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

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$GL_2(\mathbb{C})$ acting on extended complex numbers.

Let $GL_2(\mathbb{C})$ the general linear group of order two on complex. We can define a action from $GL_2(\mathbb{C})$ on $\mathbb{C}^*$ as ...
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4answers
111 views

$(-1)^{0.2}=0.8090 + 0.5878i$ how can this be?

I'm working on a numerical analysis project (working with matlab a lot) and I noticed that when I ask for matlab to compute the exponent of a negative number, it gives wrong output when the exponent ...
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2answers
45 views

Find $c$ if $a,b, \; c$ satisfy $c = (a+bi)^3 - 107i$

Find $c$ if $a,b, \; c$ are positive integers which satisfy $c = (a+bi)^3 - 107i$ I can try expanding the cube, but that seems too direct. What other ways are there to go about this?
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3answers
117 views

Computing complex number [duplicate]

"Compute $(1 + i)^{1000}$. So far I have: $(1+i)^{4 (2^2 5^3)} $ but I am not sure how to proceed. Ideas?
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2answers
62 views

Motivational example for complex numbers

Years ago I was introduced to complex numbers. In class we had been talking about the cubic polynomial and its solutions. At one point we saw an example where, when using the formula, one had to stop ...
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1answer
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Interesting examples of Cauchy's Integral formula [closed]

Question : What are some interesting and, albeit, counter intuitive examples of real integrals that are solved using Cauchy's integral Formula. Cauchy integral formula can magically transform some ...
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2answers
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2x2 inverse of a complex matrix with complex determinant

Firstly, my question may be related to a similar question here: Are complex determinants for matrices possible and if so, how can they be interpreted? I am using: $$ \left(\begin{array}{cc} a&b\\ ...
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2answers
103 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
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2answers
98 views

Complex series radius convergence

How to find the values for which $z$ converges, $z\in\mathbb{C}$, in the serie $$\sum_{n=1}^{\infty}\frac{1}{(1+|z|^{2})^{n}}$$ I know I have to use the convergence radius expression, but what I ...
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4answers
68 views

Complex Numbers: Im$(\frac{12}{z-7})=1$

Sketch and describe the set of complex numbers satisfying $$Im(\frac{12}{z-7})=1$$ where $z=x+iy$ The answer should be in circle form. Here is what I have so far: $$Im(12)=z-7$$ $$Im(12)=x+iy-7$$ ...
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complex ordinary differential equation of a real variable

reading a paper i have found the following differential equation: f''[z] - (q^2)*f[z] == i*DiracDelta[z] here f[z] is a complex function of the real variable z ...
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3answers
51 views

Square root of a squared number changes sign, which to apply first?

Heres something Ive always found interesting. Supose we have a variable $x$, and $x$ equals a negative number: Say: $$x=-17$$ Now, I can apply a square to both sides of the equation and preserve ...
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2answers
34 views

find the cube roots of -8i and plot them on a plane

I can't figure out the angle of this equation. I set it up like this: $$z^3=0-8i$$ I find that the r value is 2, but when I try to find the angle I'm stuck. I can't divide by 0, so where did I go ...
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1answer
137 views

Quick complex number proof question:

How would I go about proving the following identity: $$\frac{1}{\left|z\right|} = \left|\frac{1}{z}\right|$$ I keep finding myself going in circles. I've tried using this identity: $|z|^2 = z^*z$ ...
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7answers
224 views

Is $\mathbb{C}$ equal to $\mathbb{R}^2$?

Complex numbers are usually formally defined as pairs of real numbers. Although there are operations on $\mathbb{C}$, such as complex multiplication, which are not found in operations usually applied ...
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1answer
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Solving $|z-3| \leq|z-1-i|$

I was trying to solve graphicly: $$|z-3| \leq |z-1-i|$$ I plugged x and y in proper places as real componenets of the comlex number yielding in the end $-4x+2y+7 \leq0$ this might be tackled if ...
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1answer
38 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
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1answer
48 views

Evaluate complex number ratio

$$ \frac{35887+j(1050)}{-2824+j(-17)} \ = \ ? $$ This above number is supposed to be the sprung mass response factor to road input at frequency of 6.91 radians/second for the front suspension of a ...
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3answers
59 views

On a certain series of complex numbers

Is it possible that the above infinite series is equal to ?
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1answer
20 views

The cross-ratio of four points is a real number exactly when the four points lye on a line or a circle

Let $z,z_1,z_2,z_3$ be four points on the extended plane. Their cross-ratio $(z,z_2,z_3,z_4)$ by definition is the image $Tz$ of $z$ under the Möbius transformation $T$ that sends $z_1,z_2,z_3$ to ...
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2answers
27 views

Express $\sin 3\theta$ and $\cos 3\theta$ as functions of $\sin \theta$ and $\cos \theta$ using Euler's identity

Using Euler's identity ($e^{in\theta}=\cos n\theta+i \sin n\theta$), express $\sin 3\theta$ and $\cos 3\theta$ as functions of $\sin \theta$ and $\cos \theta$. Any ideas?
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1answer
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Complex exponent integral - prove $\int_a^b{e^{\lambda x} \text{dx}}=\frac{1}{\lambda}\left(e^{\lambda b}-e^{\lambda a}\right) $

How to prove the exponent integration rule: $$\int_a^b{e^{\lambda x} \text{dx}}=\frac{1}{\lambda}\left(e^{\lambda b}-e^{\lambda a}\right) $$ In the complex version of it - that is, when $\lambda \neq ...
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1answer
61 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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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 $res(f, -2)$??
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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 ...
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1answer
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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
40 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 ...
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1answer
36 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 ...
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1answer
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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 ...
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1answer
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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
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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 / ...
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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
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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 ...
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1answer
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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 ...
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32 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}\\ ...
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2answers
53 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, ...
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4answers
131 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 ...
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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 ...
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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 ...
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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 ...
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1answer
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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) ...
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2answers
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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 ...
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1answer
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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
32 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
128 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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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
92 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 $ ...
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1answer
77 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 ...
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1answer
51 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 ...