Chapter 2: Complex Numbers

Solve for all possible values of the real numbersand in the following equations.

$\mathbf{2}\mathit{i}\mathit{x}\mathbf{+}\mathbf{3}\mathbf{=}\mathit{y}\mathbf{-}\mathit{i}$

Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.

$\left|\frac{1+i}{1-i}\right|$

Find each of the following in the x+iyform and check your answers by computer.

$\mathit{c}\mathbf{os}\mathbf{\left(}\mathbf{i}\mathbf{\pi }\mathbf{\right)}$

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

$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}\mathbf{=}\mathbf{0}$

Solve for all possible values of the real numbersand in the following equations.

$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}\mathbf{=}\mathbf{2}\mathit{i}\mathbf{-}\mathbf{7}$

Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.

$\left|\frac{e^{\mathrm{i\pi }}}{1+i}\right|$

The functions$\mathit{s}\mathit{i}\mathit{n}\mathit{t}\mathbf{,}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{t}\mathbf{,}\mathbf{·}\mathbf{·}\mathbf{·}$,are called “circular functions” and the functions$\mathbf{sinh}\mathbf{}\mathit{t}\mathbf{,}\mathbf{cosh}\mathbf{}\mathit{t}\mathbf{,}\mathbf{·}\mathbf{·}\mathbf{·}$, are called “hyperbolic functions”. To see a reason for this, show that$\mathit{x}\mathbf{=}\mathbf{cos}\mathbf{}\mathit{t}\mathbf{,}\mathit{y}\mathbf{=}\mathbf{sin}\mathbf{}\mathit{t}$, satisfy the equation of a circle${\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$, while$\mathit{x}\mathbf{=}\mathbf{cosh}\mathbf{}\mathit{t}\mathbf{,}\mathit{y}\mathbf{=}\mathbf{sinh}\mathbf{}\mathit{t}$, satisfy the equation of a hyperbola

${\mathit{x}}^{\mathbf{2}}\mathbf{-}{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$.

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

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

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

$\sqrt[\mathbf{5}]{\mathbf{-}\mathbf{4}\mathbf{-}\mathbf{4}\mathbf{i}}$

Test each of the following series for convergence.

$\mathbf{\sum }\mathbf{1}\mathbf{/}\mathbf{(}\mathbf{1}\mathbf{+}\mathit{i}{\mathbf{)}}^{\mathbf{n}}$