Problems on complex questions

Question

I'm a senior in my high school studying complex numbers. I have a test coming up in a few days. I want to explore more questions outside my textbook and school questions. Can someone suggest a few resources where I can get good questions of complex numbers from?

The type of questions I want to practice are not very advanced complex numbers(no calculus, matrices, functions, etc).

Below is one question which I considered the most "advanced/hardest" among the questions I have.

\begin{array}{l}{\text { By considering } \sum_{k=0}^{n-1}(1+i \tan \theta)^{k}, \text { show that }} \\ {\qquad \sum_{k=0}^{n-1} \cos k \theta \sec ^{k} \theta=\cot \theta \sin n \theta \sec ^{n} \theta} \\ {\text { provided } \theta \text { is not an integer multiple of } \frac{1}{2} \pi .}\end{array}

Answer 1

$\sum\limits_{k=0}^{n-1} (1+i\tan \theta)^{k}=\frac {1- (1+i\tan \theta)^{n}} {1- (1+i\tan \theta)}$ from the formula for a geometric sum. Now $\sum\limits_{k=0}^{n-1} (1+i\tan \theta)^{k}=\sum\limits_{k=0}^{n-1} (e^{i\theta})^{k}\sec^{k} (\theta)$. So the required result can b derived by taking real parts in the equation $\frac {1- (1+i\tan \theta)^{n}} {1- (1+i\tan \theta)}=\sum\limits_{k=0}^{n-1} e^{ik\theta}\sec^{k} (\theta)$.

I will let you try the other two questions yourself.

Answer 2

There are some problems similar to the one you quoted in Chapter 1 of Schaum's Outline of Theory and Problems of Complex Variables With an Introduction to Conformal Mapping and Its Applications by Murray R. Spiegel.

Chapter 1 is a short chapter, only 32 pages of a 313-page book, so it may not seem worthwhile to buy the book for just one chapter, but on the other hand, the book is very reasonably priced for a math book, and it also contains a lot of good information if you eventually go beyond the first chapter. The emphasis of the book is on applications, not theory.

Answer 3

It is difficult to find questions involving complex numbers that do not also involve the topics that you wish to avoid (i.e. calculus, matrices, functions). Similar to what others have suggested, find a book on complex analysis and confine yourself to the book's chapter 1.

I can suggest three specific problems that you might tackle, assuming that you are familiar with https://en.wikipedia.org/wiki/De_Moivre%27s_formula:

(1) Using the quadratic equation, derive the 3 roots of $0 = (z^3 - 1) = (z - 1)(z^2 + z + 1).$
Then compare the results with $[(\cos 2k\pi/3) + i(\sin 2k\pi/3),\;$ where $\;k = 0, 1, \;\text{or}\; 2.$
Considering (for example) $\;\cos (2\pi/3),\;$ which is closely related to the special angle $\;\cos (\pi/3),\;$ you should observe that geometric analysis (+ DeMoivre's theorem) yield the identical results as the 3 roots of the corresponding cubic equation.

(2) Prove to yourself that $\;(\cos \alpha + i\,\sin \alpha)(\cos \,\beta + i\,\sin \beta) = \cos \,[\alpha + \beta] + i\,\sin [\alpha + \beta].$

(3) A slightly harder problem than #1 above: use complex analysis to derive the $\;\cos 2\pi/5\;$ as follows:

Let $\;z_1 = [(\cos 2\pi/5) + i(\sin 2\pi/5)] \Rightarrow$
$z_1\;$ is a root of $\;0 = (z^5 - 1) = (z - 1)(z^4 + z^3 + z^2 + z + 1).$
Conjecture that $\;(z^4 + z^3 + z^2 + z + 1)\;$ can be factored into $\;(z^2 +az + 1)(z^2 + bz + 1)\;$
and see what constraints this places on $a$ and $b$.
Use the quadratic equation to determine values for $a$ and $b$ that satisfy the constraints.
Having done this, you should be able to conclude that $z_1$ is one of the following 4 roots:
$(1/2) [-a \pm (\text{some stuff})] \;\text{or}\; (1/2) [-b \pm (\text{some stuff})].$
Since the imaginary part of $z_1$ equals $(\sin 2\pi/5),$ which is not equal to zero,
for whichever one of the 4 roots pertains to $z_1$, the some stuff portion must represent a non-zero imaginary number.
Therefore, the real portion of $z_1$ is either $(1/2)(-a)\;$ or $\;(1/2)(-b).$
Since $\;0 < cos (2\pi/5) = \;$ the real portion of $z_1,$
all you have to do is determine which of $(1/2)(-a)\;$ or $\;(1/2)(-b)$ is positive.