Chapter 2: Complex Numbers

Describe geometrically the set of points in the complex plane satisfying the following equations.

$\mathit{z}\mathbf{-}\overline{\mathbf{z}}\mathbf{=}\mathbf{5}\mathit{i}$.

Describe geometrically the set of points in the complex plane satisfying the following equations.

The angle$\mathit{z}\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{2}}$.

Describe geometrically the set of points in the complex plane satisfying the following equations.

$\mathit{R}\mathit{e}\mathbf{\left(}{\mathit{z}}^{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{4}$.

Describe geometrically the set of points in the complex plane satisfying the following equations.

$\mathbf{R}\mathit{e}\mathbf{\left(}\mathit{z}\mathbf{\right)}\mathbf{>}\mathbf{2}$.

Describe geometrically the set of points in the complex plane satisfying the following equations.

$\left|z+3i\right|\mathbf{=}\mathbf{4}$.

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

$\mathit{t}\mathit{a}\mathit{n}\mathit{h}\mathbf{\left(}\frac{\mathbf{\pi i}}{\mathbf{4}}\mathbf{\right)}$

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.

$\mathit{z}\mathbf{=}{\mathit{z}}_{\mathbf{1}}\mathit{t}\mathbf{+}{\mathit{z}}_{\mathbf{2}}\mathbf{(}\mathbf{1}\mathbf{-}\mathit{t}\mathbf{)}$Hint: See Problem 4; the straight line here is through the points ${\mathit{z}}_{\mathbf{1}}$and${\mathit{z}}_{\mathbf{2}}$

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.

5. ${\mathit{e}}^{\mathbf{5}\mathbf{\pi i}}$

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.

${\mathit{e}}^{\left(i\pi /4\right)\mathbf{+}\left(ln2\right)\mathbf{/}\mathbf{2}}$.

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

$\mathit{l}\mathit{n}\mathbf{(}\mathbf{-}\sqrt{\mathbf{2}}\mathbf{}\mathit{i}\mathbf{-}\sqrt{\mathbf{2}}\mathbf{)}$