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

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3
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0answers
91 views

Minimum number of real multiplications to multiply two quaternions

Karatsuba multiplication of two complex numbers can be performed with just three real multiplications (instead of four) as follows: $$(a+bi)(c+di) = (ac-bd) + i ((a+b)(c+d) - ac-bd)$$ We only need the ...
8
votes
4answers
83 views

$z^n=(z+1)^n=1$, show that $n$ is divisible by $6$.

we are given $z^n=(z+1)^n=1$, $z$ complex number. we want to prove that $n$ is divisible by $6$. I showed that $|z|=|z+1|=1$. Hence $z$ is on the intersection of two unit circles, one centered at ...
0
votes
1answer
36 views

Why is $\left|e^{iat} e^{-st}\right| \le \left| e^{iat}\right| \left| e^{-st}\right| \le e^{-st}$ true?

My text states: $\left|e^{iat} e^{-st}\right| \le \left| e^{iat}\right| \left| e^{-st}\right| \le e^{-st}$ as $\left| e^{iat} \right|=1$ where $a,s,t \in \mathbb{R}$ I thought the last ...
0
votes
0answers
31 views

Shading on Argand Diagram

Still new to the topic but can someone help me out Shade the region whose points represent complex numbers satisfying the inequalities |z-2+2i|<=2 and arg z <= -pi/4 and Re z>=1 where Re ...
1
vote
2answers
29 views

complex numbers- proving the equality part in the Cauchy–Schwarz inequality using Lagrange identity

I need to discuss the equality case of: $$ \left | \sum_{k=1}^{n} z_{k}w_{k} \right |^{2} \leq \left ( \sum_{k=1}^{n}\left | z_{k} \right |^{2}\right )\left ( \sum_{k=1}^{n}\left | w_{k} \right ...
0
votes
1answer
33 views

Why $dz\wedge d\bar{z} = d|z|\wedge d\phi$ for $z \in \mathbb{C}$?

Let $z \in \mathbb{C}$ and $\bar{z}$ its complex conjugate. Then, a Kahler form can be written as $k = dz \wedge d\bar{z}$. If we re-write $z=|z|e^{i\phi}$ how do we get that $$dz\wedge d\bar{z} = ...
1
vote
2answers
67 views

Proving convergence of Newton's method

Consider the recursive sequence $$ z_{n+1} = {1 \over 2}\left ( z_n + {1 \over z_n} \right )$$ where we start at some point $z_0 = x_0 + i y_0 \in \mathbb C$. This is Newton's formula to find the ...
0
votes
1answer
17 views

find the laurent series of f (z) = 1/(z-1)(z-2) extended.

So i understand everything with regards to how to find the laurent series as was detailed in this post already: Finding the Laurent series of $f(z)=1/((z-1)(z-2))$ My question extends on this in how ...
1
vote
2answers
38 views

The mapping of complex variable function

So I have the complex variable function $f(Z)=\frac{Z}{Z-1}$ where $|Z|<1$ I have to represent it in $w$-plane. It is how far I got: $w=\frac{Z}{Z-1} => Z=\frac{w}{w-1}$ and then ...
2
votes
0answers
29 views

Question based on geomerical properties of complex number.

Let $z$ and $w$ be two complex numbers such that $\left| z\right| \leq 1$, $\left| w\right| \leq 1$ and $\left| z+iw\right|= \left| z- i\bar{w} \right|=2$, then z equals (A) $1$ or $i$ (B) $i$ or ...
1
vote
2answers
28 views

polar coordinates and complex numbers

Prove that $\dfrac{-1+i\sqrt{3}}{2}$ is a cube root of $1$. I believe I must use polar coordinates to solve this. Perhaps $z=r\cos(\theta)+i\sin (\theta)$. Any help would be great!
0
votes
0answers
44 views

Why is the argument of complex number determined up to integer multiple of $2 \pi$?

I have just started learning about complex number and came across to this argument of complex number Let's say we have a complex number $z$. Then the argument of $z$ can be represented by this: $arg ...
0
votes
1answer
36 views

Show that $\alpha^n+\beta^n+\gamma^n=2^{2n+1}\,cos\,\frac{n\pi}{3}+\left(-\frac12\right)^n$

$$z^n=r^n(\cos{n}\theta+i\sin{n}\theta)$$ $$\alpha=2+2\sqrt3\,i$$ $${\Rightarrow}\,(2+2\sqrt3\,i)^n=\left(\sqrt{2^2+(2\sqrt{3})^2}\right)^n\left(\cos\frac{n\pi}{3}+i\sin\frac{n\pi}{3}\right)$$ ...
2
votes
3answers
43 views

Find the Laurent series of an indeterminate function

$$ f(z) = \frac{z}{(\sin z)^2} $$ at $z_0 = 0$ (for the first four terms). So I thought I knew what to do, but I don't. Since it appears to be an indeterminte form, could I by L'hopital turn it into: ...
0
votes
1answer
26 views

What are the real and imaginary parts of the complex function?

So, it is asked to find the real and imaginary parts of the specific complex function: $f(z)=sin(z)+i(3z+2) $ So I use $z$ as $z=x+iy$ everything seemed clear till I met Mr. Sinus: $u+iv= ...
1
vote
1answer
31 views

Complex partial fractions

Could anyone help me separate this into partial fractions: $$\frac{\cos(z)}{z^2+1}$$ where $z=x+iy$. I've factored the denominator to get $$\frac{\cos(z)}{(z+i)(z-i)}$$ but I'm not really sure where ...
0
votes
1answer
30 views

How do I convert a complex number to a point on the Riemann sphere?

How do I convert a complex number to a point on the Riemann sphere? Where coordinates on the sphere are $(x,y,z)$ constrained $x^2+y^2+z^2=1$? There is a formula on Wikipedia going the other ...
0
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3answers
30 views

boundedness of a function in complex space

locate the singularity and tell whether it is removable, a pole, or essential: $$\frac{e^z - 1}{z}$$ so I know the singularity is at 0, but how can I determine what kind of singulairty? For it to be ...
0
votes
1answer
32 views

What is the norm of a complex vector?

I have two arrays $a$ and $b$ containing complex values. Now I one of my target operations is the following: $$||a-b||$$ The result should be a single real number. Now I am a bit confused how to apply ...
0
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0answers
11 views

Norm (modulus) of the derivative of complex function and Newton Method

I am implementing a function that approximates a root of a complex function, say $f(z)$. As we know, at iteration $i$ we ave $$z_i = z_{i-1} - \frac{f(z_{i-1})}{f'(z_{i-1})}$$ The division of ...
0
votes
0answers
28 views

Why the origin of this complex is moving away from the origin (0,0)?

Why does the origin of the complex line z is moving away from the origin? $$Let\;z=x+i\cdot y \\\;z-1\;=\;\;(x-1)+i\cdot y\;$$ Following that I would say that the coordinates of the origin of z are ...
0
votes
0answers
51 views

Number of 1s in after converting number to base -1+i

Regarding to Base conversion: How to convert between Decimal and a Complex base? Let $s(a,b)$ is a number of $1$ after converting complex number $a+bi$ to base $-1+i$. It's easy to implement that ...
1
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1answer
43 views

Why is the principal value of $\tan^{−1}θ$ always in $(\frac{\pi}{2}, \frac{\pi}{2})$? And, why is the principal value of $\arg z$ always in $(−π,π]$?

Why is the principal value of $\tan^{−1}θ$ always in $(\frac{\pi}{2}, \frac{\pi}{2})$? Also, why is the principal value of $\arg z$ always in the interval $(−π,π]$? Is this a convention? Why these ...
1
vote
3answers
111 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
vote
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
votes
5answers
71 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
votes
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 ...
2
votes
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 = ...
-5
votes
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 $$z^4 = 8\bar{z}$$
1
vote
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 ...
0
votes
1answer
33 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
votes
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
votes
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
votes
0answers
19 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
votes
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
votes
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
votes
1answer
42 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
vote
2answers
71 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 ...
0
votes
2answers
36 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
vote
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
votes
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 $$v^3 + a_2v^2 + a_1v + a_0 = 0$$ ...
1
vote
1answer
32 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
votes
1answer
20 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 ...
0
votes
0answers
21 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 ...
-1
votes
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 ...
0
votes
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
votes
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 ...