# Tag Info

1 vote

### Cosine of a multiple of arctangent

The method is basically the same as the one for computing "more standard" expressions like $\cos(\arctan x)$ for instance (cf. here). Start by constructing a right triangle with catheti of ...
• 8,676
1 vote

### Why is ${(e^{(i×1)})}^π$ different from $e^{(i×π)}$

First, \begin{align*} \mathrm{e}^{\mathrm{i}} &= 0.54\,030\,230\,586\,813\,971\,740\,093{\dots} \\ &\qquad + 0.84\,147\,098\,480\,789\,650\,665\,250{\dots}\mathrm{i} \end{align*} The ellipses ...
• 67.6k
1 vote

• 32.5k
Accepted

### Probable typo in a book for the complex limit $\lim_{z \to 1} \frac{z}{1+ \bar z}$

You do not need to write $z$ polar form. We have $$\lim_{z \to 1} \frac{z}{1+\bar z} = \frac{\lim_{z \to 1}z}{\lim_{z \to 1}1+\bar z} = \frac 1 2 .$$ If your book claims that the limit is $1$, then ...
• 77.8k
Accepted

### Solve simultaneous equation with 3 variable using Vieta relations

From $x+y+z=3$, we deduce then $$\implies \cases{x^2 =y^2+(3-x-y)-1\\y^2 =z^2+(3-y-z)-1\\z^2 =x^2+(3-z-x)-1}\implies \cases{x^2+x =y^2-y+2\\y^2+y =z^2-z+2\\z^2+z =x^2-x+2}\tag{1}$$ Subtract the first ...
• 16.2k
Accepted

### In complex number system, sin z and cos z are unbounded and periodic. But they are continuous also. How can that be possible?

Put simply, they are periodic in one direction and unbounded in the other direction. Specifically, their values repeat if you shift by $2\pi$ along the real axis, but increase unboundedly along the ...
• 11k

### Solve $z^2+|z^2| = 1+2i$

Hint: Let $z^2=u$ $$u=1-|u|+2i$$ Equating the imaginary parts, $u$ can be written as $a+2i$ where $a$ is real $$\implies a=1-\sqrt{a^2+4}$$ $$\implies a^2+4=(1-a)^2\iff2a=-3$$ $$\implies z^2=u=?, z=?$$...
Accepted

### Why is this particular substitution made? ($y=tx$)

It is a common trick to solve equations that have terms of the form $x^ky^{n-k}$ (the sum of powers of $x$ and $y$ is the same across all the terms). Do the $y=0$ case separately, then divide by $y^n$ ...
• 6,646
1 vote

### Solving a System of Equations Involving Complex Variables and Their Magnitudes

You can start by eliminating $y = c_1 - x$. The remaining equation is $$|x| + |c_1 - x| = r$$ Think of this geometrically in the complex plane. $|x| = s$ says $x$ is on the circle of radius $s$ ...
• 453k
Accepted

### Example of a complex function $f: \mathbb{C} \rightarrow \mathbb C$ differentiable only at $z_1=1+i\,, z_2=1-i\,, z_3=-1+i\,, z_4= -1-i$

This is easy to do using the Wirtinger derivatives. Let$$\varphi(z)=(z-1-i)(z-1+i)(z+1-i)(z+1+i)=z^4+4.$$Then you are after a function $f$ such that $\frac\partial{\partial\overline z}f(z)=\varphi(z)$....
1 vote

### Example of a complex function $f: \mathbb{C} \rightarrow \mathbb C$ differentiable only at $z_1=1+i\,, z_2=1-i\,, z_3=-1+i\,, z_4= -1-i$

These $4$ points in $\mathbb R^2$ are characterized by an equation such as $$(x^2-1)^2+(y^2-1)^2=0.$$ If we set $u(x,y)=f(x)$ and $v(x,y)=g(y)$ then we already have $u_y=-v_x=0$ everywhere. Now choose ...
• 1,918
1 vote

### why does $|z|=\sqrt{\Re^2(z)+\Im^2(z)}$, which elements to $\mathbb{R}$?

This is an unusual situation, where the original poster is asking for an intuitive explanation for the formula for $~|z| ~: ~z \in \Bbb{C}.~$ Given this situation, I don't see how that the OP (i.e. ...
• 36.9k
1 vote
Accepted

### Relationship between the parts of a complex number with its conjugate

This requires some algebraic background, but yes. In general, if $L/K$ is a finite field extension, there are two special functions from $L$ to $K$ called the field trace $\text{tr}_{L/K}$ and the ...
• 424k
Accepted

### How does this solution involving complex numbers work on this inequality?

Choose $\omega=e^{2\pi i/5}$, we verify the three equations. First equation: \begin{align*} \sum_{j=1}^5|z_j|^2&=\frac15\sum_{j=1}^5\left|\sum_{k=1}^5a_k\omega^{-jk}\right|^2\\ &=\frac15\sum_{...
• 2,740
Accepted

### Area & Perimeter of region $S$ containing points of the form $a+b\omega+c\omega^2$ where $a,b,c \in [0 , 1], \omega=-\frac 12+i\frac{\sqrt 3}{2}$

I won't give you a complete answer, but I will give you an outline for reasoning in an efficient way. Suppose you have two vectors $\vec u$, $\vec v$ that are not parallel. Then for $b, c \in [0,1]$,...
• 140k
Accepted

### Let $a,b,c,d\in\mathbb{C}$ such that $|a|+|b|\leq 1$ and $|c|+|d|\leq 1$. Show that $|3a+b+3c-d|+|a+3b-c+3d|\leq 7$.

I think I have a full solution. Thanks to River Li for pointing me in the right direction. We can reparameterize as $$a=r_1e^{i\theta_1},\\b=r_2e^{i\theta_2},\\c=r_3e^{i\theta_3},\\d=r_4e^{i\theta_4}$$...
• 101
Accepted

### How to show that the abstract complexification of a real matrix Lie algebra (vector space) coincides with complex linear combinations?

I don't think this has anything to do with Lie algebras, this is just linear algebra. Let $K\subseteq L$ be a field extension, let $V$ be an $L$-vector space and $W$ be a $K$-subspace of $V$. We have ...
• 35.8k
1 vote

• 116k
Accepted

• 140k

### Divergence of the generalized continued fraction $1+ \frac{-1\mid}{\mid1}+\frac{-1\mid}{\mid1}+\frac{-1\mid}{\mid1}+\dots$

$$a=1-\frac{1}{a}$$ $$a^{2}=a-1$$ $$a^{2}-a+1=0$$ Using quadratic equation: $$a=0.5\pm \frac{i\sqrt{3}}{2}$$ Hence I think this converges?
First of all, the roots can be $3$ and $5$ (real but not equal) and so you need to consider $D\neq0$ not $D<0$. One way to proceed is: Since $z_1$ and $z_2$ are roots of the given quadratic, $z_1+2$...