Chapter 2: Complex Numbers

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

Solve for all possible values of the real numbersand in the following equations ${\left(x-iy\right)}^{\mathbf{2}}\mathbf{=}\mathbf{2}\mathit{i}\mathit{x}$.

Solve for all possible values of the real numbersand in the following equations ${\left(x+iy\right)}^{\mathbf{2}}\mathbf{=}{\left(x-iy\right)}^{\mathbf{2}}$.

Solve for all possible values of the real numbers xand y in the following equations $\frac{\mathbf{x}\mathbf{+}\mathbf{i}\mathbf{y}}{\mathbf{x}\mathbf{-}\mathbf{i}\mathbf{y}}\mathbf{=}\mathbf{-}\mathit{i}$.

Solve for all possible values of the real numbers x and y in the following equations.

${\left(x+iy\right)}^{\mathbf{3}}\mathbf{=}\mathbf{-}\mathbf{1}$

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

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

$\mathbf{(}\mathbf{1}\mathbf{+}\frac{\mathbf{\pi i}}{\mathbf{2}}\mathbf{)}$

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

cosh z=cosh x cos y+i sinh x sin y

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

$\underset{\mathbf{}}{\mathbf{\sum }{\mathbf{z}}^{\mathbf{n}}}$

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.

4. ${{\mathit{e}}^{\left(1/3\right)}}^{\left(3+4\mathrm{\pi i}\right)}$