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

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Functions of complex numbers

Let $f(z) = \sqrt z$. Using the branch that is defined everywhere except where $z = x + iy$ with $y = 0$ and $x < 0$. What are the formulas for real valued functions $u(x, y)$ and $v(x, y)$ such ...
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27 views

Numerically find all zeros of multivariate function

How do I find all zeros of a multivariate function , i.e. f(x1,x2,x2,...xn)=0 numerically? I don't know exact analytic form of f , but can numerically compute f at every point on its domain. ...
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35 views

Calculus/Complex Numbers exercise question.

Let $f$ be an increasing function in $\mathbb{R}$ and $f(-1)>0$. Let $z=\frac{f(-1)f(0)}{2}+ \frac{4\sqrt{3}}{f(1)}i$ with $|z|=\frac{z}{2}(1-\sqrt{3}i)$. Show that: i) $f(-1)f(0)f(1)=8$ ii) ...
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Problem on Complex numbers involving a point on a Circle

Question: The Complex number $z$ is represented by the point $T$ in the Argand Diagram.Given that $$z =\frac{1}{3+it}$$ where $t$ is a variable, show that i) as $t$ varies, $T$ lies on a circle, and ...
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53 views

Sequence of alternating $0$'s and $1$'s in terms of $i$?

How to redefine the function $f(n) = \begin{cases} 1, & \text{if $n$ is even} \\ 0, & \text{if $n$ is odd} \end{cases}$ in terms of arithmetic operations using ⅈ?
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24 views

product of the lengths of the segments $|AC|\bullet |BC|$ will be maximal.

Let there be a segment $AB$ the Diameter of the circle $S(0,1)$. Find all the points $C$ that belong to the closed circle $D^-(0,1)$ such that the product of the lengths of the segments $|AC|\bullet ...
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2answers
59 views

Show that $\Sigma_{n=1}^\infty |z_n| $ converges.

Assume $z_k = |z_k|e^{i\alpha_k}$ are complex numbers and that exists $0<\alpha<\pi/2$ s.t $\forall k -\alpha < \alpha_k<\alpha$ assume that $\Sigma_{n=1}^\infty z_n $ converges. we ...
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20 views

Show that $f'_n \rightarrow f'$ uniformly on every compact set. [duplicate]

Assume that $f_n,f:D(0,1)\rightarrow C$ are holomorphic. and $f_n \rightarrow f$ uniformly on every compact set. we need to show that $f'_n \rightarrow f'$ uniformly on every compact set. I kind of ...
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9 views

Find all Mobius transfers $T$ s.t $T(\Re)=\Re$ and $T(i\Re)=i\Re$

The question is: Find all Mobius transfers $T$ s.t $T(\Re)=\Re$ and $T(i\Re)=i\Re$ Now I've done some calculations and got that $T(z)=1/z$ but that is not enough..there are more and I'm not sure how ...
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4answers
122 views

$z^{10}+\frac{1}{z^{10}}=?$

$z$ is a complex number and $z^2+z+1=0$. $$z^{10}+\frac{1}{z^{10}}=?$$ For the solution: the roots of $z^2+z+1$ are: $z_1=-\frac12+\frac{\sqrt3}{2}i$ and $z_2=-\frac12-\frac{\sqrt3}{2}i$ ...
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465 views

What does a complex root signify?

What does it tell me when I find that a polynomial has complex roots, except for the obvious fact that it crosses zero for these values? To me it seems that the existance of complex roots must have ...
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1answer
48 views

A number theoretic inequality

Let $a, b$ be any two positive real numbers such that $a\geq lb$ where $l\geq 1.$ Suppose $\gamma $ is any real such that $0\leq \gamma\leq 2\pi.$ Is it true that $$|l+e^{i\gamma}|(a+mb)\leq ...
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12 views

Showing the Summation of $(\frac{w}{2})^k$ where w is a complex root

I got the correct answer for (i) and (ii) and the problem is with third part. I cant find my mistake. Since the third part is related to the second part, I will mention its answer. The ...
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1answer
24 views

Prove that $\bar z_1 z_2+z_1 \bar z_2=2\Re(\bar z_1 z_2)$

Let us consider $z_1, z_2\in \mathbb C$; we have: $$\bar z_1 z_2+z_1 \bar z_2=2\Re(\bar z_1 z_2)$$ it is easy to prove if we put $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$. But suppose we do not want to use ...
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1answer
31 views

Solve for $z$ in $\sin(iz) = 4i$

When solving this equation, I reach a point where $$e^{2z} =\frac{1-8i}{65}$$ Now I'm pretty sure this is correct since this is what the memo has as well. But what I would like to know is why I ...
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2answers
29 views

how to find cube roots of complex number

$c^3 = -1 + i$ How does one calculate the complex number(s) that satisfy above. Would like a general method, if there is one?
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6answers
50 views

Which of the following is equivalent to the expression? $i^{22}$

Which of the following is equivalent to the expression? $i^{22}$ A.) $-1$ B.) $i$ C.) $1$ D.) $-i$ What is $i$? How could it have a exponent if it's an imaginary number?
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72 views

for $z,w\in\mathbb{C}$, $\sqrt{zw} = \sqrt{z}\sqrt{w}$?

for $z,w\in\mathbb{C}$, $\sqrt{zw} = \sqrt{z}\sqrt{w}$ I started by writing $z$ and $w$ in polar coordinates, and writing it out, giving another form for the question: $$ \begin{align} y &= ...
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Prove that the following function tends to infinity as z approaches i (complex analysis)?

Prove: $\lim_{z\to i} [2 /({1 + z^2})] = \infty$ My attempt: We want $M > 0$ such that if $0 < |z - i| < \delta$ then $|2/(1 + z^2)]>M$ Now $|2/(1 + z^2)|>M$ ...
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Complex exponent problem

Find all numbers in complex plane that solves equation $e^z=4i$ Since $e^u e^{iv}=re^{i\Theta}$ it must be that $e^u=r \to u=\ln4$ and $v=\Theta+n2\pi \to v=\pi/2+n2\pi$. So the equation holds for ...
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Branches of complex logarithmic function

Attach analytic branch of some complex logarithm function and find range (image) of set if possible radius $\arg z =\pi /6$ positive $y$ axis circle $|z|=1$ ring $3 \leq |z| \leq5$ I'm not sure ...
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complex identity [duplicate]

I am trying to show that: $|e^{ix}-1|\le|x|$, x is a real-valued. First I tried Taylor series: ...
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3answers
33 views

Complex numbers proof problem

If $|z|,|w| \leq 1$, show that $|z-w|^2 \leq (|z|-|w|)^2 + (\arg(z)-\arg(w))^2$, where $z,w$ are complex numbers. How can I solve such a problem?
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120 views

Does $2^i$ exist, if so how do I calculate its value?

I was working on the question, to show that $i^i$ is a real number. That was however straight forward, $$i = e^{i\frac{\pi }{2}}$$ so $$i^i = (e^{i\frac{\pi }{2}})^{i}= e^{-\frac{\pi }{2}}$$ So ...
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3answers
51 views

Complex conjugate to the power proof

How can I proof using math induction that $$\overline{z^n} =\overline{z}^n$$ where $z$ is a complex number, $n$ is a positive whole number.
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3answers
52 views

How can I simplify the polynomial $x^4+1$ into quadratic factors? [closed]

The teacher gave us a hint that this polynomial expression can be written as the multiplication or sum of quadratic factors at the most. How can I do this?
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Condition for 3 complex numbers to represent an equilateral triangle

$z_1$, $z_2$, and $z_3$ are 3 complex numbers. Prove that if they represent the vertices of an equilateral triangle then $z_1 + \omega z_2 + \omega^2 z_3 = 0$ where $\omega$ is a 3rd root of unity. ...
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Linear Algebra - args complex number question

I need to solve this problem : $$z^3-(2+2i)^2=0$$ This is what I did : $$z^3 = (2+2i)^2$$ $$z^3 = 8i$$ The formula for args is : $$\tan(args)=\frac{b}{a}$$ in this case its clear that the args is ...
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1answer
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Proving that an analytic function that maps on to {$z\in \mathbb{C}| |z-2|=1$} from some connected open set is constant

This is the approach I took to solve this but I got stuck. Suppose$f=u+iv\in $ {$z\in \mathbb{C}| |z-2|=1$} and that $f$ is analytic on an open connected set. Then we have that $(u-2)^2+v^2=1$. ...
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61 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
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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
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4answers
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$\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... ...
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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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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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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 ...
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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 ...
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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} ...
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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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
38 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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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?