Chapter 2: Complex Numbers

Evaluate $\mathbf{\int }{\mathbf{e}}^{\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{ib}\mathbf{)}\mathbf{x}}\mathbf{dx}$and take real and imaginary parts to show that:

$\mathbf{\int }{\mathbf{e}}^{\mathbf{ax}}\mathbf{cos}\mathbf{}\mathbf{b}\mathbf{}\mathbf{xdx}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{ax}}\mathbf{(}\mathbf{a}\mathbf{}\mathbf{cos}\mathbf{}\mathbf{b}\mathbf{}\mathbf{x}\mathbf{+}\mathbf{b}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{b}\mathbf{}\mathbf{x}\mathbf{)}}{{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}}$

Evaluate$\mathbf{\int }{\mathit{e}}^{\left(a+ib\right)\mathbf{x}}\mathbf{}\mathit{d}\mathit{x}$and take real and imaginary parts to show that:

$\mathbf{\int }{\mathit{e}}^{\mathbf{a}\mathbf{x}}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{b}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{ax}}\mathbf{}\left(a\sin \mathrm{bx}-b\mathrm{cod}\mathrm{bx}\right)}{{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}}$

Show that tanh z never takes the values$\mathbf{\pm }\mathbf{1}$.

Verify the formulas in Problems 17 to 24.

$\mathit{a}\mathit{r}\mathit{c}\mathit{c}\mathit{o}\mathit{s}\mathit{z}\mathbf{=}\mathit{i}\mathit{l}\mathit{n}\left(z\pm \sqrt{z^2-1}\right)$

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.

$\mathbf{3}{\mathit{e}}^{{\mathbf{i}}_{\mathbf{2}}^{\mathbf{x}}}$.

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

$\mathbf{tanz}\mathbf{=}\mathbf{tan}\mathbf{(}\mathbf{}\mathbf{x}\mathbf{+}\mathbf{iy}\mathbf{)}\mathbf{=}\frac{\mathbf{tanx}\mathbf{+}\mathbf{i}\mathbf{}\mathbf{tanhy}}{\mathbf{1}\mathbf{-}\mathbf{i}\mathbf{}\mathbf{tanx}\mathbf{}\mathbf{}\mathbf{tanhy}}$

First simplify each of the following numbers to the$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form or to therole="math" localid="1664869600880" $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

${\mathbf{(}\mathbf{0}\mathbf{.64}\mathbf{+}\mathbf{0}\mathbf{.77}\mathbf{i}\mathbf{)}}^{\mathbf{4}}$.

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.

18.${\left(\frac{1+i}{1-i}\right)}^4$

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

$\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{2}\mathit{i}\mathbf{}\mathit{I}\mathit{n}\mathbf{}\mathit{i}\mathbf{)}$

Follow steps (a), (b), (c) above to find all the values of the indicated roots $\sqrt{\mathbf{i}}$.