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

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2answers
27 views

Re z/z continuous at z=0

How would I show that Re z/z is continuous at z=0? I know that the real value of a complex number equals the sum of the real number and its conjugate divided by two, but I'm not sure where to go when ...
0
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2answers
64 views

Algebra - Gaussian integers

Let $\mathbb{Z}[i]=\{ a+bi : a,b \in \mathbb{Z}\}$ be the ring of Gaussian integers. Let $x,y \in \mathbb{Z}[i]$ with $y \neq 0$. Show that there exist $q,r \in \mathbb{Z}[i]$ such that $x = yq + r$ ...
1
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1answer
59 views

Matrices and Complex Numbers [duplicate]

Given this set: $$ S=\left\{\begin{bmatrix}a&-b\\b&a\end{bmatrix}\middle|\,a,b\in\Bbb R\right\} $$ Part I: Why is this set equivalent to the set of all complex numbers a+bi (when both are ...
2
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2answers
88 views

Given that $x$ is a rational number, is $\sin(x\pi)$ always expressible through radicals?

This is a theory I just thought of and I am wondering if there is truth to it. Here is the logic that I am working upon: Using Euler's formula, you can deduce that $$ (-1)^x = ...
0
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0answers
32 views

What is the name for the operation of swapping the two components of a complex number (rectangular form)?

I wonder if there is a name for the operation of swapping the real and imaginary part of a complex number.
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1answer
104 views

Solve the recurrence relation $f(n) = f(n - 1) + f(n - 2) + f(n - 3)$

This is a problem I was playing with that troubled me greatly. $f(n) = f(n - 1) + f(n - 2) + f(n - 3)$ $f(1) = f(2) = 1$ $f(3) = 2$ So, the goal is to try and find a solution for f(n). I tried ...
2
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2answers
445 views

Tangent at average of two roots of cubic with one real and two complex roots

I was able to easily prove that the tangent at the average of two roots of a real cubic polynomial passed through the third root of the function. But I have only done this for functions with three ...
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2answers
39 views

$\text{Im}(z)$ in equation

I'm having trouble with this equation: $$\text{Im}(-z+i) = (z+i)^2$$ After a bit of algebra i've gotten: $$1-\text{Im}(z) = z^2 + 2iz - 1$$ But i have no clue where to go from here, how do i get ...
2
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1answer
75 views

Exponentiation of imaginary operator

It is very easy to prove that if $D=\dfrac{d}{dx}$, then $(e^{nD}f)(x)=f(x+n)$ about $x=m$ in the real numbers. Proof: $$(e^{mD}f)=\sum^\infty_{n=0}\dfrac{D^nf}{n!}m^n\\ \implies ...
2
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1answer
39 views

Representation of cardiod in the complex plane

I noticed that the complex function $$f(z) = \frac{2}{(z+i)^2}$$ seems to map the real line onto the cardioid given by the polar equation: $$r = 1- \cos(\theta).$$ I was wondering if there is a simple ...
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6answers
653 views

Extending the set of complex numbers

Mathematics as a science became richer when Cantor invented the real numbers. Then scientists wanted to solve equations which were not solvable in the real numbers so they invented the complex ...
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2answers
57 views

How to compute the sine of a complex number in floating-point arithmetic?

What is the most efficient way to numerically compute the sine of a complex number? Suppose I want to calculate the sine of a complex number a + bi on a computer. Suppose that a and b are both ...
1
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1answer
46 views

The principle argument of the product of two complex numbers in the second quadrant

I would like some help to prove the following: Show that, if Re $z_1<0$, Im $z_1>0$, Re $z_2<0$ and Im $z_2 >0$, then Arg$(z_1z_2)=$Arg$(z_1)+$Arg$(z_2)-2\pi$. Thanks for any help in ...
2
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3answers
64 views

Find the locus of $2/z$ given that $|z-(1+i)| = 2$

If complex numbers $z$ satisfy the equation $|z-(1+i)| = 2$ and $\displaystyle \omega = \frac{2}{z}$, then locus traced by $\omega$ in complex plane, is ... My try I want to solve it ...
4
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1answer
75 views

Finding the period of the solution to $y'(x) = y(x) \cdot cos(x + y(x))$ with Fourier transform; how to interpret complex result?

A question elsewhere on this site asks about detecting the frequency of oscillations in a system defined by differential equations. The equation is $y'(x) = y(x) \cdot cos(x + y(x))$. The solution ...
0
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1answer
38 views

Principle root of 3+4i

Is there a neat way of writing the principle root of 3+4i? I have an answer, but it is very ugly. Thanks for any help in advance.
5
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1answer
101 views

$\lim_{x\to 2} \, \sqrt{x-2}$

$$\lim_{x\to 2} \, \sqrt{x-2}$$ When you take the right hand limit for this expression, you get $0$. However, if you take the left hand side it gives an imaginary number. However, do you consider ...
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2answers
40 views

Geometric proof and extension of |a|=|b|=|c|=a+b+c=1 => a=1 or b=1 or c=1

We have $a,b,c\in\mathbb{C}$ verifying $|a|=|b|=|c|=a+b+c=1$, we have to show that $a=1$ or $b=1$ or $c=1$. That can be rather easily proved using trigonometry formulas. Is there a way to prove it ...
0
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2answers
115 views

How to show that…, $A(C_1)=\{z:|z-1|\leq 1, \theta \in [\frac{\pi}{2},\frac{3\pi}{2}] \}$?

Let $C_1=\{z:|z|\leq 1, \theta \in [\frac{\pi}{2},\frac{3\pi}{2}] \}$ and $A(z)=z-1$. Define $A(C_1)$. How to show that $A(C_1)\neq\{z:|z-1|\leq 1, \arg(z)=\theta \in [\frac{3\pi}{4},\frac{5\pi}{4}] ...
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2answers
39 views

Contradiction with complex gaussians…

So, I am computing something seemingly simple involving complex gaussians and constants, but I am getting a big contradiction in my calculations. The setup: Let $C$ be a complex constant, that is, ...
0
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0answers
25 views

Surjective complex map.

Show the map $(a + bi) \mapsto (a-bi)$ is surjective. Attempt: By definition, for every $(a - bi)$ in the complex set, there exists an $(a + bi)$ in the complex set such that $f[(a + bi)] = a - bi$. ...
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2answers
183 views

Simplifying $\exp {- i 2 \pi / N}$.

a and b are complex numbers and I know the equation below. $$X_{N} = a + e^{-i2\pi /N}*b$$ I wanted to simplify it. Here is what I've tried. I know $e^{-i\pi} = -1$ $X_{N} = a + \left ( e^{i\pi} ...
2
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0answers
86 views

Imaginary number in relativistic speeds

I am layman in field of mathematics but when I was reading about theory of special relativity I have come across speed limit of light and the book said that no one can cross that limit because ...
2
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2answers
66 views

Complex Numbers Exercise [closed]

If $a,b,c$ are complex numbers with $a+b+c=0$ and $\|a\|=\|b\|=\|c\| = r>0$ then prove that $$a^{2^n} + b^{2^n} + c^{2^n} = 0$$ Any ideas? Thanks!
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2answers
35 views

Find the roots of the simple equation?

x^{2}= 0 What are the roots? are they in complex plane, but how? Answer seems trivial in real numbers ain't it? Does this evolve a new system like it was with iota?
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1answer
38 views

Please refer book on complex numbers - especially covering equations of complex variables topic

I am searching for a good book to cover topics of complex numbers. Please refer book on complex numbers - especially covering equations of complex variables topic . Example : If $\alpha$ is a ...
3
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1answer
55 views

Cyclic sum inequality involving five numbers with modulus one and zero sum

When working on this MSE question, I was led to conjecture the following : If $z_1,z_2,z_3,z_4,z_5$ are five complex numbers with modulus $1$, such that $z_1+z_2+z_3+z_4+z_5=0$, then $$ ...
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2answers
30 views

locus of complex $z=(λ+3) + i\sqrt{3-λ^2}$

if $z=(λ+3) + i\sqrt{3-λ^2}$, for all real $λ$, then the locus of $z$ is ? Please help. Options are (A) circle (B) parabola (C) line (D) none of these
0
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0answers
42 views

Complex Number and Geometry

Given $A(3+4i)$, $B(-4+3i)$ and $C(4+3i)$ be the vertices of a triangle $ABC$ which is inscribed in a circle $S=0$. Let $AD, BE, CF$ be altitudes through $A, B, C$ which meet the circle S=0 at ...
1
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1answer
187 views

Consider a quadratic equation $az^2+bz+c=0$ where a,b,c are complex numbers. Prove that the equation has one purely imaginary root is given …

Problem : Consider a quadratic equation $az^2+bz+c=0$ where a,b,c are complex numbers. Prove that the condition such that the equation has one purely imaginary root is given by ...
2
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2answers
184 views

True or False: $i^2 = -1$, $\mathbb{C} = \mathbb{R}^2$

The complex numbers are typically defined as the set of all ordered pairs $(x,y),$ where $x,y \in \mathbb{R},$ along with the usual operations of addition and multiplication (which I won't write out ...
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3answers
35 views

Finding roots of complex quadratic equation

I'm trying to solve for the following equation: $$|(1+50*i*x)^2|$$ I keep getting the form $$-2500x^2 + 100ix + 1 $$ when the problem needs to have the following form: $$2500x^2 + 1$$ What steps ...
2
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1answer
43 views

Prove that $\bar{P}_{\bar{z}}=P_z,\ (P_z,P_z)=(P_{\bar{z}},P_{\bar{z}})$ with $P_z=\dfrac{\partial{P}}{\partial{z}}$

I have a problem: For $P$ is a nonzero real valued homogeneous polynomial of degree $k$: $$P(z,\bar{z})=\sum_{j=1}^{k-1}a_jz^j\bar{z}^{k-j}$$ where $a_j \in \Bbb C,\ a_j=\bar{a}_{k-j}$. ...
1
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3answers
66 views

What does $\theta = \text{arg}(a,b)$ mean?

I have this equation where an angle is calculated using following formula: $$\theta = \text{arg}(C_1, C_2)$$ where $C_1, C_2$ are some numerical values. What exactly does it mean?
2
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2answers
78 views

Is $f$ constant if $e^f$ is constant?

How would one go about showing that $f$ is constant if $e^f$ is constant by Liouville's theorem? The original problem asked to prove the entire function $f = u + iv$ was constant if the real part of ...
5
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2answers
507 views

Complex Analysis: Liouville's Theorem

I am stuck on the following question. Suppose $f = u+iv$ is entire and there exists $M > 0$ such that $|u(z)| \leq M$ for all $z\in C$. Show that $f$ is constant. I would figure we ...
1
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1answer
205 views

Complex Analysis: Isolated Singularities, Poles, and Residues

I was given the following question. Show that the isolated singularities of the function $f(z) = \frac{z}{z^4+4}$ are poles. Determine the order of each pole and find the corresponding ...
4
votes
3answers
104 views

Solve $z^4+16=0$ where $z$ is a complex number

The following exercise is related to complex numbers so $z$ is a complex number. Can you please check whether I solved correctly the exercise. $$z^4+16=0$$ $$z^4=16i^2$$ $$z^2=4i$$ I transformed the ...
3
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3answers
93 views

Solve: $(\cos x+i\sin x)(\cos 2x+i\sin 2x)(\cos 5x+i\sin 5x)={i+1\over \sqrt 2 }$

Can you please give me a hint for the following exercise: $$(\cos x+i\sin x)(\cos 2x+i\sin 2x)(\cos 5x+i\sin 5x)={i+1\over \sqrt 2 }$$ Thank you!
1
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1answer
170 views

Complex Analysis: Cauchy Integral Formula

I came across this problem. Let $C$ denote the circle {|z| =2}, parametrized as a positively oriented simple closed curve. Evaluate $\int_c\frac{1}{z^2-1}dz$ I want to approach this ...
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4answers
155 views

Is there such thing as an imaginary (imaginary number)?

In other words... is there such a thing that is to imaginary numbers what imaginary numbers are to real numbers? And could this be expressed as a "complex" type number? If a complex number is in the ...
0
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1answer
25 views

Complex polynomial inequality

Let $P(z) = 2z - z^3 + 6z^4 + z^6$ with roots $\alpha_1, \alpha_2, \cdots, \alpha_6$. Using the identity $ ||z|-|\omega|| \leq |z+\omega| \leq |z| + |\omega|$ show that if $|z| > 3$, then $|2z - ...
6
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1answer
133 views

Do there exist equations that cannot be solved in $\mathbb{C}$, but can be solved in $\mathbb{H}$?

Excluding polynomials (whose solutions are covered by the Fundamental Theorem of Algebra), do there exist any univariable equations that cannot be solved in the complex numbers, but can be solved ...
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0answers
18 views

When is $\sum_{n,m=-\infty}^\infty \frac{1}{(n\omega_1+m \omega_2)^\alpha}\in \mathbb{R}$?

This came up when reading about elliptic functions, where $\frac{\omega_1}{\omega_2}\notin\mathbb{R}$, and $\alpha>2$ for $$S(\omega_1,\omega_2,\alpha)=\sum_{\begin{matrix} n,m=-\infty\\ (n,m)\ne ...
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1answer
15 views

how do i transform this equation into the form a+bi

how do can i get $(-2+i)$ e^$\pi$i/3 in the form $a+bi$ where $a$ and $b$ are in the set of real numbers $R$ any help would be greatly appreciated thank you
0
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1answer
195 views

Convergence rate of PageRank, the problem when the second eigenvalue is complex

As far as I know the Google matrix used to calculate the PageRank is not symetric, that means that some eigenvalues can be complex, furthermore, we know that the second eigenvalue is equal to the ...
1
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1answer
64 views

Complex Analysis: Sketching a Region

How would I sketch the region: $|z + 1| = 4|z - 1|$? I decided to manipulate the expression and square both sides of the expression to get the follow. I think I am on to something, compared to what I ...
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1answer
24 views

Plotting circles with complex numbers

How do I plot this circle ? what does it look like radius 3 and centre 3/i ?
0
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2answers
85 views

Question on transformations in the complex plane

In the image (part (b)), Since $z < |3|$ before the transformation, does that simply imply that the region to be shaded after the transformation is definitely the inside of the circle and not it's ...
7
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9answers
2k views

Positive and negative complex numbers?

Can there be such a thing as positive and negative complex numbers? Why or why not? What about positive or negative imaginary numbers? It seems very tempting to say $+5i$ is a positive number ...