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Question 1

0votes

0answers

55views

If $a,b,c,d$ are complex number with the same modulus such that $a+b+c=d$ then show that one of them is equal to d

the problemIf $a,b,c,d$ are complex number with the same modulus such that $a+b+c=d$, then $d$ is equal to one of the other three.my ideaso i thought that the case where d is zero is easy to solve ...

Pam Munoz Ryan

508

asked18 hours ago

Question 2

-2votes

0answers

95views

Construction of complex numbers [closed]

We suppose first that there is an extension field $K$ of $\mathbb{R}$ and some $i\in K$ such that$i^2=-1$. Of course, $i\notin\mathbb{R}$.Since $K$ is a field, if $x,y\in\mathbb{R}$, then $z := x + ...

Andrew_Ren

1,299

askedNov 25 at 16:05

Question 3

1vote

0answers

86views

Help: Proof of Ptolemy's Inequality with Complex Numbers

This was the original problem statement:Let $ABCD$ be a quadrilateral, where $A, B,C$ and $D$ are points in anti-clockwise direction corresponding to $z_1, z_2, z_3, z_4\in\mathbb{C}$ respectively. ...

mkcW

19

askedNov 23 at 19:18

Question 4

0votes

1answer

95views

Finding the real part of some complex number using elementary functions [closed]

To find the real part of some complex number the easiest way is to take the mean of the number with its complex conjugate. Is there a way to find the the real part of some complex number z using ...

Anant S. Malviya

67

askedNov 21 at 18:31

Question 5

1vote

6answers

176views

Calculating $i^{3/2}$ in two ways gives different results [closed]

I have very little familiarity with complex numbers, but I now have to work with$$i^{3/2}$$where $i$ is the imaginary unit.The on-line WolframAlpha machine gives$$i^{3/2} = (-1)^{3/4} = -\frac 1 {\...

Alecos Papadopoulos

10.6k

askedNov 20 at 17:45

Question 6

2votes

0answers

54views

How to deform contour in the complex plane correctly

I have a physics related problem where I need to calculate integrals of following type $\displaystyle\int_{-\infty}^{\infty} d\epsilon\, A(\epsilon) g^R(\epsilon + \omega/2)g^A(\epsilon - \omega/2)$, ...

Xian-Zu

83

askedNov 20 at 16:39

Question 7

2votes

0answers

110views

Contour integral with four branch points on the unit circle

I want to compute the contour integral$$\oint_{|z|=2} z \sqrt{z^4-1}\text{d}z,$$where the path is positively oriented (it is the blue one below).It is non-zero thanks to the four branch-points $\...

94thomas

31

askedNov 20 at 13:57

Question 8

1vote

2answers

233views

Does $\sqrt x\cdot\sqrt{x}=x$ or $\sqrt{x}\cdot\sqrt{x}=|x|$

I wanted to solve this problem:$$\sqrt x + \sqrt{x+42}=\frac{7}{\sqrt x}$$I multiplied it by $\sqrt x$ to simplify the fraction. It resulted in me having$$\sqrt x \cdot \sqrt x + \sqrt x \cdot \...

hlebdsua

21

askedNov 19 at 18:39

Question 9

-1votes

0answers

141views

How many independent branches proofs of Euler's formula are there?

I'll admit it, I'm very bad at explaining. If you have any suggestion, or you wonder what I tried to say, please comment your thoughts!The starting pointThere are ...

Kenay5 55

11

askedNov 17 at 21:37

Question 10

3votes

2answers

414views

The intuition behind defining trigonometric functions in the complex plane as special combinations of exponential functions [closed]

I was studying Complex Analysis from "A First Course of Complex Analysis" and the authors stated directly that sine and cosine are defined as follows (without any intuition):$$ \sin\left(z\...

Ocean003

31

askedNov 17 at 5:47

Question 11

0votes

2answers

66views

Show that if $(z_1+\frac{a}{z_2})(z_2+\frac{b}{z_1}) \in \mathbb R^$. then there exists $k \in \mathbb R^$ such that $z_2=k\cdot\overline{z_1}$. [closed]

The problemLet $a,b\in\mathbb R$ with $a\cdot b<0$ and $z_1,z_2 \in \mathbb C^*$. Show that if $(z_1+\frac{a}{z_2})(z_2+\frac{b}{z_1}) \in \mathbb R^*$. then there exists $k \in \mathbb R^*$ such ...

IONELA BUCIU

1,199

askedNov 16 at 17:18

Question 12

0votes

3answers

129views

Show that if $(z_1+\frac{a}{z_1})(z_2+\frac{b}{z_2}) \in \mathbb R^$, then there exists $k \in \mathbb R^$ such that $z_2=k\cdot \overline{z_1}$. [closed]

The problemLet $a,b\in\mathbb R$ with $a\cdot b<0$ and $z_1,z_2 \in \mathbb C^*$. Show that if $(z_1+\frac{a}{z_1})(z_2+\frac{b}{z_2}) \in \mathbb R^*$. then there exists $k \in \mathbb R^*$ such ...

IONELA BUCIU

1,199

askedNov 16 at 12:46

Question 13

1vote

1answer

123views

Is this a valid proof for Euler's formula?

Let $y = e^{ix}$. Then $y'' = -y$, which is the equation for simple harmonic motion. The general solution is$$y = A\cos(x) + B\sin(x) \tag1$$where$$y(0) = e^0 = 1 = A\cos(0) + B\sin(0) = A \...

fredkno

27

askedNov 16 at 12:09

Question 14

2votes

1answer

86views

Why does Titu’s lemma lead to the circumcenter, but the minimum of the sum of squared distances occurs at the centroid?

Let $A = 0$, $B = 3$, and $C = 6i$ be three points in the complex plane.Define$$F(z) = |z|^2 + |z - 3|^2 + |z - 6i|^2.$$My reasoning:Using Titu’s lemma (Engel form of Cauchy–Schwarz), we can write...

mukund

151

askedNov 13 at 16:46

Question 15

6votes

2answers

263views

Is there a more intuitive way to prove that the locus of this point is an arc of a circle?

Given that the position of line segment $AB$ is fixed, there is a fixed line above it (the blue line in the figure), and there is a moving point $P$ on the line. Connecting $PA$ and $PB$, if $Q$ is on ...

King.Max

281

askedNov 5 at 10:19

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Questions tagged [complex-numbers]

If $a,b,c,d$ are complex number with the same modulus such that $a+b+c=d$ then show that one of them is equal to d

Construction of complex numbers [closed]

Help: Proof of Ptolemy's Inequality with Complex Numbers

Finding the real part of some complex number using elementary functions [closed]

Calculating $i^{3/2}$ in two ways gives different results [closed]

How to deform contour in the complex plane correctly

Contour integral with four branch points on the unit circle

Does $\sqrt x\cdot\sqrt{x}=x$ or $\sqrt{x}\cdot\sqrt{x}=|x|$

How many independent branches proofs of Euler's formula are there?

The intuition behind defining trigonometric functions in the complex plane as special combinations of exponential functions [closed]

Show that if $(z_1+\frac{a}{z_2})(z_2+\frac{b}{z_1}) \in \mathbb R^$. then there exists $k \in \mathbb R^$ such that $z_2=k\cdot\overline{z_1}$. [closed]

Show that if $(z_1+\frac{a}{z_1})(z_2+\frac{b}{z_2}) \in \mathbb R^$, then there exists $k \in \mathbb R^$ such that $z_2=k\cdot \overline{z_1}$. [closed]

Is this a valid proof for Euler's formula?

Why does Titu’s lemma lead to the circumcenter, but the minimum of the sum of squared distances occurs at the centroid?

Is there a more intuitive way to prove that the locus of this point is an arc of a circle?

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