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

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2answers
23 views

How do you compute an expression containing complex numbers with large powers?

$$(\frac{-\sqrt{3}}{2}+\frac{1}{2}i)^{123}=i$$ $$(\frac{-3}{\sqrt{2}}+\frac{-3}{\sqrt{2}}i)^{11}=\frac{3^{11}}{\sqrt{2}}-\frac{3^{11}}{\sqrt{2}}i$$ So I have these equations with the answers ...
0
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1answer
27 views

Solve z in an expression involving complex conjugates.

Solve for z, and give your answer in the form a+bi. $$\overline{z+2-2i} = {2z + 5 - 7i}$$ I know fully understand the concept of complex numbers and complex conjugates. I've found that the answer is ...
0
votes
2answers
43 views

Understanding complex numbers

I need to show that $$\left | \sum_{k=1}^n e^{ik}\right | $$ is bounded Now I am given that $$ \sum_{k=1}^n e^{ik} = e^i \frac{e^{in}-1}{e^i-1}$$ But have little idea of how to proceed further and ...
0
votes
3answers
45 views

Roots of Unity, Precalculus

(a) Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find $$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 ...
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0answers
26 views

Prove that if $g = r +ip$ is analytic on $C$ and $r(x,y) \leq M$, with $M > 0$, for all $(x,y)\in C$, $g$ is constant.

Let $g = r +ip$ be analytic on $C$. If for some $M > 0$ we have $r(x,y) \leq M$ for all of $C$, then $g$ is constant. The theorem is given without proof in my notes and I can't find any examples ...
0
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1answer
23 views

Shade on your Argand diagram the region $\frac{\pi}{4}\,{\le}\,\arg\,z\,\le\frac{\pi}{2}$

Is this saying the region from $\arg\,z=\frac{\pi}{4}$ to $\arg\,z=\frac{\pi}{2}$ in an anticlockwise direction? How would you represent the region from $\arg\,z=\frac{\pi}{4}$ to ...
-2
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0answers
23 views

Locally Lipschitz complex function [closed]

I want to study the property of being locally Lipschitz for the following function $$f(z)=\vert z\vert^\gamma z^2$$ with $\gamma\in\mathbb{R}$. Some hints to study this problem?
1
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1answer
40 views
+50

Understanding angle-preserving definition

My book (Real and complex analysis, by Rudin) gives the following definition: Let $A(z) = \frac z{|z|}$. Then we say $f$ preserves angles at $z_0$ if $$\lim_{r \to 0}e^{-i\theta} A[f(z_0 + ...
-4
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0answers
33 views

How to integrate complex numbers? [closed]

Complex numbers have 2 variable so does it's integration entail contour integration or can we integrate assuming one variable to be a constant in terms of the other or do we try to find a relation ...
-4
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0answers
34 views

Complex Analysis Exam tomorrow, what are some good to know facts? [closed]

The course covers differentiation, integration, series, and a lot of theorems. What would you say is crucial to know for an exam of undergrad complex analysis?
2
votes
4answers
58 views

Showing that linear transformations $1, T, T^2, T^3 ,\dots $ do not span the set of linear transformations of $ \mathbb C^n$ into $ \mathbb C^n$

Suppose $T : \mathbb C^n \rightarrow \mathbb C^n$, $n \geq 2$ is a linear transformation. Show that the linear transformations $1,T, T^2, \dots$ do NOT span $L(\mathbb C^n, \mathbb C^n)$, the set of ...
0
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2answers
31 views

Trisecting a line in the complex plane

We have $x = 11-13i$ and $y = 35-i$. $a$ is a complex number which trisects the line segment joining $x$ and $y$. $a$ is also closer to $x$ than $y$. Find $a$. I'm not sure where to start. Would a ...
0
votes
0answers
20 views

Linear interpolation of complex numbers

Does it make sense to say that, if I have two numbers $X = a + bi$ and $Y = c + di$, I can approximate a point between them as $Z = \frac{a + c}2 + \frac{b + d}2i$, interpolating the real and complex ...
0
votes
2answers
39 views

Minimising $|a+bw+cw^2|$ such that a,b,c are consecutive integers?

Suppose we are given a expression $k=|a+bw+cw^2|$ such that $w$ is cube root of unity ($w\neq1$) such that $\{a,b,c\}$ are consecutive integers , then how can we minimise value of expression ? I was ...
1
vote
1answer
21 views

Integral using complex numbers shortcut

I want to compute the following integral $$- \frac{1}{M(\lambda_1-\lambda_2)}\int\limits_{-\infty}^t(e^{\lambda_1(t-t')}-e^{\lambda_2(t-t')})(\beta\omega A\sin\omega t' +g)\;dt'$$ Here the integral ...
20
votes
8answers
4k views

Why did Euler use e to represent complex numbers?

From Euler we've learned that $z=re^{i\theta}$. And it's easy to see that $|z|^2=r^2$, since $re^{i\theta}\times re^{-i\theta}=r^2$. Why must we use e to represent these numbers correctly? It seems ...
3
votes
2answers
81 views

Can a number have both a periodic an a non-periodic representation in a non-integer base?

Fix an algebraic number $\beta$ and consider a complex number $\alpha$ which admits multiple representations in base $\beta$. If one representation of $\alpha$ is ultimately periodic, must every other ...
0
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1answer
33 views

How to graph $|z-1| <2$

Am I correct to rearrange this to $(z-1)^2 < 4$, and hence just graph as a circle or am I completely off?
3
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2answers
102 views

Complex number weird sum.

Question given in red. My working in black. $$\color{red}{\sum_{r=0}^{50}z^r=0}\iff z_k=\exp\underbrace{\left(\frac{{\cal i}2\pi k}{51}\right)}_{\theta_k},k\in\{n\mid n\le50,n\in\mathbb ...
-1
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0answers
69 views

Is it possible to take infinite sums of infinite sums?

I learned a few weeks ago that any diverging series can be given a specific value, not its limit ( for there are no limits in a diverging series), rather, "an average value" of the sum. From so, I ...
0
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1answer
34 views

Rotation group of $20$ degrees

Let $R_{20}$ be a rotation counterclockwise by $20$ degrees in the $xy$ plane. What is this group? Then re-write the group in terms of complex numbers of the form $e^{i\phi}$. Is their a special ...
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2answers
34 views

Calculate $\int_\Gamma ze^{z}dz$ where $\Gamma$ is line from point $z_1=0$ to point $z_2=\frac{\pi i}{2}$

$$ \int_\Gamma ze^{z}dz\ $$ where $\Gamma$ is line from point $z_1=0$ to point $z_2=\frac{\pi i}{2}$ Hello, pls. how correctly calculate this example? I don't know what do with exponent..
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2answers
152 views

Proving $x^{2}+1 \neq n! $,using Gaussian Integer.

I want to show that $$x^{2}+1 \neq n! $$ for $n>3$ where $x,n$ are both integers. Since $$x^{2}+1=(x+i)(x-i) $$ it follows that $x^{2}+1$ has only prime factors on the form ($4k+1$), whereas $n!$ ...
0
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2answers
53 views

How should this equation be read, $|z+1+3i|=|z-5-7i|$

$$|z+1+3i|=|z-5-7i|$$ $z$ represents a complex number right? Then if $$|z+1+3i|=0$$ $${\implies}|z|=|-1-3i|$$ In which sense does this $$|z+1+3i|=|z-5-7i|$$ imply, $$\implies|-4-4i|$$ But $z$ has ...
0
votes
1answer
15 views

Calculate the maximum value of the complex modulous |v-u|

I was asked to sketch the complex where $|u-1+i| = 2$ and $|v-(3+5i)|=1$ on the same diagram then I was asked to find the maximum value of |v-u| What is $|v-u|$ basically ? how am I going to ...
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2answers
24 views

Why does the function $f(z) = 1/\sin(\pi/z)$ have isolated singular points?

In the complex analysis text book "Complex Variables and Applications 8th edition", it states the function $1/sin (\pi/z)$ has singular points $z = 0$ and $z = 1/m; (m = \pm 1,2,3,4,\dots.) $ I sort ...
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0answers
16 views

Switching between Cartesian coordinate and polar coordinates

Under what assumption, every non-zero complex number represented in Cartesian coordinate system admits unique polar representation and vice versa ?
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2answers
35 views

When the argument of complex numbers is a well defined real valued function?

I know that the argument $\arg:\Bbb C\setminus\{0\}\to\Bbb R$ is multivalued function and also that if we consider $\arg:\Bbb C\setminus\{0\}\to{\Bbb R}/{2\pi \Bbb Z}$, then it is a well defined ...
3
votes
2answers
65 views

Reference to complete proof that integral closure of $\mathbb{Z}$ in $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$?

Where can I find a complete proof to the fact that the integral closure of $\mathbb{Z}$ in $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$ (the Gaussian integers are the integral closure of $\mathbb{Z}$ in the ...
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1answer
51 views

argument of $z = \sqrt 5 + 5i$?

I know that because both $a$ and $b$ are positive it is in the first quadrant and hence $\arg z$ should just equal to $\arctan(b/a)$, but I've been told that the answer is $\arg z= \arctan \sqrt5 $??? ...
0
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2answers
30 views

Why is $(-1)^p$, where, $0<p<1$ is a complex number?

Why is $(-1)^p$, where, $0<p<1$ is a complex number? If $p = \frac{1}{4}$, then $(-1)^{p}=(-1)^\frac{1}{4}=((-1)^{4})^\frac{1}{16}=(1)^{1/16}=1$ However, apparently $(-1)^\frac{1}{4}=a+ai$, ...
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0answers
28 views

Is every natural recursive relation necessarily holomorphic?

Define the set of algebraic primitive recursive relations as the set of functions defined by: $$ F(n,a,k) = F(n-1,F(n-1,F(n-1,a,a),a)...,a)_{\text{nested to depth k}}$$ $$ F(0,a,k) = a + k $$ Along ...
1
vote
2answers
26 views

Establishing a bound in the complex plane

so my function is $$f(z)= \frac{e^{iz}}{z^2+a^2} $$ What is getting to me and probably I should've been comfortable with this fact is how they establish this upper bound: $$\bigg|\int_{0}^{\pi} ...
0
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1answer
21 views

Solve complex number hyperbolic sine equation [closed]

What are the roots of equation: $$\sinh(z)=0$$ Thank you in advance.
0
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1answer
31 views

Proving that $\phi_a(z) = (z-a)/(1-\overline{a}z)$ maps $B(0,1)$ onto itself.

I want to prove that if $\phi_a: B(0,1) \to \Bbb C$ is given by $\phi_a(z) = (z-a)/(1-\overline{a}z)$ with $|a| < 1$, then $|\phi_a(z)| < 1$. Resist the itch on your finger urging you to close ...
1
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1answer
29 views

The action of real special linear group on the complex plane

Let $SL_2(\mathbb{R})\curvearrowright \mathbb{C}\cup\{\infty\}$ where $Az=\frac{az+b}{cz+d}$ Show that if $z=x+iy$ with $y>0$ (has positive imaginary part) then $Az$ does too. Then, considering ...
0
votes
3answers
31 views

Complex Number on Cartesian Coordinate System Question

I found this problem in a SAT Math II book and was confused by it: In this figure, if the point Z represents a complex number a+bi, which of the points could represent i · z? The figure has a x ...
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1answer
50 views

How can I show $| \sum_{k=1}^n e^{ik}|$ is bounded?

I know that we can write $ \sum_{k=1}^n e^{ik} = \frac{e^{i(n+1)} -1}{e^i - 1}$ But I am unsure how to proceed with showing there's some $M \in \mathbb{R}$ where $\forall n \in \mathbb{N} \space ...
0
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1answer
18 views

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1.

Let h : C → C be an analytic function such that h(0) = 0; h(1\2) = 5, and |h(z)| < 10 for |z| < 1. Then, (a) the set {z : |h(z)| = 5} is unbounded by the Maximum Principle; (b) the set {z : ...
0
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1answer
34 views

Help with an apparently simple proof in complex numbers

If $w=f(z)$ is a complex analytic function. I have been asked to proof that $$\frac{dw}{dz}=e^{-i\theta}\cdot\frac{\partial w}{\partial r}$$ Any hint? My main problem is the partial derivative on the ...
1
vote
2answers
37 views

Finding the zeroes of this complex polynomial

Now I thought it wouldn't be too much of an issue, but it is becoming hell to find the zeroes of: $$z^4 + 10z^2 +1 $$ Now reason I need them is for the poles of a function I am working on. So with ...
0
votes
1answer
25 views

Differential Equation with complex non-constant coefficients

I have the following equation which I am trying to solve: $$ R''(r) + 1/r · R'(r) + i·K(r)·R'(r) + G(r)·R(r) + i·F(r)·R(r) =0 $$ I want to solve this differential equation for $R(r)$. All the other ...
0
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1answer
19 views

Finding roots of complex polynomial with conjugates

I am having problem with the following question... I know that I should use De Moivre's formula somewhere... but can't quite get to it $$ (-15w + 34\bar{w})^4 = -1 $$ will be happy to get some help, ...
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0answers
31 views

Help proving:$| \dfrac{a-b}{1- a^* b}|=1$

Help proving:$$| \dfrac{a-b}{1- a^* b}|=1$$ Where $a,b \in \mathbb C$ What i have done is make $a=x+iy=(x,y), b=u+iv=(u,v)$ Then $a-b=(x-u,y-v)$ $a^*=(x,-y)$, $a^*b=(xu+yv,-yu+xv)$, ...
0
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0answers
22 views

Geometrical relationship between the points in the Argand diagram

Explain the geometrical relationship between the points in the Argand diagram represented by the complex numbers $z$ and $a + (z - a)e^{i\theta}$ (1) Write down the necessary and sufficient ...
1
vote
1answer
48 views

Complex function of class $C^m$

Let $m$ a positive integer and consider the function $$f(z)=\vert z\vert^\alpha z$$ with $\alpha>0$. I have to find the value of $\alpha$ for which $f\in C^m(\mathbb{C},\mathbb{C})$. Now if ...
0
votes
1answer
31 views

$(z-1)e^{-iα}+(z-1)^{-1}e^{i α}$ whose imaginary part is 0.What is equation on which it lies?

Let Y be the imaginary part of $(z-1)e^{-iα}+(z-1)^{-1}e^{i α}$.z is a complex number and α is real.Then Y=0 implies then what will be the equation of the circle on which z lies ? I tried taking ...
0
votes
2answers
63 views

Why does this work, and why is it wrong?

I have devised a "proof" that $i=0$. Obviously it can't be true, but I can't see why it is wrong. I've been thinking about this for a while, and my friend and I are very confused. What is wrong with ...
0
votes
3answers
44 views

Find all the complex numbers $z$ satisfying the following conditions

Find all the complex numbers $z$ satisfying the following conditions: $$\bar{z} = z^{n-1} \text{ where } n>2 \text{ and $n$ whole.}$$ I'm not sure if I'm correct but: $$|z|^2 = z^n$$ ...
0
votes
0answers
30 views

find laurent series of $\frac{1}{1 - cos z} $ [duplicate]

So it is not to solve everything with regards to the series. I was talking to my professor today and he mentioned something about since my denominator is an analytic function itself I could bring it ...