# Tagged Questions

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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### What is the benefit of representing a complex number as e^i(theta) versus e^(a+bi), what is the process of finding a solution to this example?

What is the benefit of representing a complex number as $e^{i\theta}$ versus $e^{a+bi}$? Am I correct in saying that these give the same information but offer convenience in different situations? ...
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### How to derive values for $i$ raised to negative integers? [closed]

This link states that the values of $i$ raised to the power of negative integers. How can we derive these values from the positive powers?
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### Taking Mod on both sides, mathematically correct?

When given a equation containing complex numbers such as $$\frac{a+ib}{c+id} = x + iy$$ and required to prove $$\frac{a^2 +b^2}{c^2+d^2} = x^2 + y^2$$ Is taking the mod of both sides a legal ...
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### Proof Involving Imaginary Number: Where's the wrong one? [duplicate]

Here are the propositions: $$i=\sqrt{-1}$$ $$i^2=-1$$ $$(i)(i)=-1$$ $$\sqrt{-1}\sqrt{-1}=-1$$ $$\sqrt{(-1)(-1)}=-1$$ $$\sqrt{1}=-1$$ There's an error in the propositions above. I think it's in the ...
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### modulus and argument of $(-4\sqrt{3}-4i)^3$?

Any fast method to obtain the modulus and argument of $(-4\sqrt{3}-4i)^3$? If i use the exponential form to solve it, is it good?
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### Help with De Moivre's Theorem: Complex Numbers

I have a homework problem which goes: Given $z^n=(z+i)^n$, using de Moivre's Theorem, show that $z=\frac{i}{e^\frac{i2k\pi}{n}-1}$ What steps should I take in tackling this question? It's a 2 mark ...
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### Image drawing complex analysis [closed]

$w=u+iv,z=x+iy$ are complex numbers and we have $w=z^2-2z$. Determine the image in the $w$-plane of the unit circle $x^2+y^2=1$. I have tried to answer this here Question and Answer. I have problems ...
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### Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$

Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$, if $z = a+ib$ then, I tried to do that with $|z| = (a+ib)(a-ib)$ then i multipled it $z$ with $z^-$ and then I got stuck. answer is $|z| = 5$
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### complex and decimal tetration

So recently in the blog post on tetration, it talked about tetration with nice clean powers (calling them these because I don't know the right term). But how does it work when given a complex power? ...
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### Roots of unity whose sum and product are known

Is cube root of unity is a complex number I know the sum is 0 and product is -1 but I am somewhat confused please give me some idea. Thanks in advance
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### Is there a way to prove that i²=-1? [duplicate]

I have 4 questions regarding the imaginary and complex numbers. (And some ideas) My questions are about the way that I’m trying to come up with a proof to the equation i²=-1 (and from there maybe ...
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### Number of roots of a complex exponent

There are $p$ solutions to $\sqrt[\frac{p}q]1$, if $\frac{p}q$ is a fraction in lowest terms. I have found on this website that an irrational exponent has infinite roots. But what about $\sqrt[a+bi]1$...
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nd we can consider this problem In general? if $|z_{1}|=|z_{2}|=\cdots=|z_{n}|=1$ if there exist complex $z(|z|=1)$ such $$\sum_{i=1}^{n}\dfrac{1}{||z-z_{i}||^2}\le C_{n}$$ find the best $C_{n}$? $$... 0answers 10 views ### Where does the formula \zeta_{qj} z=\frac{1}{2}H_{qj} z+\frac{1}{2}h_{qj} come from? I am reading the book space filling curves by Hans Sagan. I am reading about the complex representation of the Hilbert Curve. I came across the formula \zeta_{qj} z=\frac{1}{2}H_{qj} z+\frac{1}{2}... 1answer 47 views ### How to find the General expression of \sum_{k=0}^ {\lfloor n/3\rfloor} {n \choose 3k} [duplicate] Well as the title says I'm having problems trying to derive a general expression for this sum which involves cubic roots of unity$$\sum_{k=0}^ {\lfloor \frac n 3\rfloor} {n \choose 3k}$$Need help ... 4answers 66 views ### Linear algebra : Solving i \cdot\bar{z} = 2 +2i i\cdot\bar{z} = 2+2i I know that \bar{z} = a-bi so then i get i(a-bi)=2+2i Then ai+b=2+2i (because i^2=-1) When 2 complex numbers are equal you usually can equal their parts Ex: 2+2i=a+bi... 3answers 76 views ### e^{a 2\pi i} = (e^{2\pi i})^a. When a is any real number , Is it true e^{a 2\pi i} = (e^{2\pi i})^a ? The reason why I ask this question is that I met this situation wheter this equality hold in Calculating Integral in Complex ... 1answer 52 views ### How do you express cos(nx) + sin(nx) in terms of eulers constant [closed] The examples I have seen express \cos(nx) + i\sin(nx) which comes to e^{i\,nx}. Is there a way to use Euler constant when both the cos and sin have the same coefficients? 1answer 38 views ### How does analytic continuation lets us extend functions to the complex plane? I'm trying to understand analytic continuation and I noticed on wolfram that it allows the natural extension of the definition trigonometric, exponential, logarithmic, power, and hyperbolic ... 2answers 58 views ### Complex numbers inside determinant Let  \begin{vmatrix}6\iota & -3\iota & 1\\ 4 & 3\iota & -1\\ 20 & 3 & \iota \\ \end{vmatrix}= x +\iota y, then what are the values of x and y? 1answer 57 views ### Evaluate \cos \frac{\pi}{7} \cos \frac{2\pi}{7}\cos \frac{4\pi}{7} Evaluate$$\cos \frac{\pi}{7} \cos \frac{2\pi}{7}\cos \frac{4\pi}{7}.$$The first thing i noticed was that$$\cos \frac{\pi}{7}=\frac{\zeta_{14}+\zeta_{14}^{-1}}{2},$$where \zeta_{14}=e^{2\pi i/14}... 2answers 17 views ### Complex Conjugate roots with non real coefficients I understand that a polynomial with real coefficients must have complex conjugate roots (if complex roots exist) Is it possible for a polynomial with non-real coefficients to have complex conjugate ... 1answer 36 views ### Visualizing a complex function Ever since I learned about complex valued functions I've been wondering if there was a better visualization for them. Obviously we can't visualize four dimensions, but I was wondering if it would be ... 1answer 43 views ### Least value of complex expression If z_{1},z_{2},z_{3},z_{4} are 4 points on a circle |z| = 1 such that z_{1}+z_{2}+z_{3}+z_{4}=0\;, Then least value of expression |z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{4}|^2+|z_{4}-... 3answers 78 views ### Solve in \mathbb{C} : |z-i| = |z-1| I just had that question in my final exam Solve in \mathbb{C} : |z-i| = |z-1| and I couldn't do it. I found a similar thread here : Showing that \{z\in\mathbb{C}:|z-1|<|z+i|\} is an open ... 0answers 27 views ### Bounded real parts of the solutions of an equation I'd like to show that, with a,b>0, the real parts of the solutions z_n of the equation$$ az+\sqrt{z^2-ib}\tanh\sqrt{z^2-ib}=0 $$are bounded. An indication for that can be found if we ... 1answer 29 views ### biholomorphic on unit disk Let D be the unit disk and f: D\rightarrow G, \; p_1 the maximum value of dist(f(z),f(0))\; and p_2 the minimum value of dist(f(z),f(0)) for z\in \partial \bar G Prove that : |f(z)-... 1answer 31 views ### Sine and cosine solutions of a differential equation I have to solve a differential equation with constant coefficient such as$$ay'''+by''+cy'+dy=f(x)$$which has for a characteristic equation$$P_c(\lambda)=a\lambda^3+b\lambda^2+c\lambda+d=0$$First I ... 0answers 20 views ### Newton's method for nth roots of complex numbers Is it possible to use Newton's method to compute roots of complex numbers, say \sqrt[n]{a+ib} to any desired accuracy? If yes,for what initial values will converge? 2answers 46 views ### Proving (w-1)^m is purely imaginary. I'm having trouble trying to prove this: Let  m\in \mathbb Z, m even and w\in\mathbb C a primitive 2m-th root of unity. Prove that (w-1)^m is purely imaginary. What I've tried to do so ... 1answer 94 views ### Why is De Moivre's theorem not generalised for (\sin x+i\cos x)? A representation of the form (\sin x+i\cos x)^n can be reduced as follows$$( \sin x + i \cos x )^n( \cos (90-x) + i \sin(90-x) )^n( \cos (90n - nx) + i \sin(90n - nx) )$$Now for all ... 11answers 42k views ### How to prove Euler's formula: e^{it}=\cos t +i\sin t? Could you provide a proof of Euler's formula: e^{it}=\cos t +i\sin t ? thanks. 1answer 42 views ### Sufficient condition on open subsets to be equal Let U,V\subseteq\Bbb C connected open non empty, such that their closure in \Bbb C, say \overline U,\overline V, be simply connected. Then, is it true that, if$$ U\cap V\neq\emptyset\\ \...
I need to take a raincheck with this problem. I want to make sure I haven't messed up some fundamental idea. Convert the complex number $$-\dfrac{1}{2} + \dfrac{\sqrt3}{2}i$$ to polar form. I ...