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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15
votes
9answers
1k views

Is there an interval notation for complex numbers?

Just as $$\{x \in \mathbb{R}: a \leq x \leq b\}$$ can be written in the more-compact form $[a,b],$ is there an analogous notation for $$\{z \in \mathbb{C}:z=x+yi, x \in[a,b], y \in[c,d]\} \quad ?$$ ...
1
vote
2answers
96 views

Using complex solutions in a factorisation

I'm working through an assignment, and have become stuck understanding the question... In part (a) I am asked to solve the equation: $z^5 = -1$ I have done this, so I now have a set of solutions: ...
1
vote
1answer
81 views

Sum of the trigonometric series

I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far: \begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} ...
4
votes
3answers
451 views

The limit of complex sequence

$$\lim\limits_{n \rightarrow \infty} \left(\frac{i}{1+i}\right)^n$$ I think the limit is $0$; is it true that $\forall a,b\in \Bbb C$, if $|a|<|b|$ then $\lim\limits_{n\rightarrow ...
1
vote
2answers
48 views

Can I perform the quadratic formula on polynomial with complex coefficient?

2 weeks ago, we had a Math test on complex number. One of the question was: Let $z=x+iy$ be a non-zero complex number, where $x,y \in \mathbb{R}$. Given that $z+\frac{1}{z} = k$, where $k$ ...
1
vote
0answers
36 views

Cauchy-Riemann Equations - why $f'(z_o) = \frac{\partial f}{\partial x}(z_o)$ implies that f is differentiable at $z_o$

I'm trying to understand part b of this proof. The only line I don't understand is the sentence starting with "To prove the statement in (b)..." If someone could clarify why that line is true I ...
1
vote
1answer
36 views

Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$

Let $Ω=\{z=x+iy∈C : |y|<x\}.$ Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$ Okay. So I can find a conformal map from $Ω\rightarrow \mathbb{D}$. I used the map $f(z) = ...
5
votes
4answers
88 views

Question on definitions

I was going through some basic recap of complex numbers and in the book (M. Boas. Mathematical Methods in the Physical Sciences) she says we define $e^{ix}$ by the Taylor series with $x$ replaced by ...
0
votes
1answer
62 views

When $f=u+iv$ is a holomorphic function, the real part of $f'(z)$ is equal to $u_x(z)$

Suppose $f$ is holomorphic, and is written as $f=u+iv$ with $u,v$ real-valued. Why is the partial derivative $u_x(z)$ equal to $\operatorname{Re}(f'(z))$? Source This fact is used in the proof ...
2
votes
2answers
187 views

Are numbers like $\left ( -2 \right )^{\sqrt{2}}$ real or complex?

I know that numbers with rational power can be converted to radicals and based on the degree of the radical we can say that whether they are real or complex. But what about numbers like $\left ( -2 ...
1
vote
0answers
37 views

Computing the tangential and cross components of one quantity using gnomonic projection

I have a spin-2 field given called shape distortion of galaxies as $$\gamma=\gamma_1+i\gamma_2=|\gamma|e^{-2i\phi}$$ where $\phi$ is the orientation angle. If this quantity has been measured on ...
1
vote
1answer
37 views

complex functions inequalities plane

Given $w(z)=\frac{i-z}{i+z}$. Find the map w=f(z) of the part of the plane defined by inequalities: $|z|>1$ and $Im(z)>Re(z)$ so far: $|z|>1$ is this area from $Im(z)>Re(z)$ => ...
1
vote
3answers
90 views

Are $i,j,k$ commutative?

I am trying to understand quaternions. I read that Hamilton came up with the great equation: A) $i^2 = j^2 = k^2 = ijk = −1$ In this equation I understand that $i,j,k$ are complex numbers. Later ...
0
votes
1answer
21 views

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$, where $aj+b$ is a complex number, and $|f(x)|$ is the modulus function. In the past I've been calculating $|(aj+b)^{-1}|$ by multiplying the numerator and ...
1
vote
1answer
68 views

How to prove the formula for the argument of a complex number?

$\arg(x + iy) = 2 \cdot \arctan(\dfrac{y}{x + r})$ there is always the mark, this is derived from the 'Half-angle formula' How can I come from $\tan(\phi) = \tan(\phi + k \pi) = \dfrac{y}{x}$ to ...
0
votes
0answers
30 views

Is it possible to explicitly demonstrate complex number structures arises naturally from intersecting symplectic groups and orthogonal groups?

Is it possible to explicitly demonstrate complex number structures arises naturally from intersecting symplectic groups $Sp(2n, \mathbb{R})$ and orthogonal groups $O(n)$? My question may not be ...
2
votes
2answers
79 views

Simplify $(7-2i)(7+2i)$. Found difference between mine and solution guide's and didn't know why.

I looked up the solution guide and found out: $(7-2i)(7+2i)$ $=49-(2i)^2$ $=49+4$ $=53$ Why the unknown "$i$" just disappeared$?$ I supposed it might be: $(7-2i)(7+2i)$ $=49-(2i)^2$ $=49-4i$ does it? ...
0
votes
2answers
147 views

Complex Numbers Closed under division?

My question is very simple that is: Is Complex number closed under division? Can we consider this 0+0i as complex numbers? or it is not a complex number.
7
votes
3answers
211 views

Towards a formula for the Euler $\phi$ function?

$\Phi_n(1)$ and $\Phi_n(-1)$ for the cyclotomic polynomials are well-known. I am now looking for $$\Phi_n(i)$$ and/or $$\Phi_n(-i)$$ with $i$ the complex unit. The reason is : I suppose it is ...
0
votes
2answers
44 views

How to solve equation in complex numbers?

For $n$ odd ( e.g. with $n\equiv 1\mod 4 )$ I seek a solution $f(n)$ for this simple equation in the complex numbers $$(-1)^{f(n)}2^{\frac{n-1}{2}}=-\frac i2(1+i)^{n+1}$$ $f(n)$ is probably an integer ...
2
votes
0answers
107 views

On the criterion of convergence of infinite products of complex numbers

I have troubles in understanding the proof of the criterion which states that an infinite product exists iff the series of the complex logarithms of the terms of the product converges. In particular, ...
6
votes
0answers
98 views

Minimising a sum of roots of unity

Let $n$ be an integer, $n\ge2$. Let $m$ be a positive integer, $m\le n$, having no common factor with $n$. How can we select $m$ distinct complex $n$th roots of unity in such a way as to minimise ...
2
votes
2answers
137 views

Consider the general equation of a circle in $(x,y)$-plane and use the transformation $w = \frac{1}{z}$

Consider the general equation of a circle in $(x,y)$-plane and use the transformation $w = \frac{1}{z}$ where $w = u + iv$ and $z=x+iy$. I understand that \begin{align} u = \frac{x}{x^2+y^2}\\ v = ...
2
votes
3answers
66 views

How to prove that $1/|z^4-4z^2+3|\le 1/3$ if $z$ is a complex number with $|z|=2 $?

Show $$\left\lvert \frac{1}{z^4-4z^2+3} \right\rvert \leq \frac{1}{3},\, \text{ if } |z|=2.$$ I am sure it is pretty easy and I am overlooking something. So this is equivalent to $$3 \leq ...
5
votes
2answers
100 views

Complex Numbers - Finding Roots

Hi there I was wondering if someone could help me? I am struggling to find the roots of the polynomial $z^4+2z+3=0$ It is not a quadratic so can't use the quadratic formula so am not quite sure ...
36
votes
10answers
4k views

Is “$a + 0i$” in every way equal to just “$a$”?

I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$). I say this is practically the case, so in every calculation you ...
0
votes
1answer
58 views

equality of complex numbers: general case.

Can someone help me to understand this definition (or proposition) for complex numbers equality, of the form $w=x+\xi y$. \begin{align*} &\xi\text{ is a complex number such that } ...
1
vote
3answers
714 views

Proof that sum of complex unit roots is zero

When reading a proof of why $x^3+y^3=z^3$ has no nontriavial integer solutions I came across following identity: $$ y^3 = z^3-x^3 = (z-x)(z-\omega x)(z-\omega^2 x) \qquad \text{where } \omega = ...
2
votes
4answers
70 views

want to prove exponential identity with complex numbers

I want to prove that $e^{x+y}=e^x*e^y$, where x and y are complex numbers. I only want to use that evey complex number is on the form $z=a+bi$, where $i^2=-1$. And if z is a complex number we have ...
0
votes
2answers
118 views

Construction of complex numbers and exponent rules for them

I have some questions about the construction of the complex numbers in this Wikipedia article, especially of the exponents of complex numbers. $1$. Is it enough to define it as $a+bi$, where a,b are ...
2
votes
5answers
77 views

If $ \frac {z^2 + z+ 1} {z^2 -z +1}$ is purely real then $|z|=$?

If z is a complex number and $ \frac {z^2 + z+ 1} {z^2 -z +1}$ is purely real then find the value of $|z|$ . I tried to put $ \frac {z^2 + z+ 1} {z^2 -z +1} =k $ then solve for $z$ and tried to ...
0
votes
1answer
232 views

Mandelbrot sets and radius of convergence

While watching this Numberphile video on Mandelbrot sets, it's more or less stated that the fractal will "blow up" if it's radius of convergence is greater than 2. What is the mathematical basis for ...
1
vote
0answers
35 views

Integrate complex function

In a physics textbook I am working with, the following integral is caluclated: Define the complex elastic modulus as $\overline{M} = M_1 + i M_2$ where $\overline{M}$ is a function of the angular ...
0
votes
3answers
42 views

where am I going wrong with solving this equation?

solve $z^2=2e^{5{\pi}i/6}$. Well, clearly $z={\sqrt{2}}e^{5{\pi}i/12}$ is a root so its' conjugate $z={\sqrt{2}}e^{-5{\pi}i/12}$ is the other root. But I can also argue ...
1
vote
2answers
130 views

Calculating the reminder when dividing complex numbers

When dividing complex numbers, let us consider $\frac{26+120 i}{37+226 i}$. We multiply the numerator and denominator by $(37-226 i)$ and get the result, but how do I divide it like the normal ...
0
votes
2answers
52 views

Separating a Complex Valued Function

Is there a formula (with mathematical reasoning) for separating a complex-valued function $f(z)=f(x+iy)$ into the form $ f(z)=u(x,y) + iv(x,y)$? Thank You, C.A
0
votes
2answers
95 views

Real and imaginary parts of a complex-valued function

How do you get a complex-valued function $ f(z) = f(x+iy) = \frac{z^{s-1}}{e^{-z}-1}, $ where $s$ is a constant complex number and $z$ is a complex variable, into the form: $ f(x+iy) = a(x,y) + ...
3
votes
2answers
59 views

Rectangular to polar form using exact values.

I'm in a first year math course at university, and we've been asked to convert a rectangular form complex number into polar form, using exact values only. I have the modulus, that's all good. But I ...
3
votes
1answer
30 views

Show the convergence of the integral $F(t,x)=\int_{-\infty}^{+\infty}\exp[i\tau t-(i\tau)^{1/2}x - (i\tau)^\theta] \,d\tau$

The original problem is : Let $\theta$ be a number such that $1/2<\theta<1$. Prove that $$F(t,x)=\int_{-\infty}^{+\infty}\exp[i\tau t-(i\tau)^{1/2}x - (i\tau)^\theta] \,d\tau$$ defines a ...
2
votes
1answer
111 views

Why is the argument of a complex number measured anticlockwise (from the positive real axis), rather than clockwise?

I was going through some basic examples of complex numbers (finding the argument and modulus) with my brother yesterday, and he asked Why is the argument measured anticlockwise rather than ...
2
votes
0answers
65 views

Proof of Cauchy-Riemann equations using differentials as quotients?

In my analysis 2 book the proof goes like this: If a complex function $f = P(x,y) + iQ(x,y)$ is differentiable at a point $z$, then $$ \lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z} ...
0
votes
1answer
57 views

Using De-moivres to solve the following problem:

Part (i) I can solve and understand that the solutions are $Z=e^\frac{2ki\pi}{5}$ for $k = 0,1,2,3,4$ Its the part (ii) I cannot understand. Could someone kindly give me a ...
19
votes
8answers
2k views

Refining my knowledge of the imaginary number

So I am about halfway through complex analysis (using Churchill amd Brown's book) right now. I began thinking some more about the nature and behavior of $i$ and ran into some confusion. I have seen ...
0
votes
0answers
28 views

translating vectors in polar coordinates to the complex plane [duplicate]

These equations model circular motion. Equation R is the position vector given in polar coordinates. What I've done is represent this vector onto the complex plane via equation (1). Equation (2) and ...
0
votes
1answer
81 views

Separable Function: Alternative Representation

How does one get the following function $$ f(u) = f(x+iy) = \frac{u^{z-1}}{e^{-u}-1}, $$ where $z$ is a constant complex number and u is a complex variable, into the form: $$ f(x+iy) = v(x,y) + ...
0
votes
1answer
80 views

show an integral is bounded by a constant independent of a parameter

This is a question in Treves. Suppose $a>1$ and $\tau \in \mathbb R $, (i) show that for all $(\tau, \xi) \in \mathbb R^{n+1}$, $|(\tau-ia)^2 - |\xi|^2| \ge(\tau ^2+|\xi|^2+a^2)^{1/2}$ (ii) ...
2
votes
1answer
43 views

If $\{ z_k\}_{0\leq k < n}$ are the $n$-th roots of a unity with $z_0 = 1$ prove that $n = \prod (z_0-z_k)$

Let $z_k$ be a $n$-th root of the unity, i.e. $$z_k = e^{2\pi i\frac{k}{n}}\\ k\in \{0,1,\cdots,n-1\}$$ Prove that $$(z_0 - z_1)(z_0 - z_2)\ldots (z_0 - z_{n-1})= n$$ This problem was at the end of ...
2
votes
1answer
35 views

If $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ are complex numbers, then $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$

Let $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ be two complex numbers. Ahlfors says that $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$. I don't understand why that is. Any help would be greatly appreciated.
10
votes
2answers
303 views

History of the matrix representation of complex numbers

It is well-known to many that $\mathbb{C}$ can be represented by matrices of the form $\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$. For example, see this question or this ...
2
votes
1answer
184 views

Polar form of complex numbers.

Write the given number in polar form $re^{i\theta}$ i) $z = -8\pi (1 + \sqrt{3}i)$ So I thought that $\theta = \arctan(-\sqrt{3}/-1) = \frac{4\pi}{3}$ and it would be $z = 8\pi ...