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

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3answers
109 views

How to express $\sqrt{x} =-1$?

How would one express a solution to $\sqrt{x} =-1$? I just read that a solution to the above equation cannot be expressed in the form of complex numbers, really interested in any additional ...
1
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1answer
37 views

Prove that $e^{\lambda A}Be^{-\lambda A}=B$

Prove that $$e^{\lambda A}Be^{-\lambda A}=B$$ if $[A,B]=0$. $A$ and $B$ are operators and $\lambda$ is a complex number. Can anyone explain how I should go about this question? How do I calculate ...
3
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5answers
70 views

Find $z$ s.t. $\frac{1+z}{z-1}$ is real

I must find all $z$ s.t. $\dfrac{1+z}{z-1}$ is real. So, $\dfrac{1+z}{1-z}$ is real when the Imaginary part is $0$. I simplified the fraction to $$-1 - \dfrac{2}{a+ib-1}$$ but for what $a,b$ is ...
0
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1answer
20 views

How to Prove this Fractional Linear Transformation of $\mathbb C$ takes $S^1$ to itself?

Let $x\in\mathbb C$. I know that $|x|<1$ but I don't think that matters for what I'm about to ask. Let $f$ be the fractional linear transformation $f(z)=\frac{z-x}{1-\overline x z}$. Then I'm ...
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2answers
32 views

Finding a Laurent Series involving two poles

Find the Laurent Series on the annulus $1 < |z| < 4$ for $$R(z) = \frac{z+2}{(z^2-5z+4)}$$ So I am having a few issues with this. I know there are two poles in this problem particulaly $z = ...
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1answer
55 views

Find all complex numbers such that $z^4 = 8\bar{z}$ [closed]

This one is tricky because we need to find all $z\in \mathbb{C}$ such that $$z^4 = 8\bar{z}$$
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2answers
47 views

Problem when $x=\cos (a) +i\sin(a),\ y=\cos (b) +i\sin(b),\ z=\cos (c) +i\sin(c),\ x+y+z=0$

Problem : If $$x=\cos (a) +i\sin(a),\ y=\cos (b) +i\sin(b),\ z=\cos (c) +i\sin(c),\ x+y+z=0$$ then which of the following can be true: 1)$\cos 3a + \cos 3b + \cos 3c = 3 \cos (a+b+c)$ 2)$1+\cos ...
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1answer
32 views

$λ={41/10\left(\frac{1}{2^2-1}+\frac{1}{4^2-1}+\frac{1}{6^2-1}+..+\frac{1}{40^2-1}\right)}$ then $w+w^λ$ is equal to

Given that, $$λ={41/10\left(\frac{1}{2^2-1}+\frac{1}{4^2-1}+\frac{1}{6^2-1}+..+\frac{1}{40^2-1}\right)}$$ then $w+w^λ$ is equal to ? [$w$ is cube root of unity other than 1] I cannot understand how ...
0
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1answer
23 views

Complex Roots Of a equation - Equilateral triangle

$z_1$ and $z_2$ are the roots of $3z^2+3z+b=0$.If $O(0),A(z_1),B(z_2)$ is an equilateral triangle then what will be the value of b ? My approach:I took $z_1=m_1+in_1$ and $z_2=m_2+in_2$ and proceeded ...
0
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2answers
30 views

Proof of index laws for complex numbers

Can someone give a proof that index laws (and hence log laws) apply for complex numbers in the same way they do to reals, specifically that: $(a^{ix})^n = a^{ixn}$ Assuming $a, x, n$ are real and ...
0
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0answers
18 views

Complex number, integration

Calculate $\int_Cf(z)dz$ i)$f(z)=\frac{z+2}{z}$ and C is the semi-circle described by $z=2e^{i\theta},0\leq\theta\leq\pi$ ii)$f(z)=\frac{z+2}{z}$ and C is the semi-circle described by ...
2
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1answer
26 views

Questions on convergence of nested series

Say I have $\sum\limits_{n=1}^\infty e^{zn}$. We know that for $z \in \mathbb{C}$ and $n \in \mathbb{N}$ the series form of $e^{zn}$ converges. Does $\sum\limits_{n=1}^\infty e^{zn}$ converge then? ...
0
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2answers
25 views

Complex integration, complex number

Proof that $\int_0^{\frac{\pi}{6}}e^{i2t}dt=\frac{(\sqrt{3}+1)}{4}$ $\int_0^{\frac{\pi}{6}}e^{i2t}dt=\frac{e^{i\frac{\pi}{3}}}{2i}-\frac{-1}{2i}=\frac{e^{i\frac{\pi}{3}}-1}{2i}$ I know that ...
0
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1answer
41 views

Image of circle under linear transformation [closed]

Prove that the image of a circle under a linear transformation is a circle. Hint : - Let circle have parameterization $$x = x_o + R \cos(t),\qquad y = y_o + R \sin t \tag t$$ After finding matrix ...
1
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2answers
64 views

Complex Number use in Daily LIfe [duplicate]

What are the different properties of Complex Numbers. ? I have doubt on real life use of complex numbers. Where and in what conditions do we use complex numbers in our day to day life. My main focus ...
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2answers
35 views

Modulus of complex number

$$ |2e^{it}-1|^2$$ I don't understand how to work this out, I know if I had for example $|2ti-1|^2$ then I would square the real and imaginary parts and add them to get the modulus squared, but here ...
1
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0answers
47 views

Inequality with complex number

Let $z,z'\in\mathbb{C}$. I want to prove that $$\vert\vert z\vert^{\alpha}z-\vert z'\vert^{\alpha}z'\vert\leq C (\vert z\vert^{\alpha}+\vert z'\vert^{\alpha})\vert z-z'\vert$$ where $\alpha$ is an ...
0
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2answers
24 views

Complex number manipulation involving taking modulus

I'm trying to work through a problem which involves proving that a given ring is a euclidean ring but I'm a little rusty on manipulating complex numbers. It is given that $w=\frac{-1+\sqrt{-3}}{2}$. ...
2
votes
0answers
37 views

A (analytical) Geometric Way to solve this complex number problem?

Problem Let $a_0,a_1,a_2$ be three complex numbers that lie on the circle $C$ with center $K(2,0)$ and radius $r=1$. Let $v \in \mathbb C$ such that $$v^3 + a_2v^2 + a_1v + a_0 = 0$$ ...
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1answer
31 views

Calculate $\cos(z)/(z^2-\pi^2)$ using Cauchy integral formula on region |z|=4

I want to verify if my reasoning and answer is correct here. Since $\pi$ and $-\pi$ are both contained within the circle centered at 0 with radius 4, we can use the Cauchy integral formula to deal ...
2
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1answer
19 views

Calculate $\sin(z)/(z+i)$ using Cauchy Integral Formula on region $|z+i|=3$

I just want to know what I'm doing wrong here. So we have a singularity at $z=-i$ but this is inside the region of circle centered at $-i$ with radius 3. Hence by Cauchy Integral Formula we have ...
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0answers
20 views

Rate of convergence when substituting sequence by its limit

Let $(z_n)_{n \in \mathbb{N}}, (w_n)_{n \in \mathbb{N}}, (w_n')_{n \in \mathbb{N}}$ sequences of complex numbers with $|z_n| = 1$ for all $n \in \mathbb{N}$. Assume that $z_n \to z$ and $w_n \to w, ...
2
votes
3answers
32 views

Showing that $i\arg(e^{2z})=2iy?$

Does $i\arg(e^{2z})=2iy?$ If it does I have solved my problem, and hence it seems like it must be the case, but I don't see it. $$i\arg(e^{2z})=i\arg(e^{2x+2iy})=i\arg(e^{2x}e^{2iy})\implies ...
5
votes
2answers
40 views

Factoring a quadratic equation with complex numbers

I'm very new to complex numbers and am having some difficulty factoring a quadratic polynomial: $$x^2-2x+10.$$ Using the quadratic formula gives $$x=\frac{4 \pm\sqrt{4-2(1)10}}{2(1)}=\frac{2 \pm ...
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1answer
46 views

Proving a complex numbers result

$z$ is defined as $z=a+bi$. Show that $|z|^2=zz^*$ and $(z-ki)^*=z^* +ki$. In an argand diagram a set of points representing the complex number $z$ is defined by the equation ...
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1answer
32 views

An analytic function such that $|f^2(z)-1|=|f(z)-1|.|f(z)+1|<1.$

Let , $f$ be an analytic function such that $$|f^2(z)-1|=|f(z)-1|.|f(z)+1|<1.$$ on a non empty connected set $U$. Then (A) $f$ is constant . (B) $Im (f)>0$ on $U$. (C) $Re(f)\not = 0$ on ...
2
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1answer
23 views

d/g is a real number if the roots of the equation x^2 + dx + g^2 = 0 have the same absolute value?

$d$ and $g$ are complex numbers and $g$ is not eqaul to $0$. Prove that if the roots of the equation $$x^2 + dx + g^2 = 0$$ have the same absolute value, then $d/g$ is a real number. I tried to ...
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2answers
28 views

Is function holomorphic for all complex numbers

pls for help/check. I have a function: $f(z)=3e^{i(z)}$. Is this function holomorhics for all complex numbers? $f(x+iy)=3e^{i(x+iy)}=3e^{-y}\cos (x)+i3e^{-y}\sin (x)$ so $Re=3e^{-y}\cos (x)$ and ...
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1answer
37 views

Find the line integral of $1/(z^2+4)^2$ over region $\gamma$

I have to find: $$I=\oint_{\gamma}\frac{dz}{(z^2+4)^2}.$$ $\gamma$ in this case is a circular curve defined by $|z-i|=2$, which is a circle centered at $i$ with radius $2$. It is clear that the ...
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0answers
20 views

Find Line integral of $e^{-z} /{z-\pi/2}$ on a region $\gamma$

Let $\gamma$ be the diamond connecting points $x=2, -2$ and $y=2, -2$. and its oriented positively (counter-clockwise, I believe?). I'm not so sure if we can use the Cauchy integral formula here and ...
2
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1answer
64 views

Weird problem $z^i=i$

Weird problem $z^i=i$: $$i\ln z=i$$ Then: $$\ln z=1$$ Therefore: $$e^1=z\implies z=\cos(1)+i\sin(1)$$ is that right? I was told it's not.
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0answers
43 views

What does it mean to raise a complex number to a complex power?

What does the formula $ (a+ib)^{x+iy}$ mean? All variables are real here, and $i $ represents $\sqrt {-1}$.
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2answers
17 views

how to find the argument of a multipication of complex numbers? [closed]

I know how to find Arg(z). I'm having problem with the following expression: $$Arg((\frac{\sqrt{3}}{2}+ \frac{i}{2})^{16}(\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i)^{10})$$
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1answer
33 views

Complex numbers - addition of two modulus help

Ok, so I got the answer to part i), but however, I'm not so sure how to get the answer to part ii). The answers say its an ellipse and they specified the equation, but I can't understand how they came ...
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2answers
52 views

How can “$y = \sqrt{-x}$ ” be sketched on the x-y plane?

I was reading James Stewart's book Calculus 5th edition (international student edition) and I came across an example that seemed wrong to me. In chapter 1 section 3, he talks about transformations of ...
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2answers
29 views

Inequality with absolution value for complex number

How to show that inequality: $|1-\bar{\alpha} z| \ge |z-\alpha|$ $z$ and $\alpha$ are complex number, $\alpha$ is constans and $|z|<1$, $| \alpha| < 1$ I can proof that by using substition ...
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1answer
19 views

Does Fermat's Theorem for Stationary Points hold for functions $f: \mathbb C \to \mathbb R$

Given a function $f: \mathbb C \to \mathbb R$ ($z = x+yi, \; x,y \in \mathbb R)$ Does this hold? $f$ has an extremum at $ z_0 = x_0 + iy_0$ $f$ is differentiable at $S$ and $z_0 \in S$ ...
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2answers
38 views

Finding the minimum value of magnitude of this complex number

|z|>2 then find the minimum value of "|z+1/2|" How can I solve this using circles?
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2answers
51 views

If $\sum (2009-r)\cos(\frac{2\pi r}{2009})=-n/2;\quad 1\leq r\leq 2008,$ then the digits in the unit's …

$$\text{If}\;\sum (2009-r)\cos\left(\frac{2\pi r}{2009}\right)=-n/2;\quad 1\leq r\leq 2008$$ then the digits in the unit's place of $(9417709487)^n$ must be equal to? Well how do I proceed? Hints ...
2
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1answer
26 views

Let $A_1,A_2,..,A_n$ be the vertices of n sides of a regular polygon such that $1/A_1.1/A_2=1/A_1.1/A_3+1/A_1.1/A_4$ then value of $n$ must be?

Let $A_1,A_2,..,A_n$ be the vertices of n sides of a regular polygon such that $$\frac{1}{A_1A_2}=\frac{1}{A_1A_3}+\frac{1}{A_1A_4}$$ then value of $n$ must be? Any ideas on how to start? I'm having ...
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3answers
36 views

Trigonometry-Complex Numbers Based Problem

If $2^7\cos^5x * \sin^3x$=$a\sin8x- b\sin 6x +c\sin 4x + d\sin 2x$ where $x$ is real then what will be the value of $a^4 + b^4 + c^4 + d^4$? Even a hint will suffice... I don't know how to proceed! I ...
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0answers
49 views

Does there exist an entire function such that $f\left(n+\frac{1}{n}\right)=0$

Does there exist an entire function $f:\mathbb C \to \mathbb C$ such that $f\left(n+\frac{1}{n}\right)=0$ for all $n\in \mathbb N$ ? I tried through Taylor series expansion , also by contradictory ...
0
votes
2answers
37 views

Complex Number -A problem on conjugate

|$z_1$|=2,|$z_2$|=3,|$z_3$|=4 and |$2z_1+3z_2+4z_3$|=9 then the absolute value of $8z_2z_3+27z_3z_1+64z_1z_2$ must be equal to? ($z_1,z_2,z_3$ are complex numbers) I tried manipulating with the ...
3
votes
2answers
67 views

Complex number, how to solve

Calculate i)$(1+i)^i$ ii)$(-1)^{\frac{1}{\pi}}$ I did i)$(1+i)=\sqrt{2}e^{i\frac{\pi}{4}}$. Knowing that if $z$ and $c$ are complex numbers $z^c=e^{c\log z}$ ...
4
votes
1answer
164 views

Base conversion: How to convert between Decimal and a Complex base?

My motivation for this question is exploring beyond the ideas in Project Euler Problem 508. In that problem, it is helpful to know how to convert between a decimal number and a number in base ...
1
vote
1answer
24 views

Complex number, logarithm power proof

Proof that i)$Log(1+i)^2=2*Log(1+i)$ ii)$Log(-1+i)^2\neq2*Log(-1+i)$ What I did i)By definition $z^a=e^{a\log z}$, so if $z=(1+i)$ and $a=2$ $$Log(1+i)^2=Log(e^{2\log(i+1)})=2*log(i+1)$$ But I do ...
1
vote
1answer
10 views

Find all solutions for a complex logarithm

$\log z = 6i$ I am working on a problem very similar. What I am seeing $\log z = \ln|z| + i(\theta + 2\pi n)$ for $n\in\mathbb{Z}$ What I am curious about, as if seen obvious to me that $ \log ...
0
votes
1answer
20 views

Complex number, logarithm and exponential

Find i)$Log(-ei)$ ii)$Log(1-i)$ I'm not too sure about how to solve this, what I did is Take $z=-ei$ so $Log(z)=\log r + i\theta, \space r>0, \space -\pi<\theta\leq\pi$ ...
0
votes
1answer
26 views

Complex number, logarithm

Find i)log(e) ii)log(i) I do not know if these issues are of simple fact, that there is something behind. I did i)Since $log$ and $e$ are inverse functions so$$log(e)=log(e^1)=1$$ Knowing that ...
1
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0answers
25 views

Does there exists an automorphism of $\Bbb{C}$ that's also an exponential hom?

Is there an automorphism of the field $\Bbb{C}$ of complex numbers, $\phi$, such that for all $z, w \in \Bbb{C}$ we have in addition to being a ring hom, $\phi(z^w) = \phi(z)^{\phi(w)}$?