Chapter 2: Complex Numbers

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.

$\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{}\mathit{\pi }\mathbf{-}\mathbf{2}\mathit{i}\mathbf{}\mathit{l}\mathit{n}\mathbf{3}\mathbf{)}\mathbf{.}$

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.

2i-2.

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

$\frac{\mathbf{d}}{\mathbf{dz}}\mathbf{cos}\mathbf{}\mathbf{z}\mathbf{=}\mathbf{-}\mathbf{sin}\mathbf{}\mathbf{z}$

Find each of the following in the $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form and compare a computer solution.

$\mathit{t}\mathit{a}\mathit{n}{\mathit{h}}^{\mathbf{-}}\left(i\sqrt{3}\right)$.

Test each of the following series for convergence.

$\mathbf{\sum }_{\mathbf{}}\frac{{\mathbf{i}}^{\mathbf{n}}}{\mathbf{n}}$

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

$\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{z}}^{\mathbf{n}}}{\sqrt{\mathbf{n}}}$

Express the following complex numbers in the $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others—try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

9.$\mathbf{2}{\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{\pi }\mathbf{/}\mathbf{2}}$

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.

$\mathit{s}\mathit{i}\mathit{n}\left(\pi -i\mathrm{In}3\right)$.

Evaluate each of the following in x+iyform, and compare with a computer solution ${\left(-1\right)}^{\mathbf{i}}$.