Chapter 2: Complex Numbers

Find real $\mathit{x}$and $\mathit{y}$for which$\left|z+3\right|\mathbf{=}\mathbf{1}\mathbf{-}\mathit{i}\mathit{z}$, where $\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$.

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.

role="math" localid="1658476746206" $\mathit{(}\mathit{c}\mathit{o}\mathit{s}\frac{\mathbf{3}\mathbf{\pi }}{\mathbf{2}}\mathit{+}\mathit{i}\mathit{}\mathit{s}\mathit{i}\mathit{n}\frac{\mathbf{3}\mathbf{\pi }}{\mathbf{2}}\mathit{)}$

Verify that${\mathit{e}}^{\mathbf{i}\mathbf{\omega }\mathbf{t}}\mathbf{,}{\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{\omega }\mathbf{t}}\mathbf{,}\mathit{c}\mathit{o}\mathit{s}\mathit{\omega }\mathit{t}\mathbf{,}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathbf{sin}\mathit{\omega }\mathit{t}$satisfy equation (16.21).

Follow steps (a), (b), (c) above to find all the values of the indicated roots $\sqrt[\mathbf{4}]{\mathbf{-}\mathbf{4}}$.

Evaluate each of the following in x+ iyform, and compare with a computer solution.

${\mathit{i}}^{\mathit{2}\mathit{i}\mathit{/}\mathit{\pi }}$

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

$\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathbf{3}\mathit{z}\mathbf{\right)}\mathbf{=}\mathbf{4}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{3}}\mathbf{\left(}\mathit{z}\mathbf{\right)}\mathbf{-}\mathbf{3}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathit{z}\mathbf{\right)}$

Find each of the following in the x + iyform and compare a computer solution.

$\mathit{c}\mathit{o}\mathit{s}{\mathit{h}}^{\mathbf{-}\mathbf{1}}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}$

First simplify each of the following numbers to the$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form or to the form. Then plot the number in the complex plane. $\mathbf{5}\mathbf{(}\mathbf{c}\mathbf{o}\mathbf{s}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{5}}\mathbf{+}\mathbf{i}\mathbf{s}\mathbf{i}\mathbf{n}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{5}}\mathbf{)}$

.

Test each of the following series for convergence.

$\mathbf{\sum }_{\mathbf{}}\frac{{\mathbf{(}\mathbf{1}\mathbf{+}\mathit{i}\mathbf{)}}^{\mathbf{n}}}{{\mathbf{(}\mathbf{2}\mathbf{=}\mathbf{i}\mathbf{)}}^{\mathbf{n}}}$

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

$\mathbf{\sum }_{\mathbf{n}\mathbf{-}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}}}{\mathbf{n}}$