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

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Under what conditions on $a,b$ is $1/(a+bi)=(1/a)+(i/b)$?

Question in proofs review in the complex numbers unit. I expressed $1/(a+bi) = (a-bi)/(a^2+b^2)$ I then separated the two terms in the denominator to get $a/(a^2+b^2)-bi/(a^2+b^2)$ I then equated ...
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How to find the locus of the complex equation : $z \overline{z} +az +b\overline{z}+c=0$

How to find the locus of the complex equation : $z \overline{z} +az +b\overline{z}+c=0$ I have no clue how to find the locus of such equation in complex plane. Please guide on this thanks in ...
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Find the number of distinct elements.

Let $\omega$ denote a non-real cube root of unity. Then find the number of distinct elements in the set $\{ (1+\omega + \omega^2 + \cdots + \omega^n)^m | m,n \in \Bbb Z_+ \}$
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sum of complex number of different magnitude

Is there a systematic way to express the sum of two complex numbers of different magnitude (given in the exponential form), i.e find its magnitude and its argument expressed in terms of those of the ...
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128 views

A uniformly convergent series

How does one show that the series $$\sum_{k = 1}^\infty \left\{\frac{s}{k} - \log\left(1 + \frac{s}{k}\right)\right\}, \quad s \in \mathbb{C} \setminus \{0, -1, -2, \ldots\}$$ is uniformly convergent? ...
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Orthogonality with complex numbers

I have these 2 equations: $$\begin{align} (68-4i,44+12i,-38-2i)\mathbf x=0 \\ (-66i-18,-52i-10,42i+12)\mathbf x= 0 \end{align}$$ I need to find the span of vector $\mathbf x \in \Bbb C^3$. I'm ...
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Complex numbers exercise - homework

We have to prove that $z_1^{24n}+z_2^{24n}=2^{12n+1}$ if we know that a)$z_1z_2=2$ b)$z_1^3+z_2^3=-4$ I have tried many things but nothing worked so far
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Locus of Complex Number

Would be great to get your help in finding the locus of this complex number $z$: $|z-z_1|+\sin \alpha|z-z_2|=\sin \theta$ From this question I proceed to a refined one- What would ...
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Complex modulus? No, not the absolute value.

I was trying to make a class for complex numbers (VB.NET) but then I stumbled upon a problem. How do I define the $mod$ operator for Complex numbers? First I asked Wolfram Alpha. It didn't help much. ...
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Clarification on a step in the proof of Lagrange's identity for complex numbers.

I wrote this proof of the following identity and I want to verify that a certain step is correct. $\newcommand{\conj}[1]{\overline{\vphantom{b}#1}}$ $\newcommand{\on}[1]{\operatorname{#1}}$ ...
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How to expand out negative powers for complex numbers

I have the following expansions but I don't know how my teacher gets them. Apparently there is a formula for it (though the guy who told me didn't know it well), but I cannot find it in my notes. For ...
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About gaussian integers and orders.

I noticed $(1+i)^{16}= 256$ so $(1+i)^{16} - 1$ is a multiple of $17$. So $(1+i)^{16} - 1$ is a multiple of $(1+4i)$ or $(1-4i)$. $(1+i)^{|1+4i|}$ is congruent to $1$ or $i$ mod $(1+4i)$. I think . ...
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Finding the least value for points in a loci

The question goes like this: On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z+i|=1$ and the locus representing the complex numbers $w$ satisfying ...
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Complex Analysis Questions Compilation

All of my questions are in relation to Gamelin's Complex Analysis. How was the parametric form of the line from the North Pole on the unit sphere through a point P come to be? It is $$ N + t (P-N) ...
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Regarding Gaussian integers and primitive roots.

Can modular arithmetic be set up using gaussian integers instead of (non-complex) integers? If so is there an analogue of 'primitive roots' with Gaussian integers?
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Is it possible to extend the complex plane affinely and projectively at the same time?

Is it possible to extend the complex plane affinely and projectively at the same time? That is by adding both the positive infinity (with based on it directed infinity) AND the unsigned complex ...
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Complex numbers

3 questions, not sure how to do them. Let $z$ and $w$ be complex numbers such that $w=\frac{1}{1-z}$ and $|z|^2=1$. Find the real part of w. If $z=e^{i\theta}$ prove that ...
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Converting from non-complex cartesian form to complex exponential form

If I had a non complex function in cartesian form, e.g: $$f(t)=8\sin(3t+4)$$ How would I convert this into the exponential form: $$f(t)=|M|e^{i(\omega t +\phi)}$$ ? I understand how to do it ...
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34 views

Standard deviation on absolute square of complex average

I am calculating a quantity $q=|c|^2$ where I obtain $c\in\boldsymbol C$ as an average of a collection of estimates with errors: $\langle c\rangle=\sum_{j=1}^nc_j$, and the question is what error to ...
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Are there any applications of sedenions?

I've been interested in hypercomplex number systems for a while as fascinating little toys, all the way up to octonions; But the octonions look like they're the end of the line of interesting math as ...
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How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity $\infty$?

How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity? Are there any paradoxes arizing? What are the rules when the ...
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36 views

Root of unity paradox

Suppose $w$ is a cube root of unity. Then we know that $w^3 =1$. now suppose we want the value of $w^4$. $w^4 = (w^3)^{4/3} = 1^{4/3} = 1$ which is obviously false Why does this happen?
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Residues more than one singularity at 0

Having trouble calculating the residue at 0 for this integral within the unit circle I understand that its a pole of order 3 because both the z^2 and the sinz have singularities at 0. Is there an ...
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Complex number, strangely written

Find all the complex solutions of the equation: $$\frac{z^3}{i} = 1$$ I mean is this the same thing as $$z^3 = i$$? Because I don't understand why my teacher would put it like that on a test. At ...
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Discontinuity of principal argument in nonpositive real axis

Let $\operatorname{Arg}(z)$ be principal argument function defined in branch $(-\pi, \pi]$. Prove that $\operatorname{Arg}(z)$ is discontinuous in every point in nonpositive real axis. "Solution": ...
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Complex Eigenvalue Periodic

This is probably a simple question, but I'm having trouble finding a clear answer. Let's say we have a system, with the complex eigenvalue: $$\lambda = \alpha + i\beta$$ I know that $\alpha < 0$ ...
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Functional Equation and totally multiplicative functions

Find all $f:\mathbb{C}\rightarrow\mathbb{C}$ s.t. $\forall a, b\in \mathbb{C}, f(ab) = f(a)f(b)$. I could deduce a lot of things about what happens at the roots of unity, and 0, but I can't find out ...
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Why isn't $i$ affected by powers?

When finding roots of complex functions we can write for example: $$z=2-2i$$ Let's find complex numbers $w$ such that $$w^4 = 2-2i$$ $$\large z = \sqrt{8} e^{ \frac{- \pi }{4} i}$$ This reads: ...
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Finding isolated singularities

I am having trouble categorizing the singularities of the following complex valued function: $$f(z) = \frac{z^2}{sin(z)}$$ It seems like the isolated singularities are $2n\pi$ where ...
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Complex numbers trig form

Given that $z= \cos(x) + i\sin(x),$ show that $\frac{1}{1+z}= 1 + i\tan(x/2)$ where x is not equal to pi/2 I tried to add 1 to z and then invert and realize the denominator but didn't get anywhere ...
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Does a purely imaginary number have a corresponding “angle” in polar coordinate system?

Let's say we have a pure imaginary number with no real part, $i$. I know that complex numbers in the form $a+bi$ can be converted into the polar coordinate system using the following relations: ...
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Complex numbers

If someone could help me with this question I would really appreciate it.For some reason I am getting a weaker version of these inequalities when applying triangle inequality. Let S be the interior ...
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Rationalization of complex denominator of n-th power

I am stuck with rationalization of this expression, $$ \dfrac{(i\omega)^a}{x^a+(i\omega)^a}, $$ where '$\omega$' is frequency, '$x$' is constant, '$i$' imaginary unit, and '$a$' is non-integer ...
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Showing where complex function is analytic and differentiable.

I've been asked to show where the following function is analytic and differentiable; $$f(z) = x^4 + i(1-y)^4$$ for $z = x + iy$ First, I noted that $u(x,y) = x^4$ and $v(x,y) = (1-y)^4$. Then, I ...
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50 views

Finding the Remainder of Complex Polynomials

Suppose $f(-1 + i) = 2 + 5i$ and $f(-2 - i) = -3$ determine the remainder of $f(x)$ divided by $(x + 1 - i)(x + 2 + i)$. I don't really know where to start any help would be great. Thanks :)
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Summing complex numbers of magnitude $1$

A professor at my university gave me the following problem: For what real values of $m$ do we have $$L=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^Ne^{2\pi ik^3m}=0$$ If $m$ is rational, expressed ...
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What's the first fundamental form of a regular surface in complex coordinates and how to get it?

Precisely, the first fundamental form of a regular surface is given by $$ds^2=Edx^2+2Fdx\ dy+Gdy^2.$$ What's the form of $ds^2$ in complex coordinates $z=x+iy$.
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What values or form of values can we get for these multiplications modulo a prime?

If we have four complex values, all of the form $a + b i$, for integers $a$ and $b$, we can label them $c$, $d$, $e$ and $f$. Now if we want to find $g$ and $h$ such that $$g \equiv ce \equiv df ...
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There is only one $z\in \Bbb C$ such that $z^2=w$ and $Re(z)>0$

Let $w\in\Bbb C$, show that unless $w\in \Bbb R^-$, there exists only one $z\in \Bbb C$ such that $z^2=w$ and $Re(z)>0$. This question is related to this other question, but this is a ...
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Which basic operations are undefined even for complex numbers?

I'm aware of: $\frac{X}{0}$ (dividing by zero) $0^0$ (raising zero to the power of zero) Are there any others?
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Complex numer equation

Let $n\in\mathbb{N}$. Determine all complex numbers $z\in\mathbb{C}$ such that $z^{n-1}$ = $\bar{z}$ . I'm not sure if I'm doing this question right, but would the solutions be $+ 1,-1$ or $0$?
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Complex number proof

Let f(x), g(x) $\in \mathbb C[x].$ Prove that if f(x) | g(x) and g(x) | f(x), then there exists a nonzero $c \in \mathbb C$ such that $f(x) = c * g(x)$ (You may use the fact that for any p(x), q(x) ...
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Complex Number Question - $|z^{z}|$

Find all possible values of $$\mid z^{z} \mid$$ using the polar for of $z$. I have tried putting it into polar form but nothing comes out that seems easy to work with/looks like a reasonable simple ...
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complex expression to the power of a complex expression

I have a math exam tomorrow, and i am not sure with my solution for a exercise. can you please tell me if i am right. Question is: $$(1+i)^{(1-i)}$$ My solution is: $$\sqrt{2} e^{(i ...
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proving a limit of a function by definition

Consider $f: \Bbb{C} \to \Bbb{C}$ defined by $$ f(z) = \begin{cases} z^3 + 2z &\text{if } z \ne i \\ 3 + 2i &\text{if } z = i \end{cases} $$ Prove that $$ \lim_{z \to i} f(z) = i $$ using the ...
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Convergence of complex power series question

I need some help to solve this problem and find the domain of convergence of the following power series: $$\displaystyle\sum_{n=0}^\infty(2^n+i^n)(z-2i)^n$$ Thank you!
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Example of holomorphic function from unit disc to itself

let $f:\mathbb{D} \to \mathbb{D}$ be analytic function with $f(0)=0$,where $\mathbb{D}$ is the open disc $\{z \in \mathbb{C}:|z|<1 \}$ then $1.|f'(0)|=1$ $2.|f(\frac{1}{2})|\leq \frac{1}{2}$ ...
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Proof that there are only three two-dimensional real algebras

I was reading this article http://en.wikipedia.org/wiki/Hypercomplex_number#Two-dimensional_real_algebras I'm trying to figure out how the step from completing the square to concluding that there are ...
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Complex number contour integral

Determine the contour integral ∫Ѱ 1/z dx, where Ѱ is the positively oriented unit circle with centre at -2 given that Ѱ(t) = -2 + e^(it), 0<=t<=2pii. I understand that Ѱ is the positively ...
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Calculating the DFT of a sequence following a mathematical expression

This is homework, so please don't give a full solution. Give a formula for $F_k$ for all $k$ where $f_n=4^n$ for all $n=0,\dots ,N-1$. I ended up abusing Wolfram Alpha and getting probably way ...