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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2answers
13 views

what is the set of points in the complex plane which satisfies |z| = Re(z) + 2?

what is the set of points in the complex plane which satisfies |z| = Re(z) + 2? so $ \sqrt{x^2+y^2} = x + 2 $ this is not a circle or anything and it asks me to sketch it what should I do
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2answers
60 views

Is $ze^{z}$ differentiable?

As the title says, I am looking at the function: $$f(z) = z e^{z},$$ and I want to know whether it is differentiable - and if so, to find its derivative (which I can do). Usually I would either ...
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0answers
17 views

What other methods can you use find the roots of a complex equation without using de Moivre's Theorem?

e.g. $z^n=x+iy$ is solved by $z=r^\frac1n[\cos(2\pi k+\theta) + i\sin(2\pi k+\theta)]^\frac1n$, where $r = |z|$ and $\theta=arg(z)$ Can you find $z$ without using the above formula?
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4answers
74 views

How in this world can I simplify this $\sqrt 2\cdot(1/(\sqrt2)-1/(\sqrt2)\cdot i)^{31}$ ????

I have a problem, obviously. I am doing some maths and now I have to simplify this: $\sqrt 2\cdot(1/(\sqrt2)-1/(\sqrt2)\cdot i)^{31}$ ????. But I just don´t know how ???? I´ve started simplifying by ...
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3answers
19 views

Solutions to the complex equation z^n=w with one solution given

In an old test paper $z_0=2+i$ is given as one solution to $z^4=-7+24i$ and we are asked to find further solutions. In the solution is given $$z_1=-z_0=-2-i$$ as $$z^4_1=(-1)^4z_0^4=z_0^4$$ I ...
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2answers
26 views

Condition on $k$ if $|z-z_1|^2+|z-z_2|^2=k$

How can we prove that $$|z-z_1|^2+|z-z_2|^2=k$$ will represent a circle if $|z_1-z_2|^2 \leq 2k$ Please give me some hints to initiate this question.
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0answers
23 views

Understanding complex multiplication as vector addition

I've been studying complex multiplication and vector math lately. Below is a visual representation of what happens when one multiplies a complex number $ 1 + xi $ by itself repeatedly, plotting the ...
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1answer
25 views

Plotting numbers in the complex plane using WolframAlpha

I read this article on understanding imaginary numbers as rotations of real numbers in the complex plane. Having read it, it's easy for me to see how the real number $1$ is simply a point on the real ...
0
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2answers
61 views

The number of solutions for n raised to a complex exponent

My understanding is that there is one and only one solution when solving for $z$ when $z = n^s$, where $s$ is a complex number of the form $a + bi$. However, there are many solutions to $z$ when ...
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1answer
35 views

Evaluate $\ln[(1+i)^7], \mathrm{Re}[\cos(1+i)]$ and $|e^{3+i\pi/4}|$

I am unable to verify my answers or methods for the 3 parts of this question, any input would be appreciated. I have attached my solutions below.
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1answer
14 views

Positivity for Hermitian dot product?

One of the requirements for the Hermitian dot product is positivity, i.e. $||v||^2 \ge 0, $ and $||v||^2 = 0 \iff v=0$. I was wondering what exactly this means in the complex numbers. Does it mean ...
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1answer
26 views

Proofing de Movire without Induction and in a neat way

The "usual way" gone for proving de Movire is via the road of induction. However this road get tiresome and thus wondered, if there were another way. However I came up with a proof that relies on ...
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1answer
57 views

Explain this complex number simultaneous equation step. [closed]

The explanation appears on this web page: OPs problem question Following through, I see everything until the move linking these two steps: $15a - 10b = 7a - 6b$ $8a = 4b$ What mathematical logic ...
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0answers
197 views
+500

Bounding a sum involving a $\Re((z\zeta)^N)$ term

This is a follow up to this question. Any help would be very much appreciated. Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$. Let ...
3
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2answers
51 views

For three complex numbers we have $|z_1|=1$ ,$|z_2|=2$ ,$|z_3|=3$ and $|9z​_1z_2 + 4z_1z_3 + z_2z_3|=12$

For three complex numbers we have: $|z_1|=1$ ,$|z_2|=2$ ,$|z_3|=3$ and $|9z​_1z_2 + 4z_1z_3 + z_2z_3|=12$ Then find value of $|z_1 + z_2 + z_3|$ I took $z_1=1(\cos A+i\sin A),z_2=2(\cos ...
0
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1answer
45 views

Convergence of series $\sum 2^n \sin\left(\frac{a}{3^n}\right)$ with $a$ complex numbers. [closed]

Find the complex numbers $a$ (if exist) for which is convergent the series $$\sum 2^n \sin\left(\frac{a}{3^n}\right)$$
1
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1answer
29 views

How to prove the following for an inner product space?

How to prove $\langle x,y\rangle=ax_1\bar{y}_1+bx_2\bar{y}_2$ defines an inner product of $\Bbb{C^2}\iff$$a,b\in\Bbb{R^+}$ $\Rightarrow$ if $\langle x,y\rangle=ax_1\bar{y}_1+bx_2\bar{y}_2$ defines an ...
2
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1answer
30 views

Convergence of $\sum\frac{(n-a)^2}{(n-b)^3}$ with $a,b$ complex numbers.

Find the complex constant $a, b$ for which $\sum\frac{(n-a)^2}{(n-b)^3}$ converges and diverges.
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2answers
24 views

Find all the points where $f$ is analytic with $f(z)= \frac{z^2+1}{(3z-1)(z-i+1)}$.

I start by expanding the denominator and separating the real and imaginary but get stuck when deciding what my $u$ and $v$ should be. Thanks.
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2answers
63 views

How to solve $\textrm i^y=y$? [duplicate]

What is the value of $\textrm i^{{\textrm i}^{\textrm i \dots}}$? My try: put $\textrm i^{\textrm i ^{\textrm i \dots}}=y$, so we get $\textrm i^y=y$. Now how to solve this equation?
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4answers
41 views

Evaluate $\Re[\cos(1+i)]$

Evaluate $\Re[\cos(1+i)]$. The trigonometric function in the expression is throwing me in a loop and need some guidance on how to evaluate this. Thanks.
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2answers
73 views

Real and Imaginary Parts of tan(z)

This is where I'm at: I know $$ \cos(z) = \frac{e^{iz} + e^{-iz}}{2} , \hspace{2mm} \sin(z) = \frac{e^{iz} - e^{-iz}}{2i}, $$ where $$ \tan(z) = \frac{\sin(z)}{\cos(z)}. $$ Applying the above, with a ...
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3answers
36 views

$|z_1+z_2|>|z_1-z_2|$ implies $-\frac{\pi}{2}<arg\big(\frac{z_1}{z_2}\big)<\frac{\pi}{2}$

For two complex numbers $z_1$ and $z_2$, it is given that: $$|z_1+z_2|>|z_1-z_2|$$ How could we prove that $-\frac{\pi}{2}<arg\big(\frac{z_1}{z_2}\big)<\frac{\pi}{2}$ If I take ...
0
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1answer
38 views

what does a line over a complex number mean?

I understand that z and w are complex numbers, but I don't understand what I need to calculate. What is the w with that line above? And is all this a product?
2
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1answer
43 views

Finding all complex numbers $z$ for which $z^2+az+a^2$ and $az^2+a^2z$ are both real

During my course on complex numbers, I found this problem: Let $a\in \Bbb C$. Find all complex numbers $z$ for which $z^2+az+a^2$ and $az^2+a^2z$ are simultaneously real numbers.
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1answer
86 views

Representing $ 2 i \sin(2 \pi n z) $ as a product

I'm currently reading a surprising proof of the Quadratic Reciprocity Law, which uses the following function: $ f(z) = 2 i \sin(2 \pi z) $ One of its properties is that $$\frac{f(nz)}{f(z)} = ...
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1answer
44 views

Radius of convergence of $\sum\limits_{n}c_{n}z^{n^{2}}$ given that the radius of convergence of $\sum\limits_{n}c_{n}z^{n}$ is finite and nonzero

I know that the radius of convergence of a given power series $\sum_{n=1}^{\infty}c_{n}z^{n}$ is $R$, where $0<R<\infty$. Given this information, I need to find the radius of convergence of ...
0
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1answer
47 views

$\sin(\pi/ n)\cdot \sin(2\pi/ n)\cdot\ sin( 3\pi/ n)\cdots \sin((n-1)\pi/ n)=n/(2^n-1)$ [duplicate]

I am just a beginner in Complex Numbers...and came across these sums: If $\ 1,\omega_1\ , \omega_2\ , \omega_3\ \cdots \, \omega_{n-1}\ $ are n nth roots of unity. Then, prove $\ ( 1-\omega_1) (1- ...
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2answers
70 views

Solving third degree polynomial $x^3+2x^2+6x+5=0$

The polynomial $f(x) = x^3+2x^2+6x+5$ has one integer root. Find the integral and the other roots. If the polynomial has an integer root, $a$, we can write: $$x^3+2x^2+6x+5=(x-a)(x^2+bx+c)=0$$ ...
1
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1answer
14 views

Is it possible to clear the x using the Lambert function?

$ y = \frac{x^2}{4} - \frac{ln(x)}{2} $ Solving, I get to: $ e^{4y} = \frac{e^{x^2}}{x^2} $ But I don't know how to continue.
1
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2answers
41 views

If $ x+iy = \sqrt{\frac{a+ib}{c+id} } ,$Show that$ (x^2+y^2)^2 = \frac {a^2+b^2}{c^2+d^2} $

$ x+iy = \sqrt{\frac{a+ib}{c+id} } , $Show that $ ({x^2+y^2})^2 = \frac {a^2+b^2}{c^2+d^2} $ How do i do this ?I tried squaring both sides but x+iy expansion becomes difficult when squaring the next ...
1
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1answer
34 views

Radius of Convergence of Complex Power Series

I need to find the radius of the convergence of $\sum_{n=1}^{\infty}3^{n}z^{n^{2}}$ using the Cauchy-Hadamard formula. I'm not feeling 100% proficient at this method, however, so I'm asking 1) if what ...
0
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1answer
134 views

Conformal mapping explanation?

Well I have a question that is "identify the two interesting points in the picture for the function $f(z) = z + 1/z$. Explain your answer." where it refers to the tool here: ...
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1answer
26 views

if $f(z) = e^z$ and $C$ is any curve joining $ -i\pi/2$ and $i\pi/2 $ find $ \int_C f(z)dz $

If $f(z) = e^z$ and $C$ is any curve joining $-i\pi/2 $ and $i\pi/2 $ find $$\int_C f(z)dz.$$ So $e^{z} = e^{x}e^{iy}$ and $z=z(y)$ then $z(a) = -i\pi/2$ and $z(b) = i\pi/2$. Then $$\int_C ...
0
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1answer
21 views

Finding an inverse of a matrix with entries in $F_3$

I am stuck on a question that asks to find the inverse of the following matrix with entries in $F_3$ $$ \begin{bmatrix} 1 & 0 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \\ \end{bmatrix} $$ ...
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0answers
35 views

Fundamental Theorem of Algebra - Complex Factorisation

Hoping someone could help me. As part of a question I am asked to solve the equation: $z^5=-1$ I have done this and have a set of solutions $z_0, z_1, z_2, z_3, z_4$. For part (b) I am asked to ...
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2answers
37 views

Show that the second derivative $\Gamma''(x)$ is positive when $x>0$

Let $\Gamma(x)=\int_0^{\infty}t^{z-1}e^{-t}dt$. I know that the first derivative is positive, since $\Gamma(x)$ is increasing when $x>0$, but I don't know how to show that the second derivative is ...
0
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1answer
43 views

Evaluation of successive powers of a complex number

The problem calls for a deduction of the equation $$1+z+z^2+z^3+z^4=0$$ Where $z=\cos\frac{2\pi}{5}+i\sin\frac{2\pi}{5}$. Hence ...
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1answer
38 views

Will there be two square roots for a Complex number?

We know that in real numbers $\sqrt{x^2}=|x|$. But in complex numbers my query is we can have two square roots. For example in my book the question is to find the value of $$z=\sqrt{i}-\sqrt{-i}$$ I ...
1
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2answers
45 views

Sum of the n nth roots of unity

1) If $z=\cos\frac{2\pi}{n}+i\sin\frac{2\pi}{n}$ 2) Then apparently: $z^0+z^1+z^2...z^{n-1}=\frac{1-z^n}{1-z}=\frac{0}{1-z}=0$ Could someone please explain why: ...
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2answers
44 views

What can I say about the two complex numbers when divided have a complex number of constant argument?

Suppose there are two complex numbers $Z_1 \ and \ Z_2$ We are given that : $$arg(\frac{Z_1}{Z_2}) = k$$ k is an arbitrart constant. Is there any way to visualise the complex number ...
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2answers
16 views

Differential equation using $n$th order formula

Use the formula for the $n$th roots of a complex number to find the solution to the following differential equation. $y^{(6)}-64y=0$ $y^{(6)}$ is the $6$th derivative
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0answers
21 views

Given complex numbers raised to real powers, what do imaginary part = 0 level sets look like?

Suppose you lay flat (horizontally) the Argand plane, with x real. Consider the vectors v in that plane. Then let z, the vertical axis, be the real part of the complex solution to v^t, with t real. So ...
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0answers
37 views

Geometric interpretation of the determinant of a complex matrix

A complex $n$-dimensional vector space $V$ can be thought of as a real $2n$-dimensional vector space equipped with a map $J:V \to V$ with $J^2 = -I$. Complex-linear maps are then linear maps $V \to V$ ...
1
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1answer
34 views

Complex inner product proof

I have just solved this problem in the real inner product space with $\langle \cdot , \cdot \rangle$ as the inner product. Show that in a real inner product space we have: $\langle x,y \rangle = ...
0
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0answers
28 views

Sketch subset of $\mathbb{C}$ which satisfies $|z-3-4i|=5$

I proceeded by plotting $z$ on the complex plane, and the modulus of $z-3-4i$: From this I deduced that: ...
1
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1answer
22 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 ...
0
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1answer
39 views

The equation $ |z-1| = 5* |z-i|$ represents the equation of circle or elipse?

The equation $ |z-1| = 5* |z-i|$ represents the equation of circle or elipse ? My approach : Let $|z| = (a+ib) \implies$ $|(a-1) + ib| = 5*|a+(b-1)i|$ $\therefore$ I get $24a^2 + 24b^2 +2a -50b +49 ...
1
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1answer
58 views

Why this book says that $ 2^{1/2} = ±\sqrt{2} $?

Shouldn't it be: $ 2^{1/2} = \sqrt{2} $ ? I know the problem is that there they are working with complex numbers, but I still don't understand. The book is in the link, page 113, when they move ...
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1answer
49 views

Can anyone teach me how to answer this question??? [closed]

$\frac{x}{1-x} = \frac{{x}^{a_1}}{1-x^{d_1}}+\cdots+\frac{{x}^{a_n}}{1-x^{d_n}}$ for all $x\neq 1$.Then show that $d_1,\cdots,d_n$ cannot be all distinct.Also note that $a_1,\cdots,a_n \in \mathbb{N}$ ...