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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1
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1answer
29 views

Can principal curvatures be complex numbers in a real world situation?

Can the equation for the principal curvatures, $k^2 - 2Hk + K = 0$ (where H is equal to the mean curvature and K is equal to the Gaussian curvature), ever have complex roots? In other words, where ...
0
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0answers
13 views

Finding phase of complex function

I have come across finding the phase of function $$A(n)= e^{\frac{-j2\pi\epsilon N_d}{N}}\sum_{k=0}^{L-1}|x(k-n)|^2$$ the author state that the phase of $A(n)$ is expressed as $$\epsilon = ...
1
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2answers
44 views

Finding $f^{(m)}(i)$ where $f(z) = (1 + (z - i)^2)^{-1}$ without differentiating.

I have a question and I'm not to sure how to approach it, so any kind of help will be awesome. I was given this question in the practice final, however there are no solutions/hints to this question, ...
3
votes
3answers
57 views

Cauchy's Integral Question Complex Number

I have a question and I'm kind of stuck, I was wondering if you were able to help me move forward. The question is, Use Cauchy's integral formula to evaluate, $$ \int_{|z| = 1}\frac{e^{2z}}{z^2}dz ...
0
votes
1answer
31 views

Finding Complex Numbers to Satisfy an Equation

Sorry I am back again so soon but I am struggling to understand the following question: Find three complex numbers that satisfy the following equation: |z-2| = |z-3i| So far I believe I have two ...
2
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2answers
36 views

If $\left| \frac{z-i}{z-1}\right| = \sqrt2$ and $|z| = \sqrt 5$ and $Im(z)<0$, find $Im(z)$, $Re(z)$

I'm having a lot of problem with the first term. If I try to rationalize the fraction, I get an expression too long to work with. Similarly, substituting from the second term gives horrible results. ...
1
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0answers
13 views

Cayley-Dickson Construction

Assume $A$ is an algebra with involution $*$, and it has a norm $|x|=\sqrt{x^*x}$, which satisfies $|x||y|\geq|xy|$. By Cayley-Dickson construction, we have an algebra $B=A\times A$, which satisfies ...
2
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1answer
78 views

How to derive an explicit formula for $\sum \frac{e^{i n \theta}}{n}?$

Suppose $\theta$ is not an integer multiple of $\pi$. The series $ \left | \sum e^{i n \theta} \right |$ is bounded above by $\frac{1}{|\sin \theta|}$ and, as $\left ( \frac{1}{n} \right ) $ is ...
1
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1answer
25 views

An estimate for dyadic numbers

I would like to prove that for some positive $\delta<1/2$ we have the following inequalities $$ |\frac{1}{2^{n+1}} - \frac{1}{2^{m+1}}| \leqslant \delta\left( |\frac{1}{2^{n}} - \frac{1}{2^{m+1}}| ...
1
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1answer
49 views

Proving that $\mathbb{C}$ has a natural definition

I would like to prove a theorem in complex analysis which states: Let $K$ be a commutative field. We suppose that $L$ is a sub-field of $K$ and that $K$ is thus a vector space of finite dimension ...
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1answer
29 views

does $\sqrt[i]{x}$ have infinite solutions like the complex logarithm?

We know that $\log(x)= z+ 2πix$ where $z$ is a solution to $\log(x)$ and for any integer $x$. Can a similar derivation be made for $\sqrt[i]{x}$? Intuition says yes because of a derivation of ...
-1
votes
1answer
103 views

Couldn't there be a better definition of imaginary numbers? [closed]

It is a common known fact that i is defined as a number that satisfies the quality of $i^2 = -1$ or in other words, $i = \sqrt-1$. The intuition of this is often said to be that multiplying by $i$ ...
1
vote
1answer
33 views

Residues and poles, proof with poles

Proof that statements i)If $f_1$ and $f_2$ have residues $r_1$ and $r_2$, show that the residue of $f_1+f_2$ at $z_0$ is $r_1+r_2$. ii)If $f_1$ and $f_2$ have simple poles at $z_0$ show ...
3
votes
2answers
52 views

Complex analysis, find the residue

Find the residue of $f(z)=\frac{1}{z^2\sin z}$ at $z_0=0$ What I tried Let $g(z)=1$ and $h(z)=z^2\sin z$, both are analytics but they have zeros of different orders then $f(z)$ don't have ...
0
votes
2answers
21 views

Finding the image of a curve under a mapping

I have a mapping $$w=(\sqrt 3+i)z-1+\sqrt3 i$$ and I am trying to find the image of $$y=0$$ under this mapping and I have reached a solution but it is wrong. This is my working. I did $$u+iv=(\sqrt ...
2
votes
1answer
44 views

Complex analysis, residues of function

If $f(z)$ has residue $b_1$ at $z=z_0$, show by example that $[f(z)]^2$ need not to have residue $b_1^2$ at $z=z_0$ What I tried Suppose that $f$ is analytics in the neighborhood of $z_0$ and ...
2
votes
2answers
33 views

the root of $2^{x}$ = $2^{1.5}$ based on $2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$

Can you obtain or is it plausible to find the roots of if $2^{x^{\cos(x)}}\sqrt{\cos(x)}=2^{x}$ $x > 0$ & $cos(x) > 0$ what does $x$ equal in $2^{x}$ = $2^{1.5}$ exactly?
1
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2answers
49 views

Finding the orthogonal complement of a complex subspace

Let $i := \sqrt{-1}$ . Consider $W \subseteq \mathbb{C}^3$ defined by $W := \{(1, 0, i),(1, 2, 1)\} $. Find $W^\perp$. My biggest issue with this problem is not knowing how to extend the basis of $W$ ...
2
votes
1answer
63 views

Imaginary part of $f_m$ tends to zero

Does anybody have an idea how to show that for $|x|< \pi$ the imaginary part of the following sequence of functions $f_m$ tends to zero for $m \rightarrow \infty.$ $$f_m(x):=\left( ...
10
votes
2answers
79 views

Prove that exist $e_1,\dots,e_n\in\{-1,1\}$ such that $|e_1z_1+{\dots}+e_nz_n|\le\sqrt2$

Let $z_1,\dots,z_n\in\mathbb{C}$ such that $|z_p|\le1$ for every $p\in\{1,\dots,n\}$. Prove that exist $e_1,\dots,e_n\in\{-1,1\}$ such that $|e_1z_1+{\dots}+e_nz_n|\le\sqrt2$. I have firstly ...
3
votes
2answers
142 views

Residues and poles proof

Let the degree of the polynomials $P(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ $a_n\neq0$ and $Q(z)=b_0+b_1z+b_2z^2+\cdots+b_mz^m$ $b_m\neq 0$ be such that $m\geq n+2$. Show that if all the zeros of ...
1
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2answers
37 views

Complex analysis, residues and integrate

Let $C$ denote the circle $|z|=1$ oriented counterclockwise. Show that i)$\int_Cz^ne^{\frac{1}{z}}dz=\frac{2\pi i}{(n+1)!}$ for $n=0,1,2$ ii)$\int_C e^{z+\frac{1}{z}}dz=2\pi ...
2
votes
1answer
35 views

When is $1+e^{-i\pi(a+b)}+e^{-i\pi(b+c)}+e^{-i\pi(a+c)}$ non-zero, $a$, $b$ and $c$ being integers?

I am trying to find the conditions on the integers $a$, $b$ and $c$ such that $$1+e^{-i\pi(a+b)}+e^{-i\pi(b+c)}+e^{-i\pi(a+c)}$$ is not equal to zero. I think that the conditions for which it is equal ...
2
votes
3answers
62 views

Complex number $\frac{z}{z+1}=2+3i$

Given that $\frac{z}{z+1}=2+3i$, find the complex number $z$, giving your answer in the form of $x+yi$. Can someone give me some hints for solving this question? Thanks
2
votes
1answer
63 views

Help with the proof that the sum of all the roots of a complex number is zero

If a complex number $z \neq 0$ has n roots, then each root can be expressed as: $$z_j=(\sqrt[n]{r}) e^{ {i (\theta +2\pi j) }/{n} } $$ For $j=0,1,2,...,n-1$ Thus, the summation of all the roots ...
1
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1answer
31 views

Complex integrate and residues

Evaluate the integral of that $f(z)=\frac{z+1}{z^2-2z}$ around the circle $|z|=3$ oriented counterclockwise First I found that singularity points are $z=0,z=2$ ...
0
votes
1answer
26 views

Complex analysis, residues

Find the residue at $z=0$ of $f(z)=\dfrac{\sinh z}{z^4(1-z^2)}$. I did \begin{align} \frac{\sinh z}{z^4(1-z^2)} & =\frac{1}{z^4}\left[\left(\sum_{n=0}^\infty ...
0
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1answer
16 views

Question about a simple rule of the complex logarithm

According to the Wikipedia page on complex logarithms: Also, the identity $\log(xy) = \log x + \log y$ can fail: the two sides can differ by an integer multiple of $2\pi i$. Does the same hold ...
0
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1answer
32 views

Complex Numbers and their Conjugates

How many complex numbers are there that are conjugates of their own cubes? Is there some simple way to find this?
2
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2answers
337 views

Mathematical Way To find how many complex numbers [closed]

Suppose that $a$ and $b$ are integers and that $ |a + bi| \leq 5 $, then how many complex numbers $a + bi$ are there? Is there a mathematical way to do this?
0
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2answers
49 views

What does this circle look like?

I have been given an integral to evaluate over a circle $$|z-i|=3$$ I am trying to work out what this circle looks like. Is the radius of the circle $3$ centred on the origin?
0
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0answers
18 views

how to find singularities of complex functions.

how to find singularities of this function; i) $f(z)= 1+\frac{1}{\sqrt{z}}$ and ii) $f(z)=e^{1+\frac{1}{\sqrt{z}}}$ As the z is in the denominator the function (i) has pole at z=0 but what's ...
1
vote
2answers
85 views

The Imaginary number [duplicate]

Sometimes one writes $i^2=-1$ to characterize the imaginary number and sometimes as the root of $-1$. So when I resolve the first equation, I get for the imaginary number two roots of $-1$ and thus ...
1
vote
1answer
71 views

$e^{2\pi i x} = (e^{2\pi i})^x$: What happens if x is rational? [duplicate]

I'm a bit embarrassed that I've had difficulty on getting around this one: $$e^{2\pi i x}$$ Solving it by itself, we can reduce it down to $(e^{2\pi i})^x = 1^x$ such that $e^{2\pi i x} = 1$ for all ...
1
vote
1answer
27 views

Norm of a complex cross product

Let $c=(c_1,c_2,c_3)$ be a complex vector. How can we see that $\|c\|^2=\|c\times \bar{c}\|$? Here the bar means component wise complex conjugation, the norm is the Hermitian norm, and the cross ...
2
votes
2answers
44 views

Complex Number, Quaternions and Octonions [duplicate]

There are complex $\mathbb C$, quaternions $\mathbb H$ and octonions $\mathbb O$. Is there any higher dimensional generalization of them, such in the $\mathbb R^{16}$? Or why do we just study three ...
5
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1answer
62 views

Real Manifold … Complex Coordinates?

I'm working in an earlier edition of John Lee's book on smooth manifolds, and he has a number of problems where he represents a real manifold using complex variables. For instance in chapter 3 ...
0
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1answer
27 views

What would the multifunctional inverse of $F(x)=|x|$ be?

What would the multifunctional inverse of $F(x)=|x|$ be, assuming $x$ is on the complex plane. Also, how would this usually be represented? Note that this won't be a 'true' function. (But assume a ...
0
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3answers
60 views

Complex numbers? [duplicate]

There are plenty of questions out there asking what complex numbers mean and I never seem to get any of them. I have a few specific questions i want to ask about complex numbers. 1) what is the ...
0
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1answer
34 views

Proving relation for square root of complex number

How do I represent $\sqrt{1 + ja}$? I'm trying to show that it's approximately equal to $\pm(1 + \frac{ja}{2})$ when $a \leq 1$.
0
votes
2answers
28 views

Ignoring the pole?

I have the integral $$\int^{2 \pi}_0 \frac{z}{z+2} dz$$ where $$|z|=3$$ I parametrise the integral and get $$\int^{2\pi}_0 \frac{9 ie^{2i\theta}}{3e^{i\theta}+2}$$ and this gives the required answer ...
0
votes
1answer
25 views

Evaluating a complex integral in punctured plane.

I am trying to evaluate the complex integral $$\int \frac{z}{z+2} dz$$ And $$|z|=3$$ We can see there is a pole at $z=-2$. How do I go about solving this, what is the strategy?
1
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1answer
47 views

Arithmetic progression with complex common difference?

Suppose we have the following sequence: $$\{0,i,2i,3i,4i,5i\}$$ Can we call this sequence an arithmetic progression with first term $0$ and common difference of $i$ ? Clarification: Here, $i$ is ...
0
votes
2answers
35 views

Is there such a thing as complex rational numbers and does it have the same properties as the usual complex numbers as extension of the real numbers?

I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$). I immediately see a problem with ...
0
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0answers
19 views

Find a generator for vectorial subspace

S = {$(a, b, c, d) ∈ C^4 : 2ia = b, c + d − ib = 0$} $c+d-i(2ia)=0$ $c+d+2a=0$ $c=-d-2a$ $(a,2ia,-2a-d,d)=a(1,2i,-2,0)+d(0,0,-1,1)$ Is this solution correct?
3
votes
3answers
68 views

Why imaginary numbers axis is plotted perpendicular to the real numbers axis?

Negative numbers axis is plotted to the opposite side of the positive real number axis that make sense but i do not understand why imaginary numbers are plotted perpendicular to the real numbers axis. ...
2
votes
4answers
295 views

Cauchy integral formula

Can someone please help me answer this question as I cannot seem to get to the answer. Please note that the Cauchy integral formula must be used in order to solve it. Many thanks in advance! ...
0
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0answers
38 views

Complex function

Can anyone give me a hint to approach this question? I haven't done anything like this before so I'm bit confused about what this question is asking. Thank you very much for all your help.
0
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1answer
45 views

Midpoint of two complex numbers in polar form

Say we have two complex numbers: $re^{i\theta}$ and $se^{i\phi}$ Is there a straightforward way to find the polar form of the midpoint of these two complex numbers? I think I'm correct in saying ...
2
votes
2answers
54 views

Help solve ${{z}^{3}}=\overline{z}$ ($z\in \mathbb{C}$) [duplicate]

Me and my friend try to solve $${{z}^{3}}=\overline{z}$$ where $z \in \mathbb{C}$. My way to solve it was: $\operatorname{cin}(\theta )=\cos(\theta)+\sin(\theta)i$ \begin{align} & z=r ...