Questions tagged [complex-numbers]

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

A complex number is a number in the form $z=a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, or alternatively, $z=r\cdot e^{i\theta}$, with $r$ called the magnitude and $\theta$ called the argument.

The complex conjugate, $\overline z$, is $a-bi$ or $r\cdot e^{-i\theta}$.

Read more about complex numbers and their properties here.

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The derivative $\frac{\mathrm d}{\mathrm dx} x^x=x^x\left(\ln x+1\right)$ is problematic for $x<0$

To take the derivative of $x ^ x$, we write $$\dfrac {\mathrm d}{\mathrm dx} x^x=\dfrac {\mathrm d}{\mathrm dx} e^{\ln x^x}=\dfrac {\mathrm d}{\mathrm dx} e^{x\ln x}= e^{x\ln x}× \dfrac {\mathrm d}{\mathrm dx}(x\ln x)=x^x\left(\ln x+1\right)$$ Here…
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Using Fundamental Theorem of Algebra to find $z_0$ such that $|p(z_0)| < |p(0)|$

A question from the book Hubbard's Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach (5th edition): I was able to easily find some point $z_0 = i/3$ (just by trying out points), but am unsure how to use their equation in…
user246678
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Multiplicity of Complex Conjugates of Repeated Complex Eigenvalues

I know that for a real-valued matrix, complex eigenvalues come in complex conjugate pairs. However, I'm wondering what happens for repeated complex eigenvalues (i.e. complex eigenvalues with multiplicity greater than 1). In that case, does the…
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Determine all complex numbers which satisfy conditions - $|z|=2$ $\space$ and $\space$ Im$(z^6)=8$ Im$(z^3)$

Determine all complex numbers $z$ which satisfy following conditions: $|z|=2$ $\space$ and $\space$ Im$(z^6)=8$ Im$(z^3)$ I first calculated $z^3$ and…
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A method for solving cubic equation

So I'm reading Beardon's Algebra and Geometry, and in chapter on complex numbers, author gives the following method for solving cubic equation: Suppose we want to solve cubic equation $p_1(z)=0$, where $p_1(z)=z^3+az^2+bz+c$. Now…
Sarunas
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When does equality hold in $\Bigr\lvert\sum_{k=1}^n a_kb_k\Bigr\rvert^2 \le \left(\sum_{k=1}^n |a_k|^2\right)\left(\sum_{k=1}^n |b_k|^2\right)$?

I'm reading Ahlfors' complex analysis. During the proof of Cauchy's inequality, the author uses the following equation: $$ \sum_{k=1}^n \bigr\lvert a_k - \lambda \overline{b_k}\bigr\rvert^2 = \sum_{k=1}^n |a_k|^2 + |\lambda|^2\sum_{k=1}^n |b_k|^2 -…
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If $a_i\in\mathbb{R}$, $\omega^2+\omega+1=0$, and $\sum_{i=1}^n\frac{1}{a_i+\omega^k} =2\omega^{2k}$ for $k=1,2$, find $\sum_{i=1}^n\frac{1}{a_i+1}$.

In this question, $\omega$ is the complex cube root of $1$ and $a_i \in \mathbb R$. If $$\sum_{i=1}^n \frac{1}{a_i + \omega} =2\omega ^2$$ and $$\sum_{i=1}^n \frac{1}{a_i + \omega ^2} =2\omega\,,$$ then find $$\sum_{i=1}^n \frac{1}{a_i +…
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Geometry solution involving complex numbers from USAMO

Quadrilateral $AP BQ$ is inscribed in circle $ω$ with $∠P = ∠Q = 90^{\circ}$ and $AP = AQ < BP$. Let $X$ be a variable point on segment $P Q$. Line $AX$ meets $ω$ again at $S$ (other than $A$). Point $T$ lies on arc $AQB$ of $ω$ such that $XT$ is…
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Well-ordering of positive Gaussian integers under lexicographical ordering?

I am reading a paper by Richard Weimer called "Can the complex numbers be ordered?" and he makes the following claim. Let $G^+=\{a+bi : a,b$ are positive integers $ \}$ and let $<$ denote lexicographical ordering. It is easily demonstrated that…
jamie
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Why does it make sense to talk about the 'set of complex numbers'?

In my complex analysis course we've discussed quite a few times the idea that $\mathbb{C}$ is really 'the same thing' as $\mathbb{R}^2$ with the added complex multiplication operation. I've also read a number of the popular posts here including this…
masiewpao
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combination of points in the open unit disk also lie in the unit disk

Suppose $a,b$ are points in the open unit disk $\{z\in\mathbb{C}:|z|<1\}$. Then, does the combination $$\frac{(1-|a|^2)b+(1-|b|^2)a}{1-|ab|^2}$$ also lie in the unit disk (open)? I think yes, but am unable to prove. The triangle inequality does not…
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Different interpretations of imaginary number

I went through a linear algebra course and I'm a bit confused.. I think I understand the geometric interpretation of imaginary numbers - multiplying by $i$ results in rotation by $90$ degrees in so that $1$ becomes $i$ and so forth. And this is…
user315648
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Closedness of a subset of complex numbers under addition

If we have two angles $$\phi_1,\phi_2\in[0,2\pi]$$ such that $$\phi_1\le\phi_2$$ and we perform standard addition on complex numbers from the subset of $$S=\{ z\in \mathbb{C} : arg(z) \in[\phi_1 ,\phi_2]\}$$ do we get a closed algebraic structure…
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Complex conjugations

I'm stumped on an equation from a coursera course (Intro to DSP) that has to do with complex exponential multiplication: In line 2, by simple rules of complex multiplication, it should be (h+k). I understand it being (h-k) has to do with complex…
erap129
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Can someone explain the solution to "If $x^2+ax−3x−(a+2)=0$ has real and distinct roots, then minimum value of $(a^2+1)/(a^2+2)$ is?"

Could we not have directly skipped to the last line of the solution? The conclusion seems disjointed from the previous lines. [Question]: If $x^2+ax−3x−(a+2)=0$ has real and distinct roots, then minimum value of $(a^2+1)/(a^2+2)$ is...? [Given…
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