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

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2
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4answers
30 views

Complex number, series

Show that $$\frac{1}{z^2}=1+\sum_{i=1}^\infty (n+1)(z+1)^n$$ when $|z+1|<1$ I'm having problems to resolve this type of exercise since my book has virtually no exercises of this type, these ...
0
votes
0answers
4 views

How to calculate the variance of the argument of a complex number?

Given a number $s \in \mathbb{C}$ and the (Gaussian) variances of its components $\sigma^2(\Re(s))$ and $\sigma^2(\Im(s))$, how can I calculate the variance $\sigma^2(\arg(s))$ and the covariances ...
0
votes
1answer
24 views

Complex number, series representation

Show that for any finite value of $z$ $$e^z=e+e\sum_{n=1}^\infty \frac{(z-1)^n}{n!}$$ For $z=1$ $$f(z)=f(z_0)+\sum f^{(n)}(z_0)\frac{(z-z_0)^n}{n!}$$ equality is checked, but I do not know how to ...
2
votes
2answers
144 views

Calculating $\sum^{10}_{k=1}\left(\sin\frac{2k\pi}{11}+i\cos\frac{2k\pi}{11}\right)$

Find the value of $$\sum^{10}_{k=1}\left (\sin\left (\frac{2k\pi}{11} \right )+i\cos\left (\frac{2k\pi}{11}\right ) \right)$$ My approach: Since $\cos\theta + i\sin\theta = e^{i\theta}$, we can ...
10
votes
3answers
143 views

Does $\sqrt{i + \sqrt{i+ \sqrt{i + \sqrt{i + \cdots}}}}$ have a closed form?

I've been brushing up on my complex analysis recently, and I've come across a problem that's stumped me: What are the real and imaginary parts of $$\sqrt{i+\sqrt{i+\sqrt{i+\sqrt{i+\cdots}}}} ?$$ I ...
0
votes
2answers
23 views

Complex Polynomials

I have this question on a practice final, Find all the solutions to the equation $$2z^2 = √2 − i√2$$ I'm not quite sure how to solve this! Should I approach this with synthetic division? Any help ...
1
vote
2answers
39 views

Complex Number inqualities

Although the inequalities are not defined on complex numbers. But does the inequality $x < 4 + 5i$ be said to possess any solutions ? Where $ i = \sqrt{-1}$.
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2answers
41 views

Can some one explain why the answer to part a describes a circle, or part of it?

Problem Statement: The transformation $T$ from the complex $z$-plane to the complex $w$-plane is given by $w=\frac{z+1}{z+i}, z\neq i$. a) Show that $T$ maps points on the half-line $\arg ...
13
votes
3answers
9k views

What is the “standard basis” for fields of complex numbers?

What is the "standard basis" for fields of complex numbers? For example, what is the standard basis for $\Bbb C^2$ (two-tuples of the form: $(a + bi, c + di)$)? I know the standard for $\Bbb R^2$ is ...
0
votes
2answers
28 views

Have I Correctly Defined the Set of Nonzero Complex Numbers $\mathbb{C^*}$?

If the set of complex numbers $\mathbb{C} = \{a+bi\mid a,b \in \mathbb{R}\}$, then what would be the definition of the set of nonzero complex numbers? Am I right in defining such a set as ...
317
votes
21answers
57k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
1
vote
4answers
31 views

Find Solution of trigonometric complex equation

Find the solutions of $\sin z = 3$ There are 2 ways to solve this, I know how to do this with: $\sin z = \frac{1}{2i}(e^{iz}-e^{-iz}) = 3$ Now, I am now doing in the way: $\sin z = \sin x \cosh y+i ...
1
vote
1answer
28 views

Why is (-1)^(2/3) equal to -1/2+(i sqrt(3))/2

Can someone please explain to me how $(-1)^{\frac2 3}$ can be written as $\frac {-1}{2}+\frac{i \sqrt3} 2$ ? Do you use the corrolation $(-1)^c = e^{(i c \pi)}$, where ${c}$ is a constant?
2
votes
3answers
64 views

Is $(a+bi)(a-bi) = a^2 + b^2 $ solely a real number or a complex number?

I have not dealt with complex numbers for a while now, but I was wondering if I multiplied the complex number $a+bi$ by its conjugate $a-bi$ to obtain $$(a+bi)(a-bi) = a^2 + b^2 $$ where $a,b \in ...
0
votes
2answers
28 views

Simple math Question concerning the natural logarithm of Complex Number

There is this simple exercise, in which the complex number is given in polar form as z= mod=|10|,arg=322.75 degrees and i must find the ln of it. So to do that i must first convert the complex number ...
2
votes
1answer
32 views

Why is $z^4-1-i=0$ a polynomial equation which does not have real coefficients?(complex-number)

Why is $z^4-1-i=0$ a polynomial equation which does not have real coefficients? Its coefficient is $1$ and $1$ is a real number, isn't it?
2
votes
3answers
51 views

Finding the sum of the trigonometric serie:

There are two series: $$1) 1+\dfrac{\cos{x}}{p}+\dfrac{\cos{2x}}{p^2}+...+\dfrac{\cos{nx}}{p^n}=\sum_{k=0}^{n}{\dfrac{\cos{kx}}{p^k}}$$ $$2) ...
0
votes
1answer
38 views

Sum of bits in range of twindragon curve

http://blog.garritys.org/2012/12/base-i-1-there-be-dragons.html As the link above shows, it's possible to represent every Gaussian integer by converting a number N into its binary representation and ...
1
vote
3answers
41 views

What is a basis for the vector space $ \Bbb{C}^{n} $ (a complex vector space)?

I know that a basis for $ \Bbb{C} $ is $ \{ 1,i \} $. This set is linearly independent in $ \Bbb{C} $ and spans $ \Bbb{C} $. I think that the dimension of $ \Bbb{C}^{n} $ may be $ 2 n $, but I’m just ...
7
votes
2answers
2k views

Simplest examples of real world situations that can be elegantly represented with complex numbers

Mathematics could be defined as the study of formally defined abstractions. These abstractions may or may not be useful for describing real world phenomenon. Indeed, Physics could be defined as the ...
5
votes
4answers
79 views

Find all solutions to the following equation: $x^3=-8i$

Find all solutions to the following equation: $$x^3=-8i$$ I found the modulus, $$r=8$$ $$\operatorname{arg}(x)=\arctan(-8/0)=-π/2+2πk$$ By De Moivre's Theorem: ...
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votes
0answers
19 views
+50

Pisier's $\epsilon$-net condition

I'm reading a book about Sidon sets and I'm stuck on the following proof. In order to facilitate the comprehension of my problems I will give the full proof and the context. Let $G$ be a compact ...
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 ...
0
votes
0answers
13 views

equvivalence resistance of hexagonal infinite

I am trying to evaluate equivalence resistance between two nodes of hexagonal infinite grid, I am stuck at the integral at end of the image attached. pl see if the integral could be evaluated. Let ...
3
votes
2answers
37 views

Is every Pisot-like integer the product of a Pisot integer and a root of unity?

For lack of better terminology, let's call an algebraic integer $\beta$ Pisot-like if $|\beta|_{\mathbf{v}} > 1$ for the place $\mathbf{v}$ of $\Bbb{Q}(\beta)$ corresponding to the embedding $\beta ...
0
votes
1answer
59 views

Number of 1s in after converting number to base -1+i

Regarding to Base conversion: How to convert between Decimal and a Complex base? Let $s(a,b)$ is a number of $1$ after converting complex number $a+bi$ to base $-1+i$. It's easy to implement that ...
2
votes
1answer
30 views

Complex integration on circle

Calculate the integral of $g(z)$ along the closed path $|z-i|=2$ in the positive direction when i)$g(z)=\frac{1}{z^2+4}$ ii)$g(z)=\frac{1}{(z^2+4)^2}$ First I checked the described area ...
0
votes
1answer
25 views

Complex function, analyticity domain

Find the function domain of analyticity i)$f(z)=\frac{z^2}{z-3}$ ii)$f(z)=ze^{-z}$ To check the domain of analyticity of a function, I only need to replace $z=x+iy$ and check the conditions of ...
5
votes
5answers
293 views

Confused with imaginary numbers [duplicate]

In 9th grade I had an argument with my teacher that ${i}^{3}=i$ where $i=\sqrt{-1}$ But my teacher insisted (as is the accepted case) that: ${i}^{3}=-i$ My Solution: ${i}^3=(\sqrt{-1})^3$ ...
0
votes
1answer
24 views

Complex integration and theorems

If $C$ is a closed path oriented in the positive direction and $$g(z_0)=\int_C \frac{z^3+2z}{(z-z_0)^3}$$ show that $g(z_0)=6\pi iz_0$ when $z_0$ is in interior of $C$ and $g(z_0)=0$ when $z_0$ is out ...
3
votes
3answers
63 views

Zeroes of sin(x)

Consider the function f = $\sin(x)$ defined as $$ \sin(x) = \frac{e^{ix}- e^{-ix}}{2i} $$ How to prove that the only zeroes of this function lie on the line $i = 0$ in the complex plane and ...
0
votes
1answer
30 views

Equivalent forms of expressions with complex numbers

Which expressions are equivalent to $ {1\over{(9i+z)^4}} + {1\over{(9i-z)^4}}$ Select all that apply. $ {18i\over{(81−z)^8}}$ $ {−18i\over{(81+z)^8}}$ $ {18i\over{(81+z)^8}}$ $ ...
0
votes
2answers
22 views

Integral of strictly real function has imaginary component

Intuitively and informally speaking, $\int_{a}^{b}f(x)dx$ is summing all of the values $f(x)$ yields for $x\in [a,b]$. So it would make sense that if $f(x)$ is strictly real over $[a,b]$, then ...
-1
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1answer
33 views

Solve $3x³ + 3y³ + 2x² - 32 = 0$, $4x² + 2 = 0$ and $10y² + 2x² + 12 = 12x³$.

Hi my friend asked this to me, i'm not good at math. $$3x³ + 3y³ + 2x² - 32 = 0$$ $$4x² + 2 = 0$$ $$10y² + 2x² + 12 = 12x³$$ remove 2x² $$2x² = -1$$ $$3x³ + 3y³ - 1 - 32 = 0$$ $$10y² - 1 + 12 = ...
2
votes
1answer
146 views

Find all solutions for a complex equation: $(1+i)z^2 - (6+i)z + 9+7i=0$

There is this math assignment that we've been given to find all the answers for some diffrent math problems. The problem is: $(1+i)z^2 - (6+i)z + 9+7i=0$, find all the solutions and answer in ...
5
votes
3answers
102 views

Does $1^i$ and $1^{\frac{0}{0}}$ also give $1$ again? [duplicate]

This is the property of Real number $1$ that, $1^n=1$ does this property only hold $\forall n \in \mathbb R$ or also $1^i=1$ and $1^{\frac{0}{0}}=1$ If it is; explain how? I think that it should ...
1
vote
1answer
17 views

$(N(\alpha), N(\beta)) = 1 \rightarrow (\alpha, \beta) = 1$ and backwards?

Let us have $\alpha, \beta$ arbitrary Gaussian integers. Is it true, that if $(N(\alpha), N(\beta)) = 1 \rightarrow (\alpha, \beta) = 1$? Is it true backwards? I know when a Gauss-integer is prime, ...
0
votes
1answer
38 views

Solve Trigonometric Complex Equation

Find all solutions of $\sin (z) = 2$. Here are the things I did: 1) By definition: $\sin z =\dfrac{e^{iz} − e^{−iz}}{2i}= 2$. Multiply $2i$ to the equation and make it quadratic: $e^{2iz} ...
1
vote
2answers
72 views

Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?

Is there anything special with the form: $$\left|\frac{z-a}{1-\bar{a}{z}}\right|$$ ? With $a$ and $z$ are complex numbers. In fact, I saw it in a problem: If $|z| = 1$, prove that ...
0
votes
2answers
48 views

Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $[z_1,z_2,z_3,z_4]\in\mathbb{R}$

Let $[z_1,z_2,z_3,z_4]$ denote the cross ratio of the complex numbers $z_1,z_2,z_3,z_4\in \mathbb{C}$. Show that the distinct points $z_1,z_2,z_3,z_4\in\widehat{\mathbb{C}}$ lie on a generalized ...
1
vote
1answer
17 views

There exists a continuously differentiable bijection, $g:[a,b]\to [c,d]$ satisfying $g'(k)>0$ with $z(k)=w(g(k))$

Let $z:[a,b]\to \mathbb{C}$ and $w:[c,d]\to \mathbb{C}$ such that there exists $t(s):[c,d]\to [a,b]$ which is a continuously differentiable bijection with $t'(s)>0$ and $w(s)=z(t(s))$. Then I ...
1
vote
2answers
53 views

Simplify $Im \left(\frac{az+b}{cz+d}\right)$

Let $z \in \mathbb{H}$, where $\mathbb{H}$ denotes the half plane $\mathbb{H}=\{z \in \mathbb{C}:Im(z)>0\}$. Let \begin{equation*} f(z)=\frac{az+b}{cz+d} \end{equation*} which is called a Mobius ...
0
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3answers
52 views

Why does the complex equation $z=Ae^{it}+Be^{-it}$ represents an ellipse?

Why does the complex equation $z=Ae^{it}+Be^{-it}$ represents an ellipse?, being $A,B \in \mathbb C$ How can it be described?
0
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3answers
47 views

Proving that $|z-1|-|z+1|=1$ its an hyperbola, and $\Re(1-z)=|z|$ its an ellipse.

Proving that $|z-1|-|z+1|=1$ its an hyperbola, and $\Re(1-z)=|z|$ its an ellipse. If $z\in \mathbb C$ I cant see why there are a hyperobla and an ellipse respct.
1
vote
1answer
15 views

Difference of roots of unity in polar form

I want to write the difference between $n$-th roots of unity in the form $re^{i \theta}.$ It is enough to find the polar form of $1 - \zeta^k$. By thinking geometrically, I can guess the formula $$1 ...
3
votes
0answers
47 views

Why is there only one type of imaginary number? [duplicate]

We've defined the square root of -1 as an imaginary number i (or j, if you're a physicist). Is there any reason why we can't/haven't made other systems of imaginary numbers for other "impossible" ...
1
vote
1answer
22 views

Exponent identities with imaginary exponents$\left(a^i\right)^i$

I've been trying to understand how imaginary exponents work, and I think I mostly understand it, but I'm confused by something like $\left(a^i\right)^i$ (where $a$ is real). According to the normal ...
12
votes
3answers
1k views

Finding the square root of a complex number - why two solutions instead of four?

I want to find the square roots of a complex number, $w = a+ib \in \mathbb{C}$, i.e. I'm looking for solutions, $z = x + iy$, for the equation $z^2 = w$. This question has been asked here a couple of ...
4
votes
1answer
25 views

Proving for $w \in \Bbb C$ with modulus $1$ and argument $2 \theta$ that $\frac{w-1}{w+1}=i\tan \theta$

The complex number w has modulus $1$ and argument 2$\theta$ radians. Show that $$\frac{w-1}{w+1}=i\tan \theta.$$ Attempted solution: I just assumed that $w=1(\cos 2\theta +i \sin 2\theta)$ and ...
0
votes
1answer
20 views

In what ways can I extend the error function to accept complex arguments?

What are the different approaches to extending the error function to accept complex arguments? When should I favor using one approach over another?