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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1answer
141 views

Let $\cos (bx)-\sin ^2(x)-1=0$ has no zero point except 0. What's the value of b?

Let $\cos(bx)-\sin^2(x)-1=0$ has no zero point except 0. What's the value of b? I have plotted many graphs of the function for several $b$. I think b can only be complex number. Is that right?
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2answers
112 views

Prove that $4w^2 + 5z^2 = 4z^2 + 5w^2$ if $|z| = |w|$ and $4z^2+5w^2=azw$

I know that: $$|z| = |w| = p$$ $$4z^2 + 5w^2 = azw ,\qquad a \in R$$ I need to prove that $4w^2 + 5z^2 = azw$ How I solved it is: $$|z| = p \implies |z|^2 = p^2 \implies zz^* = p^2$$ then solved ...
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2answers
2k views

What is $(1 − i)e^{{i\pi}/4}$ equal to?

I don't know where to start... It's a multiple-choice question: I can choose from $\sqrt{2}, 0, 2, 1$ Thank you!
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3answers
242 views

Convergence of the arguments of a sequence of complex numbers

Suppose the sequence $z_{n}$ converges to a nonzero limit $A$ and let $\Phi_{n}$ be any sequence of values of $Arg (z_{n})$ satisfying the inequality $$|\Phi_{m}-\Phi_{n}|<\pi$$ for $m>N, ...
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2answers
1k views

Grassmann Variables and Complex Conjugate

While dealing with Grassmann Variables, the complex conjugate is defined as $$ (\phi \psi)^{\dagger} = \psi^{\dagger} \phi^\dagger $$ and why not $ \phi^{\dagger} \psi^\dagger $. I want to know the ...
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1answer
568 views

Complex Numbers in Fractal Algorithms

I am a high school freshman who is undertaking a small development project on fractals. I do not want to get too in depth, but I would love to blow my math teacher's socks off. Having looked through ...
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4answers
660 views

complex number question involving modulus

Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then $$\frac{|z − w|}{|1 − z^*w|} = 1$$ [Hint: Note that $|a|^2 = aa^*$.] Hey guys, couldn't get my ...
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1answer
157 views

Prove the equation has no root in the circle $|z| < 1$

Suppose that $0 < a_0 \le a_1 \le \dots \le a_n$. Prove that the equation $$P(z) = a_0z^n + a_1z^{n-1} + \dots + a_{n-1}z + a_n = 0$$ has no root in the circle $|z| < 1$.
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2answers
241 views

Solving complex equation

How do i further solve the following complex equation: $$ z\cdot \bar{z} + z + \bar{z} + i\cdot z - \overline{i \cdot z} = 9 + 4i $$ $$ a^{2} - b^{2} + 2a - 2b = 9 + 4i$$ How do i solve from here ...
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2answers
154 views

Getting a real number from a complex number

I'm attempting to program a formula to say how full an horizontal cylinder is with liquid. Here is the formula I am using with variables from measurements I took: When I use Wolframalpha to solve ...
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1answer
107 views

Can a convergent sum using only integers produce a complex result?

We use this function to define the boundaries for the product in the denominator: $$f(\text{n$\_$})\text{:=}\frac{1}{8} \left(2 n (n+2)-(-1)^n+1\right)$$ We calculate the infinite sum: $$\sum ...
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3answers
903 views

How to plot $\{z \in \mathbb{C} : |z-i|>|z+i|\}$

How would I draw the set $\{z \in \mathbb{C} : |z-i|>|z+i|\}$ and $\{z \in \mathbb{C} : |z-i|\not=|z+i|\}$? Im not sure how to solve the second one, and for the first one, I tried squaring both ...
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1answer
1k views

Finding roots of unity?

The $n$th roots of unity are the complex numbers: $1, w,w^2,...,w^{n-1}$, where $w=e^{\frac{2\pi i}{n}}$. Why is this true? I understand why $w$ is 1 root of unity, but why are $w^0,..., w^{n-1}$ ...
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1answer
189 views

A way to define the imaginary axis

There are some ways to define the imaginary axis.Some are obvious, like $\space Re(z)=0 \space$ others not. I set up a condition that I think defines the imaginary axis. Let $x$ be a real number, ...
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2answers
117 views

complex equation

In Gaussian plane draw solution of equation |z-(1+2i)|=2 My solution: Wolfram solution: I don't understand, why my solution is not right. Could anyone help me, please?
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2answers
62 views

Is there a similar function in complex number system corresponding to logarithim in real number system?

i notice that there are $e^{i\theta}$ in math,so is there a similar function in complex number system corresponding to logarithim in real number system?
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2answers
94 views

Induction (concerning $1+z+\dots+z^n$) and follow up question

I am doing a review of stuff from earlier in the semester and I can't prove this by induction: Use induction on $n$ to verify that $1+x+\cdots+z^n= \frac{1-z^{n+1}}{1-z}$ (for $z\not=1)$. Use this ...
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5answers
376 views

Finding the n-th root of a complex number

I am trying to solve $z^6 = 1$ where $z\in\mathbb{C}$. So What I have so far is : $$z^6 = 1 \rightarrow r^6\operatorname{cis}(6\theta) = 1\operatorname{cis}(0 + \pi k)$$ $$r = 1,\ \theta = \frac{\pi ...
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1answer
235 views

Why are the coefficients of the base states of a qubit complex numbers?

Why are qubits represented as $$\left|{q}\right\rangle = \alpha\left|{0}\right\rangle+\beta\left|{1}\right\rangle\equiv\alpha\left[{1 \ 0}\right]^T+\beta\left[{0 \ 1}\right]^T; ...
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2answers
140 views

Finding Complex Number $z$ in $\frac{z+2i}{z-2i}=\frac{7-6i}{5}$

What I did: Cross Multiply, try to expand out the mod and args, but they all seem to lead to dead end (probably I am not seeing something)
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3answers
974 views

Find opposite vertices of a rhombus, given the other 2

I am stuck with this problem. I posted an earlier problem with a square, where rotation with i of 90 degrees was possible. This one is a rhombus, how should I proceed? Given ABCD is a rhombus with ...
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2answers
484 views

How to express in closed form?

How to express this function in closed form without condition verfication and Re and Im functions (only with absolute value function)? $$f(z)= \begin{cases} - ...
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2answers
149 views

Complex inequality $||u|^{p-1}u - |v|^{p-1}v|\leq c_p |u-v|(|u|^{p-1}+|v|^{p-1})$

How does one show for complex numbers u and v, and for p>1 that \begin{equation*} ||u|^{p-1}u - |v|^{p-1}v|\leq c_p |u-v|(|u|^{p-1}+|v|^{p-1}), \end{equation*} where $c_p$ is some constant dependent ...
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1answer
24 views

Dense on the unit circle

I am reading: "It is sufficient to show that the points $z_n = e^{2\pi in \xi}$ $\:\:n = (1, 2, 3...)$ are dense on the unit circle. ( $\xi$ is an irrational number)" How is this possible? Can ...
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1answer
43 views

Limits and L'Hopital

$$\lim_{z \to i} \frac{z^4-1}{z-i}$$ I'm reading in a bunch of places that I can't use L'Hopital's rule for this problem. Why is this so? And if I can't use this rule then how would I go about ...
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1answer
42 views

Sketch the region described by $\text{Im}\left[\frac{z-z_{1}}{z-z_{2}}\right] =0$

This question is similar, but please do not mark this as a duplicate, because it is not exactly the same as what is being asked in that question, and besides, the answers given to it do not answer my ...
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1answer
30 views

Sketch the region defined by $Re z + Im z < 1$

I have to sketch the region given by $Re z + Im z < 1$, and I'm stuck. For any complex number $z$, $Re z = x$ and $Im z = y$. Is this as simple as graphing the inequality $x + y < 1$, then? If ...
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1answer
36 views

Write $e^{\ln(5)}i$ in polar and rectangular form

Is there something I'm missing? Below is my attempt, but I feel as though I might have missed something to learn about complex numbers when $r=e^n$. $$|ie^{\ln(5)}|=e^{\ln(5)}$$ ...
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1answer
40 views

How do you find the sum: $\sum_{r=1}^6 \tan^2\left(\frac{r \pi}{n}\right)$

How do you find the sum: $$\sum_{r=1}^6 \tan^2\left(\frac{r \pi}{n}\right)$$ I managed to solve this question using complex numbers so I thought I'd share the solution. If you know of any better ...
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2answers
69 views

Is there a quicker way to compute this?

A complex sequence $(z_n)$ is given by: $$z_n=\frac{e^{in^2}}{1+in^2}$$ I found it's absolute value, $|z_n|=\frac{1}{\sqrt{1+n^4}}$, by multiplying the top and bottom by the conjugate of the complex ...
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2answers
30 views

Argument of $\pi e^{-\frac{3i\pi}{2}}$

Find the argument of $\displaystyle \pi e^{-\frac{3i\pi}{2}}$ I thought the formula for the argument was $\arg{z} = i\log\frac{|z|}{z}$ In this case $|z|= \pi$, so it turns out that $-i ...
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1answer
34 views

How can I find the conjugate of $\frac{z}{1+z^2}$ uniquely in function of $z$?

I'm a bit blocked actually, how can I find the conjugate of $\frac{z}{1+z^2}$ uniquely in function of $z$? Is anyone is able to give me a hint?
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1answer
61 views

Is there an extension of the complex numbers in which $|z|^2 = -1$ admits a solution?

Just as $x^2 = -1$ required the invention of imaginary numbers, I can think of another equation ie $|z|^2 = -1$ where $z=a+ib$. For this, don't we need to go to a higher dimension space (i.e., 3) and ...
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3answers
45 views

Solving equation involving complex numbers

(a) Find real numbers $a$ and $b$ such that $(a+bi)^2 = -3-4i.$ (b) Hence solve the equation: $z^2+i\sqrt{3}z+i = 0$. Original Image In the above question, I have solved part (a), with ...
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2answers
23 views

If $z$ and $\omega$ are two complex no. and $\theta = \arg\left(\frac{\omega -z}{z}\right)\;,$ Then Max. of $\tan^2 \theta$

If $\omega$ and $z$ are two complex number such that $|\omega| = 1$ and $|z|=10$ and Let $\displaystyle \theta = \arg\left(\frac{\omega -z}{z}\right)$ Then Maximum possible value of $\tan^2 ...
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1answer
27 views

Suppose $f$ and $g$ are holomorphic in $G$ and $\gamma \sim_G0$. Prove that if $f(z)=g(z)$ for $z\in\gamma$, then $f(z)=g(z)$ for $z$ in $\gamma$.

Suppose $f$ and $g$ are holomorphic in the region $G$ and $\gamma \sim$ $_G$ $0$. Prove that if $f(z)=g(z)$ for all $z\in\gamma$, then $f(z)=g(z)$ for all $z$ inside $\gamma$.
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1answer
51 views

Integration over complex plane

I have a problem with the following integral $$\int_{-\infty}^{\infty}\frac {x\sin x}{x^4+1}$$ Can someone please help me with the way the solution goes? I would highly appreciate it Thanks in ...
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1answer
39 views

Evaluate this integral using residue theorem

So we have $\int_{0}^{+\infty}\dfrac{x^2-a^2}{x^2+a^2}\cdot\dfrac{\sin x}{x}dx$. ($a>0$). I considered that we can just calculate the half of the imaginary part of ...
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3answers
53 views

Complex numbers problem

I'm trying to use the relevant rules and definitions but making little progress.
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1answer
32 views

Show that $-\log(1-\mathrm{e}^{\mathrm{i}x}) = -\log\left(2\sin\left(\frac{x}{2}\right)\right) + \mathrm{i}\dfrac{\pi - x}{2}$

Show that $$ -\log(1-\mathrm{e}^{\mathrm{i}x}) = -\log\left(2\sin\left(\frac{x}{2}\right)\right) + \mathrm{i}\dfrac{\pi - x}{2}. $$ This is a last step in one of my problems, and I know the two ...
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2answers
34 views

A general approach for this type of questions

I've come across multiple questions like these. $\left(\iota=\sqrt{-1}\right)$ If $f\left(x\right)=x^4-4x^3+4x^2+8x+44$, find the value of $f\left(3+2\iota\right)$. If ...
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1answer
23 views

Adding waves of equal frequency using phasors

Given the following equation I have to find $a$ and $\phi$: $$a_1 \cos(\omega t + \phi_1) + a_2 \cos(\omega t +\phi_2) = a \cos(\omega t + \phi)$$ Is there an easy way to construct $a$ and $\phi$? I ...
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2answers
101 views

How to check if a set of vectors span a space? [duplicate]

how do I check if the following set of vectors $[3+i,0,i],[0,1+3i,2],[6+2i,3+9i,6+2i]$, or in general any set of complex vectors span the space $\Bbb C^3$, not using any matrix methods? Thanks for ...
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1answer
67 views

How does a complex graph of $x^y$ look?

What would a graph of a function $f: \mathbb{R}^2 \to \mathbb{C}$ look like if $$f(x,y) = x^y$$ This is a question that's been on my mind since I was first introduced to exponential functions. I know ...
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1answer
30 views

How to get the geometric shape of an amount with complex numbers?

I am trying to solve the following equation. M is an amount $$M=\left\{z\in\mathbb{C}\colon \left|\left(\sqrt 2 - i\sqrt 2\right)z+\sqrt 2i\right|=2\right\}$$ I understand that I will get a circle ...
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2answers
65 views

Intuition behind Re(z) not being analytical

Is there any way to "see" the reason for which the Re(z) is not analytical? Edit: what I need is intuition(and maybe something graphic) and not definition of analytic function. So, the question is, ...
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2answers
40 views

If z and w are complex numbers can we use the proof in $\mathbb{R}$ to demonstrate that $|z w|=|z||w|$?

If yes could you explain why? Sorry if the question is trivial but I'm new to complex numbers and I see lots of examples where properties of real numbers are used in complex without to prove it . This ...
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2answers
77 views

Solving $z^6=64$

$$z^6=64$$ My attempt: $$\stackrel{\text{Euler}}{=}[r(\cos(\theta)+i\sin(\theta))]^{6}=64$$ $$\stackrel{\text{De moivre}}{=}r^6(\cos(6\theta)+i\sin(6\theta))=64$$ ...
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2answers
65 views

How to interpret the plot of a function of a complex variable?

I know what a complex number is: $a+bi$. But I have seen these functions that make no sense to me, something such as this: $$f(z)=z^2+1$$ where $z$ is a complex number. Does this have to do with ...
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3answers
62 views

Logarithm of imaginary numbers?

How do I solve this question? I tried using the quadratic formula on the question equation and got $x_1 = 0.25 +1.089724..i = \ln r$ $x_2 = 0.25 -1.089724..i = \ln s$ I know $\ln x = \log_ex$, ...