Questions tagged [complex-numbers]
Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.
19,780 questions
0 votes
0 answers
69 views
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. ...
- 9
1 vote
6 answers
167 views
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 $$i^{3/2} = (-1)^{3/4} = -\frac 1 {\...
- 10.6k
2 votes
0 answers
103 views
Contour integral with four branch points on the unit circle
I want to compute the contour integral $$ \oint_{|z|=2} z \sqrt{z^4-1}\text{d}z, $$ where the path is positively oriented (it is the blue one below). It is non-zero thanks to the four branch-points $\...
- 31
-1 votes
0 answers
138 views
How many independent branches proofs of Euler's formula are there?
I'll admit it, I'm very bad at explaining. If you have any suggestion, or you wonder what I tried to say, please comment your thoughts! The starting point There are ...
- 11
0 votes
2 answers
65 views
Show that if $(z_1+\frac{a}{z_2})(z_2+\frac{b}{z_1}) \in \mathbb R^*$. then there exists $k \in \mathbb R^*$ such that $z_2=k\cdot\overline{z_1}$. [closed]
The problem Let $a,b\in\mathbb R$ with $a\cdot b<0$ and $z_1,z_2 \in \mathbb C^*$. Show that if $(z_1+\frac{a}{z_2})(z_2+\frac{b}{z_1}) \in \mathbb R^*$. then there exists $k \in \mathbb R^*$ such ...
- 1,199
1 vote
1 answer
122 views
Is this a valid proof for Euler's formula?
Let $y = e^{ix}$. Then $y'' = -y$, which is the equation for simple harmonic motion. The general solution is $$y = A\cos(x) + B\sin(x) \tag1$$ where $$y(0) = e^0 = 1 = A\cos(0) + B\sin(0) = A \...
- 27
2 votes
1 answer
85 views
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 ...
- 151
5 votes
6 answers
306 views
Show that $x^2+y^2-2ixy$ is not an analytic polynomial.
To show that $x^2+y^2-2ixy$ is not an analytic polynomial. We assume that it is an analytic polynomial and try to reach a contradiction. First we write $$x^2+y^2-2ixy=\sum_{k=0}^N \alpha_k(x+iy)^k.$$ ...
