Messages

16

MAT334--Lectures & Home Assignments / Re: Complex number visualized tool(From Night Section)

« on: October 02, 2018, 02:59:55 PM »

Here is a very good online version of the complex function plotter, based on David Bau's version. It allows you to zoom and pan the graph, and contains many different visualization options. You can even take a screenshot off of it!

Link: https://people.ucsc.edu/~wbolden/complex/#z

17

MAT334--Lectures & Home Assignments / Re: Section 1.4 Question 1

« on: October 02, 2018, 01:27:44 AM »

This sequence converges to 0 because the absolute value converges to 0. Note ${\sqrt{\frac{2}{3}}} < 1$, so powers of ${\sqrt{\frac{2}{3}}}^n$

Proof:
The Arg of $\frac{(1+i)}{\sqrt{3}}$ is $\frac{\pi}{4}$. That can be verified as $(1+i)$ "points northeast", and the $\sqrt{3}$ denominator is irrelevant to the Arg.
The magnitude $|\frac{(1+i)}{\sqrt{3}}|$ is ${\sqrt{\frac{2}{3}}}$. That can be verified as $|1+i|$ = $\sqrt2$, and $\frac{\sqrt2}{\sqrt3}$ = ${\sqrt{\frac{2}{3}}}$.

Therefore, $\frac{(1+i)}{\sqrt{3}} = {\sqrt{\frac{2}{3}}}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})$
And by DeMoivre's law, $[\frac{(1+i)}{\sqrt{3}}]^n$ = ${\sqrt{\frac{2}{3}}}^n(\cos \frac{n\pi}{4} + i \sin \frac{n\pi}{4})$

Intuitively magnitude of $[\frac{(1+i)}{\sqrt{3}}]^n$ can only spiral down counterclockwise as $n$ increases, and eventually approaches 0.

18

MAT334--Misc / Re: Quiz

« on: September 12, 2018, 04:30:12 PM »

From the course webpage, it says

"We plan to give 7 Quizzes, each worth 4 points, worth together 20 points. They will be offered either on the Lectures, or on Tutorials (the arrangements could differ for different quizzes).
If we reduce the total number of quizzes to 6, each will be 5 points worth."

19

MAT334--Lectures & Home Assignments / Re: About the definition of Argument (in book)

« on: September 12, 2018, 04:24:46 PM »

The $\arg$ of a complex number $z$ is an angle $\theta$. All angles $\theta$ have an infinite number of "equivalent" angles, namely $\theta =2k\pi$ for any integer $k$.

Equivalent angles can be characterized by that they exactly overlap when graphed on a graph paper, relative to the $0^\circ$ mark (usually the positive $x$-axis). Or more mathematically, they have the same sine and cosine. It also makes sine and cosine a non-reversible function, as given a sine or cosine, there are an infinite number of angles that satisfy this property.

$\Arg$, on the other hand, reduces the range of the possible angles such that it always lie between $0$ (inclusive) to $2\pi$ (exclusive). That is because one revolution is $2\pi$, or $360$ degrees. That is called the principal argument of a complex number.

We will later discover that complex logarithm also have a similar phenomenon.