Chapter 2: Complex Numbers

Find and if$\mathit{z}\mathbf{=}\frac{\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{i}\mathbf{t}\mathbf{)}}{\mathbf{2}\mathbf{t}\mathbf{+}\mathbf{i}}$ .

Question: Find vand a if $\mathit{z}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathbf{2}\mathit{t}\mathbf{+}\mathit{i}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{t}$, can you describe the motion.

Find one or more values of each of the following complex expressions and compare with acomputer solution.

${\mathbf{(}\mathbf{-}\mathbf{e}\mathbf{)}}^{\mathbf{x}\mathbf{i}}$

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find$\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{r}\mathbf{,}\mathit{\theta }$in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

-4i.

Verify each of the following by using equations (11.4), (12.2), and (12.3).

$\mathbf{cos}\mathbf{}\mathbf{2}\mathit{z}\mathbf{=}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{z}\mathbf{-}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathbf{}\mathit{z}$

Find each of the following in rectangular form x+iyand check your results by computer. Remember to save time by doing as much as you can in your head.

$\mathit{c}\mathit{o}\mathit{s}\left(i\mathrm{In}5\right)$.

Find the disk of convergence for each of the following complex power series.

$\mathbf{\sum }_{\mathbf{n}\mathbf{-}\mathbf{1}}^{\mathbf{\infty }}{\mathit{n}}^{\mathbf{2}}{\mathbf{\left(}\mathbf{3}\mathit{i}\mathit{z}\mathbf{\right)}}^{\mathbf{n}}\mathbf{.}$

Question: Evaluate each of the following in x+yform, and compare with a computer solution.\

$\ln \left(\frac{1-i}{\sqrt{2}}\right)$

Test each of the following series for convergence.

$\underset{\mathbf{}}{\mathbf{\sum }{\displaystyle \frac{\mathbf{1}\mathbf{+}\mathbf{i}}{{\mathbf{n}}^{\mathbf{2}}}}}$

First simplify each of the following numbers to the $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$form. Then plot the number in the complex plane.

${\left(\frac{1+i}{1-i}\right)}^{\mathbf{2}}$.