Chapter 2: Complex Numbers

Verify the formulas in Problems 17 to 24.

$\mathit{s}\mathit{i}\mathit{n}{\mathit{h}}^{\mathbf{-}\mathbf{1}}\mathit{z}\mathbf{=}\mathit{l}\mathit{n}\mathbf{(}\mathit{z}\mathbf{\pm }\sqrt{{\mathbf{z}}^{\mathbf{2}}\mathbf{+}\mathbf{1}}\mathbf{)}$

Verify the formulas in Problems 17 to 24.

${\mathbf{cosh}}^{\mathbf{-}\mathbf{1}}\mathbf{z}\mathbf{=}\mathbf{In}\left(z\pm \sqrt{z^2-1}\right)\mathbf{=}\mathbf{\pm }\mathbf{In}\left(z+\sqrt{z^2-1}\right)$

Find each of the following in rectangular $\mathit{a}\mathbf{+}{}_{\mathbf{z}}\mathit{b}$ form $\mathit{i}\mathit{f}\mathbf{}\mathit{z}\mathbf{=}\mathbf{2}$if ; if $\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}{}_{\mathbf{z}}\mathit{y}$ .

role="math" localid="1658486330935" $\frac{\mathbf{1}}{\mathbf{z}\mathbf{+}\mathbf{1}}$.

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.

21.${\left(\frac{1-i}{\sqrt{2}}\right)}^{40}$

Evaluate each of the followingin$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form, and compare with a computer solution.$\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathbf{i}\mathbf{l}\mathbf{n}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{\right)}$

Use a computer to plot graphs of sinh(x), cosh(x), and tanh(x).

Follow steps (a), (b), (c) above to find all the values of the indicated roots.

$\mathbf{2}\mathbf{+}\mathbf{2}\mathbf{i}\sqrt{\mathbf{3}}$

Verify the formulas in Problems 17 to 24.

${\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{z}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{In}\left(\frac{1+z}{1-z}\right)$

Find each of the following in rectangular $\mathit{a}\mathbf{+}\mathit{z}\mathit{b}$ form if $\mathit{z}\mathbf{=}\mathbf{2}\mathbf{-}\mathit{z}\mathbf{3}$; if $\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}\mathit{z}\mathit{y}$.

$\frac{\mathbf{1}}{\mathbf{z}\mathbf{-}\mathbf{1}}$

Find the values of the indicated roots.

$\sqrt[\mathbf{3}]{\mathbf{2}\mathbf{i}\mathbf{-}\mathbf{2}}$