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

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2
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1answer
62 views

Singularities of complex functions.

How do I determine the singularities of a function? What is a singularity? In the functions below which are the singularities? a)$$f(z)=\frac{1}{(z^4+2z)}$$ b)$$f(z)={e^{1/z}}$$
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3answers
42 views

Determining Laurent Series expansion and residues

Determining Laurent Series expansion and residues of $f(z)=\frac{z}{(z+1)(z+2)}$ around $z = -2$. What is the validity of the expanded region? What is $res(f, -2)$??
2
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1answer
67 views

Alternative definition of complex number, showing it is equivalent to the tradidional one.

The author of a book makes an alternative definition of the complex numbers, later he shows that this definition is equivalent to the ordinary definition where we define $i^2=-1$. Here is his ...
3
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1answer
73 views

Why $\ln(1)\neq 2\pi ik$

Given that $e^{2\pi ik}=1$ for all $k \in \mathbb{Z}$, why isn't $\ln{e^{2\pi ik}}=2\pi ik$? On the other hand $\ln(1)=0$. What am I missing here?
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4answers
62 views

Which way will produce the following integral?

Which way $\gamma$ will produce the following integral? $$\int\limits_{\gamma}\frac{3+i}{z^5 - z}dz = 0$$
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3answers
41 views

Complex solutions of polynomial question

$2z^3-6z^2+mz+n = 0$ $m, n$ are real and $1+\sqrt{ 2} i$ is a solution. Find $m$ and $n$. Attempt to solve : Giving the known theorem $1-\sqrt{2}i$ is also a solution, so we can substitute each time ...
0
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1answer
37 views

Power sum of two complex numbers

Let $a + b i$ be a complex number whose absolute value is greater than $1$ and whose argument is not a rational multiple of $\pi$ . For $n = 1, 2, 3, \cdots$ define $f(n) =| (a + b i ...
0
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1answer
24 views

complex conjugate pairs of a quartic

I tried my hand at this question, which included finding the partial fractions of $\frac{x^2}{1-x^5}$. I found a factor of $1-x$ for the denominator, but I do not know how to work out the complex ...
0
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1answer
32 views

Solve: $z^4 +2\sqrt3 +2i = 0$

Solve: $z^4 +2\sqrt3 +2i = 0$ I'm already trying to solve this exercise for $20$ minutes, no luck. I got up to here: $z^4 = -2i -2\sqrt3 = -2(\sqrt3 + i)$ but it's impossible to compute from here. ...
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1answer
32 views

Problem calculating the argument of a complex variable

In Signals & Systems 2nd Ed. written by A. V. Oppenheim, there is a result of Fourier transformation: $ \begin{align} H (j \omega) = \frac{1 + (j \omega / \omega_{0})^2 - 2 j \zeta (\omega / ...
-1
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3answers
60 views

complex roots calulation question

How can we find the roots of an equation such as:$z^2 +z +1=0 ,z \in \mathbb{C} $ ?
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2answers
86 views

Find the roots of the equation $(1+xi)^n+(1-xi)^n=0$

Find the roots of the equation $f(x)=(1+xi)^n+(1-xi)^n=0$. I'm having problems finding the roots...this is what I've done: First I expressed $(1+xi)^n$ and $(1-xi)^n$ in trigonometric form and ...
0
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1answer
27 views

Prove that a Möbius transformation $T$ sends the imaginary line to the circle $\{z: |z|=2\}$,

Problem Let $T:\overline{\mathbb C} \to \overline{\mathbb C}$ be a Möbius transformation such that $T(1+2i)=1$, $T(-1+2i)=4$ and $|T(0)|=2$. Show that $|T(bi)|=2$ for all $b \in \mathbb R$. The ...
0
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2answers
35 views

Cartesian $-10i$ to Polar form

I am trying to convert the following problem to polar form: $$z=-j10.$$ Using this equation, where $|z|=r=\sqrt{x^2+y^2}$ and $\arg z=\theta=\arctan(y/x).$ $$\eqalign{z&=|z|e^{j\arg z}\\ ...
1
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2answers
54 views

Solve $(2z-1)^5 - i = 0$

Solve $(2z-1)^5 - i = 0$ I started by saying that $(2z-1)^5 = i$ $(2z-1) = \sqrt[5]i$ $z =$ $(\sqrt[5]i +1) \over 2$ $z^5 =$ $(i +1) \over 32$ $z^5 =$ $1 \over32$$ *(i +1)$ From there, ...
3
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4answers
139 views

Picture/intuitive proof of $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$?

Is there a nice geometric, intuitive or picture proof as to why the easily algebraically provable identity $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$ is true? Note I'm not looking for a ...
1
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2answers
49 views

Product of $n$ complex numbers in rectangular form.

Given a complex number $z_j$ such that $$z_j\in\{a_1+b_1 i,\ a_2+b_2i, \ ...\ ,a_n+b_ni\}$$ is there formula for calculating $$z_1 \cdot z_2 \cdot \dots \cdot z_n =\prod_j z_j?$$ For two complex ...
0
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3answers
55 views

Complex Numbers and Linear Algebra

Explain why there does not exist a $\lambda $ in the Complex Field such that $$\lambda \left(2-3i, 5+4i, -6+7i \right) = \left(12-5i, 7+22i, -32-9i \right)$$ Can someone help me figure out how to go ...
2
votes
4answers
66 views

$f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set $f(A)$

Let $f:\mathbb R^2\to\mathbb R^2$ be defined by the equation $$f(x,y)=(x^2-y^2,2xy).$$ Show that $f$ is one to one on the set $A$ consisting of all $(x,y)$ with $x>0$. What is the set ...
1
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1answer
46 views

An Integration Calculation

I'm just having a bit of difficulty understanding the last couple of steps made in the paper Horowitz & Hubeny - Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium (p.8) ...
2
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2answers
46 views

Complexes question $(z-1) \over (z+1)$

Given $z \ne -1$. Prove that $(z-1) \over (z+1)$ is an Imaginary number if and only if $|z| = 1$. I tried computing $(z-1) \over (z+1)$ by multiplying like that: $(z-1) \over (z+1)$$(z-1) \over ...
1
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1answer
25 views

Help on a complex number equality problem

Find the general value of $\theta$ which satisfies the equation $\displaystyle (\cos\theta+i\sin\theta)(\cos2\theta+i\sin2\theta)...(\cos n\theta+i\sin n\theta)=1$ My thoughts: Simplest answer is ...
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2answers
36 views

Proving a Complex number equality

To Prove: If $\displaystyle p=\operatorname{cis}\theta =\cos\theta+i\sin\theta$ and $\displaystyle q=\operatorname{cis}\phi =\cos\phi+i\sin\phi$, then show that $\displaystyle ...
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3answers
134 views

Complex number equality

To Prove: $\displaystyle (\cos\theta +i\sin\theta)^4(\sin\theta-i\cos\theta)=\cos 8\theta+i\sin 8\theta$ My Attempt: $\displaystyle (\cos4\theta ...
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2answers
21 views

Help on Algebraic manipulation of a complex number

If $\displaystyle \frac{1}{x+iy}+\frac{1}{u+iv}=1$; x,y,u,v being real quantities, express v in terms of x and y. My Attempt: $\displaystyle \frac{(u+x)+i(y+v)}{(x+iy)(u+iv)}=1$ $\displaystyle ...
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3answers
124 views

Express a complex number in modulus amplitude form

Express a complex number in modulus amplitude form $\displaystyle 1+\sin \alpha +i\cos \alpha $ My Attempt: $\displaystyle r\cos \theta= 1+\sin \alpha $ $\displaystyle r\sin \theta= \cos \alpha $ ...
3
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1answer
79 views

Fourier series without Fourier analysis techniques

It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula ...
3
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1answer
54 views

Rewriting this complex square root

for some reason I can to figure out how to rewrite this square root. I have: $\sqrt{2+i}$ And I need to rewrite it into: $\frac{\sqrt{2(\sqrt{5} + 2)} + \sqrt{-2(\sqrt{5} - 2)}}{2}$ Can anybody ...
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1answer
58 views

Find the 6th root of $-3+4i$ and plot on complex plane

So I have a rough idea on how to get the answer but I'm getting stuck on the angle or argument for the equation. The question is: Find the 6th root of $-3+4i$. I first find the $r$ value which ...
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0answers
25 views

Curves composition with holomorphic function

Statement $(i)$ Let $\gamma:\mathbb R \to \mathbb C$ a $C^1$ curve. Let $v={\gamma}'(t_0)$ the complex number that one obtains from translating to the origin the tangent vector to $\gamma$ at ...
8
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5answers
467 views

What's the importance of a formula for the real and imaginary parts of a complex number?

I've learned that $$\bbox[8px,border:1px solid black]{\operatorname{Re}(z)= \frac{z+\overline{z}}{2} \qquad \qquad \operatorname{Im}(z)=\frac{z-\overline{z}}{2i}} $$ And that in the number $z=a+bi$, ...
2
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1answer
49 views

Do sine and cosine of complex numbers have anything to do with right-triangles or circles?

I've recently been working on a web application that draws iterating function generated fractals. I've noticed that the sine and cosine functions can be used to draw exquisite plots using an ...
6
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5answers
219 views

How do i find $(1+i)^{100}?$

How do I find $(1+i)^{100}$ without expanding $(1+i)$ 100 times? Is there a quicker way to do this? The hint was to find the modulus and argument of $1+i$ which I've got as $\sqrt{2}$ and $\pi/4$ ...
0
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1answer
21 views

Quadratic factor to complex numbers

How to convert this quadratic factor to complex number form? (With steps please) Reference: $Z = a + bi$, $i = \sqrt{-1}$ $$-3 + \frac{\sqrt{-12}}{2}$$ Thanks!
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0answers
41 views

Why Does $e^{ix}=\cos(x)+i\sin(x)$? [duplicate]

Something I've always wondered, but never had a good answer too (I accept there may not be one). I fully understand how to derive this, so I'm not looking for an analytic proof. But rather I cannot ...
3
votes
1answer
51 views

Finding Möbius transformations that satistfy certain conditions

Problem Find Möbius transformations that send $(i)$ the circle $|z|=2$ to $|z+1|=1$, and $-2$ to $0$, $0$ to $i$. $(ii)$ the upper half-plane $Im(z)>0$ to $|z|<1$ and $\lambda$ to $0$ (where ...
0
votes
1answer
34 views

Complex conjugate root theorem question

From the Complex conjugate root theorem we get that if a polynomial in one varaible with real coefficients has as solution $a + bi$ , than $a-bi$ must also be a solution...however, what happens if ...
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4answers
352 views

Is it true that “there is no such thing as the square root of minus one”?

Is the statement "there is no such thing as the square root of minus one" a true statement? It seems to me that we need to be careful about the word "the" as it appears in the statement. If we see it ...
0
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1answer
65 views

Complex numbers: How to solve the “contradiction”? [duplicate]

$$-1 = i\cdot i = \sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1$$ $$-1 = 1$$ Obviously, something is wrong here, but I can't put my finger on it. How to solve this "contradiction"?
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2answers
30 views

Limit of complex numbers' sequence (related to Möbius transformation)

Problem Let $T(z)=\dfrac{7z+15}{-2z-4}$. Let $z_1=1$ and $z_n=T(z_{n-1})$ for $n\geq 2$ Find $\lim_{z_n \to \infty}z_n$ I am having a lot of difficulties trying to solve this. I've tried to find a ...
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2answers
42 views

Sketch the set $\{ z \in \mathbb{C} | \left|\frac{z-i}{z+i}\right|<1 \}$

My question is to sketch the set $\{ z \in \mathbb{C} | \left|\frac{z-i}{z+i}\right|<1\}$ in the complex plane. I substituted $z$ for $a+bi$, but did not get anywhere: ...
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0answers
21 views

Is alpha and endomorphism of C considered as avector space over R? Is it an endomorphism of C considered as avector space over itself?

Let alpha:C$\to$C be the function defined by alpha:a+bi$\to$ -b+ai.(1) Is alpha and endomorphism of C considered as a vector space over R?(2) Is it an endomorphism of C considered as a vector space ...
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0answers
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Usefulness of alternative constructions of the complex numbers

Complex numbers $\mathbb{C}$ are usually constructed as $\mathbb{R}^2$ together with a suitable multiplication. But this is not the only possible way, one can get to the complex numbers. One ...
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0answers
29 views

Controlling the Sum of a Set of Complex Numbers

Consider a set of N previously fixed angles $\phi_i$. Let $p$ be a positive integer. If $\sum^N_{i=1} e^{ip\phi_i} = 0$, what if any restriction does this place on the value of $p$? If $\phi_i = 2\pi ...
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2answers
52 views

Finding the modulus of a complex number that satisfies a polynomial relation

Consider $z\in\mathbb{C}$ such that $z^2-2z+3=0.$ Find the modulus of $$f(z)=z^{17}-z^{15}+6z^{14}+3z^2-5z+9$$ My attempt: $z^2-2z+3=0\Leftrightarrow\left[ ...
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1answer
40 views

Cross ratio and symmetric points exercise

Problem Let $C$ be a circle or a line belonging to $\overline{\mathbb C}$ and let $z_2,z_3,z_4$. Two points $z$ and $z^*$ are said to be symmetric with respecto to $C$ if ...
4
votes
2answers
39 views

Find $zw, \frac{z}{w},\frac{1}{z}$ for $ z=2\sqrt{3}-2i, w=-1+i$

Find $zw, \frac{z}{w},\frac{1}{z}$ for $ z=2\sqrt{3}-2i, w=-1+i$ I went wrong somewhere, this is what I have so far (this is in polar): ...
0
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1answer
45 views

Möbius transformations on $\space \overline{\mathbb R}$

Prove that a Möbius transformation $T(z)=\dfrac{az+b}{cz+d}$ maps $\overline{\mathbb R}$ to $\overline{\mathbb R}$ if and only if it can be written with real coefficients. If it can be written with ...
0
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2answers
54 views

$\int_0^\pi\sin(2t)e^{-in2t}dt$ complex number integral for integer values of n

$$\int_0^\pi\sin(2t)e^{-in2t} \, dt$$ wolfram alpha say the answer is $$\frac{1-e^{-2 i n π}}{2-2 n^2}$$ although using the integral trig identity $$\int ...
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2answers
63 views

Solve for x if $z$ is a complex number such that $z^2+z+1=0$

I was given a task to solve this equation for $x$: $$\frac{x-1}{x+1}=z\frac{1+i}{1-i}$$ for a complex number $z$ such that $z^2+z+1=0$. Solving this for $x$ is trivial but simplifying solution ...