Chapter 2: Complex Numbers

Express the following complex numbers in the 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.

3.$9e^{3i\mathrm{\pi }12}$

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

$\mathit{l}\mathit{n}\mathbf{(}\mathit{i}\mathbf{+}\sqrt{\mathbf{3}}\mathbf{)}$

Find the power series for ${\mathit{e}}^{\mathbf{x}}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{x}$and for ${\mathit{e}}^{\mathbf{x}}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathit{x}$from the series for $\mathit{e}\mathit{z}$in the following way: Write the series for ${\mathit{e}}^{\mathbf{z}}$; put $\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$. Show that ${\mathit{e}}^{\mathbf{z}}\mathbf{=}{\mathit{e}}^{\mathbf{x}}\left(cosy+isiny\right)$; take real and imaginary parts of the equation, and put $\mathit{y}\mathbf{=}\mathit{x}$.

In each of the following problems, z represents the displacement of a particle from the origin. Find (as functions of t) its speed and the magnitude of its acceleration, and describe the motion.

$\mathit{z}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{+}\mathit{i}\mathbf{)}{\mathit{e}}^{\mathit{i}\mathit{t}}$

Find each of the following in rectangular form $\mathbf{x}\mathbf{+}\mathbf{iy}$and check your results by computer. Remember to save time by doing as much as you can in your head.

${\mathbf{e}}^{\mathbf{-}\left(i\pi /4\right)\mathbf{+}\mathbf{ln}\mathbf{3}}$.

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

$\mathbf{sinh}\mathbf{}\mathbf{z}\mathbf{=}\mathbf{sinh}\mathbf{}\mathbf{x}\mathbf{}\mathbf{cos}\mathbf{}\mathbf{y}\mathbf{+}\mathbf{i}\mathbf{}\mathbf{cosh}\mathbf{}\mathbf{x}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{y}$

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 $\mathbf{1}\mathbf{-}\mathit{I}\sqrt{\mathbf{3}}$.

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

$\mathbf{1}\mathbf{-}\frac{{\mathbf{z}}^{\mathbf{2}}}{\mathbf{3}\mathbf{!}}\mathbf{+}\frac{{\mathbf{z}}^{\mathbf{4}}}{\mathbf{5}\mathbf{!}}\mathbf{-}\mathbf{·}\mathbf{·}\mathbf{·}$

Solve for all possible values of the real numbers $\mathit{x}$and $\mathit{y}$in the following equations.

$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}\mathbf{=}\mathbf{3}\mathit{i}\mathbf{-}\mathit{i}\mathit{x}$

Solve for all possible values of the real numbers $\mathit{x}$and $\mathit{y}$in the following equations. $\left(2x-3y-5\right)\mathbf{+}\mathit{i}\left(x+2y+1\right)\mathbf{=}\mathbf{0}$