}*\sqrt{-3}=\sqrt{-1*5}*\sqrt{-1*3}$$ $$\sqrt{-1*-1}*\sqrt{5*3}=\sqrt{5*3}$$ $$=\sqrt{15}$$ But we all know that this below is right, $$\sqrt{5}i*\sqrt{3}i=-\sqrt{15}$$ So, please explain ...
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1answer
39 views

$|1-e^{2\pi i n z}|<1$ for $z$ in the upper half plane

If $z$ is in the upper half plane, then $|e^{2 \pi ni z}|<1$ for every $n\in\mathbb{N}$. But why is also $|1-e^{2\pi i n z}|<1$? I just get $$|1-e^{2\pi i n z}|\leq1+|e^{2 \pi ni z}|<1+1=2....
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29 views

Find all the solutions in the complex field of the following equation:

Firstly my apologies for anything that should be in LaTex format correctly, I gave it a valiant effort. I have been asked to solve the equation: $(1-z)^6 = (1+z)^6$ A hint given states: do not ...
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1answer
25 views

General sum of $n$th roots of unity raised to power $m$ comprime with $n$

I am trying to find a reference for the following proposition: Let $m$ and $n$ be coprime. Then, $$ \sum_{k=0}^{r-1} \exp\left( i \frac{2\pi}{n} k m \right) = 0 $$ if and only if $r$ is an integer ...
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35 views

Geometrical Applications of Comple Numbers, roots of Unity

If $\omega$ is a non-real cube root of unity, prove that $$\left|\dfrac{x+y\omega+z\omega^2}{x\omega+z+y\omega^2}\right|=1$$ $$$$ Despite trying for a long time, I haven't been able to come up with ...
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65 views

Solve an equation with complex variable

I have tried to solve the following equation: $$ (z+1)^4+4(z-1)^4=0 \quad,\quad z=(a+bi) $$ If I could leave the $z^4$ alone I could solve it using the formula, but I do not know how to operate with ...
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34 views

Show that $f(z)=az+b$ if $f$ is of the form $g(x)+h(y)$ and $f$ analytic

Suppose that $f$ is analytic in a domain $\Omega$ and $f(z)=g(x)+h(y)$ for $z=x+iy \in \Omega$, where $g,h$ are complex-valued functions. Show that $f(z)=az+b$, where $a,b \in \mathbb{C}$ are ...
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63 views

Real part of $cis (\theta)$

I have a question on a proof regarding this: The question is "Show this is true," although I am not sure if the last minus sign should be an equal sign, or else it doesn't make sense. I know that ...
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50 views

Find and sketch the image of the straight line $z = (1+ia)t+ib$ under the map $w=e^{z}$

I need to find and sketch the image of the straight line $z = (1+ia)t +aib$, where $-\infty < t < + \infty$, $a,b\in \mathbb{R}$, and $a \neq 0$, under the map $w = e^{z}$. In order to ...
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64 views

Complex derivative of $\cos(x) \cosh(y)-i \sin(x) \sinh(y)$

The problem: Determine the derivative of the following function $f(z)=\cos(x) \cosh(y)-i \sin(x) \sinh(y)$ The original exercise can be found at 2.17 (e) page 36 Should i try to rewrite the function ...
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86 views

Equation in the complex plane $8z=i|z|^3\bar{z}$?

I do not know what to do with this equation. I tried to make both sides in trigonometric form, but after I don't know how to move forward to solve it. $$8z=i|z|^3\bar{z}$$ In trigonometric form $$8\...
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66 views

Find Complex Roots

Question: Find the complex roots of $$ {(z^{12} -1)\over (z^4-1)(z^3-1)} = 0 $$ What I have attempted: $$ {(z^{12} -1)\over (z^4-1)(z^3-1)} = 0 $$ $$ {(z^{6} -1)(z^{6} +1)\over (z^2-...
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1answer
82 views

What is ${\rm Im}(z)+{\rm Re}(z)=1$?

I've started studying complex numbers and I am having conceptual difficulties when asked to plot the equation $${\rm Im}(z)+{\rm Re}(z)=1$$ How do I graph it ?I can't see the analogy with the ...
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105 views

Using Demoivre's Theorem prove that $ {\cos5 \theta} = 16{\cos^5 \theta} - 20{\cos^3 \theta} + 5{\cos \theta} $ .

$ {\cos5 \theta} = 16{\cos^5 \theta} - 20{\cos^3 \theta} + 5{\cos \theta} $ . Demoivre's Theorem $$ \{\cos \theta + i \sin \theta \}^n = \cos n\theta + i\sin n\theta $$ Where n is an integer . I ...
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43 views

How does taking the logarithm of negative numbers compare to taking logarithm of positive numbers?

Is this an accurate statement: The logarithm of negative number is not unique but the logarithm of a positive number is unique. I know the first clause is true but I'm not sure if it applies to ...
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38 views

Trying to prove whether $\lim_\limits{z \to 0} e^{-1/z^4}$ exists or not.

This problem involves complex analysis, and the limit is as follows: $\lim_\limits{z \to 0} e^{-1/z^4}$ I have to either show that this limit exists, or that it does not. This is how I've ...
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75 views

If $w$ be nth root of unity , then $1+2w+3w^2+\dots+nw^{n-1}$ is equal to?

I tried it by letting expression $1+2w+3w^2+\dots+nw^{n-1}= x$ and then multiplying $w$ both sides . I subtracted equation 1 from 2 but it does not seems to help me because i have just started ...
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62 views

If $1+x^2=\sqrt{3}x$ then $\sum _{n=1}^{24}\left(x^n+\frac{1}{x^n}\right)^2$ is equal to

I tried this problem by different methods but i am not able to get the answer in easy way . First i found the roots of equation and then represented it in polar form of complex number . I got $\cos \...
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38 views

Let $A=\left\{a\in R:\text{the equation}(1+2i)x^3-2(3+i)x^2+(5-4i)x+2a^2=0\right\}$ has atleast one real root.

Let $A=\left\{a\in R:\text{the equation}(1+2i)x^3-2(3+i)x^2+(5-4i)x+2a^2=0\right\}$ has atleast one real root.Find the value of $\sum_{a\in A}a^2$. What should i do in this question to find the ...
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101 views

why is the value of $(-1)^\frac{2}{3}$ not 1?

$(-1)^2$ equals 1 also $(-1)^\frac{1}{3}$ equals -1 any way you arrange it, it should be one. how ever, typing this into Wolfram or google,gives me an irrational complex number. I would love some ...
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68 views

Let $a\ne1$ be an nth root of identity, show $1+2a+3a^2+\dots + na^{n-1} = \frac{n}{a-1}$.

Let $a\ne1$ be an nth root of identity, show $1+2a+3a^2+\dots + na^{n-1} = \frac{n}{a-1}$. I took $S= 1+2a+3a^2+\dots+na^{n-1}$. I have a hint that says I should compute $(1-a)S$ and then use a ...
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67 views

Extending the complex numbers by the solution of $|x| = -1$

I don't think I've ever encountered a situation where I've wanted to solve equations of the form $|x| = -1$, but you often hear that mathematics should be explored for the sake of mathematics. I'm ...
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99 views

Complex number identity by trigonometry

Show that $\lvert e^{i\theta} - 1\rvert = 2\lvert\sin(\theta/2)\rvert$ by using the geometry of the triangle with vertices 0, 1, and the midpoint of the line joining 0 and $e^{i\theta}$. I have been ...
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47 views

Solving $\frac{df}{dt}=\frac{i\cdot f}{|f|}$ where $f: \mathbb{R^+} \mapsto \mathbb{C}$

I've never seen a complex DE before, so this is uncharted territory for me. But it's separable so it's easy to turn it into an integral: $$f(t) = \int_0^t\frac{i \cdot f}{|f|} dt$$ Can this be solved? ...
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723 views

$\mathbb{C}$ is a one-dimensional complex vector space. What is its dimension when regarded as a vector space over $\mathbb{R}$?

$\mathbb{C}$ is a one-dimensional complex vector space. What is its dimension when regarded as a vector space over $\mathbb{R}$? I don't understand how $\mathbb{C}$ is one-dimensional. Please help me ...
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26 views

Simple proofs of the following identity related to the roots of units

I'm looking for simple proofs of the following identity $$ z_k\Pi_{j\neq k}(z_k-z_j)=n, $$ where each $z_j$ is the n-th root of the unit, i.e., $z_j=e^{\frac{2\pi i j}{n}}$,$j=1,2,\ldots,n$. Thanks ...
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94 views

Prove that the integral of $\sin^2(x)/(5+3\cos(x))$ from $0$ to $2\pi$ is $2\pi/9$

I'm not really unsure of how to approach this problem. I was thinking of reparametrizing the sin and the cos to its exponential form but I realize that it becomes even messier and leads sort of ...
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48 views

A seemingly simple property of complex numbers that won't submit.

Let $a,b\in\mathbb{Z}[i]$ such that $|b-a|>|b|$ and $|a|>|b|$. I want to show that the absolute value of the real part of $\frac{a}{b}$ is greater than $\frac{1}{2}$. For example, let $a=4+4i$...
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63 views

Expansion of imaginary numbers

If $(1+i)^{100}$ is expanded, what is the value of the real part of the result? I know that this has to do with binomial theory and Pascal's triangle, but I don't know how to use it here.
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95 views

Proving convergence of Newton's method

Consider the recursive sequence $$ z_{n+1} = {1 \over 2}\left ( z_n + {1 \over z_n} \right )$$ where we start at some point $z_0 = x_0 + i y_0 \in \mathbb C$. This is Newton's formula to find the ...
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40 views

Show that $e^\mathbf{iA} + e^\mathbf{iB} = 2e^\frac{i(A+B)}{2}\cos(\frac{A-B}{2})$

Where $i=\sqrt{-1}$ and $A,B\in \mathbb{R}$ are constants. I've tried already with Euler's formula, but cannot prove the equation above. Best Regards, Thanks.
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48 views

Is the set $\{z\neq0:Arg(z)>0\}$connected

Is the set $\{z\neq0:Arg(z)>0\}$connected, where $Arg(z)$ is the principal value of the argument. (The principal value, is the value in the open-closed interval $(−\pi, \pi]$) I think the set is ...
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3answers
64 views

Find the value of $z^n+1/z^n$ if $n$ is arbitrary natural number

If $z$ is a complex number satisfying $$z + \frac{1}{z} = \sqrt{3}$$ then for arbitrary natural number $n$, determine the value of $$z^n + \frac{1}{z^n}$$ I have tried it with $n=2,3,4$ but it ...
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29 views

Find the locus of $w$

$$ \text{Find the locus of $w$, where $z$ is restricted as indicated:} \\ w = z - \frac{1}{z} \\ \text{if } |z| = 2 $$ I have tried solving this by multiplying both sides by $z$, and then using the ...
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1answer
379 views

Limit of complex numbers

What would be the limit of following term? $$\lim_{n \to \infty} \frac{e^{inx}}{2^n}$$ I tried to convert the $e^{inx}$ into trigonometric form and tried to do some simplification but got stuck ...
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1answer
114 views

Classification of Discrete Subrings of $\mathbb C$

I am interesting in classifying the subrings of $\mathbb C$ which are discrete with respect to the standard topology (that is, the topology induced by the standard absolute value). Here, I am using ...
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1answer
74 views

Find $(1+i)^i$ in simpler terms, without imaginary exponents. [duplicate]

I was asked to find $(1+i)^i$, I don't know what to do when there is an imaginary component in the exponent. since $1+i=\sqrt{2}e^{-\frac{1}{4}i \pi}$ then $(1+i)^i = \sqrt{2}^i e^{\frac{1}{4} \pi}$ ...
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2answers
176 views

How one can implement the equation with “$i$” in it?

I have an equation: $$f(t)=c(e^{i2\pi\frac{n}{T}t}+e^{-i2\pi\frac{n}{T}t})$$ ...for $t\in(-\pi,\pi)$, and with $T=2\pi$. I have to draw a plot of the function $f(t)$ for $n\in\left \{0,1,2,5 \right \}...