Chapter 2: Complex Numbers

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.

${\left(1+i\right)}^{\mathbf{2}}\mathbf{+}{\left(1+i\right)}^{\mathbf{4}}$

Describe the set of points $\mathit{z}$for which $\mathbf{Re}\left(e^{i\pi /2}z\right)\mathbf{>}\mathbf{2}$.

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

$\frac{\mathbf{1}}{\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{(}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{40}\mathbf{°}\mathbf{+}\mathbf{i}\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{40}\mathbf{°}}$.

In the following integrals express the sines and cosines in exponential form and then integrate to show that

${\mathbf{\int }}_{\mathbf{-}\mathbf{\pi }}^{\mathbf{\pi }}\mathit{s}\mathit{i}\mathit{n}\mathbf{3}\mathit{x}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{4}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}\mathbf{0}$

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

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

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

$\mathbf{}\mathbf{tani}\mathbf{}\mathbf{z}\mathbf{=}\mathbf{i}\mathbf{}\mathbf{tanhz}$

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly findin 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.

$\mathbf{5}\mathbf{(}\mathbf{cos}\mathbf{0}\mathbf{-}\mathit{i}\mathbf{sin}\mathbf{0}\mathbf{)}$.

Express the following complex numbers in the $x+iy$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.

16.$(i-\sqrt{3})(1+i\sqrt{3})$

Follow steps (a), (b), and (c) above to find all the values of the indicate droots $\sqrt[\mathbf{6}]{\mathbf{-}\mathbf{1}}$.

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

${\left(\frac{1+i\sqrt{3}}{2}\right)}^{\mathbf{\text{'}}}$