Chapter 2: Complex Numbers

Show that tan z never takes the values $\mathbf{\pm }\mathit{i}$. Hint: Try to solve the equation tan z= iand find that it leads to a contradiction.

Verify the series in (7.3) by computer. Also show that it can be written in the form $\mathbf{\sum }_{\mathbf{n}\mathbf{-}\mathbf{0}}^{\mathbf{\infty }}{\left(-1\right)}^{\mathbf{n}}{\mathit{z}}^{\mathbf{2}\mathbf{n}}\mathbf{\sum }_{\mathbf{k}\mathbf{-}\mathbf{0}}^{\mathbf{n}}\frac{\mathbf{1}}{\left(2k+1\right)\mathbf{!}}$. Use this form to show by ratio test that the series converges in the disk $\left|z\right|\mathbf{<}\mathbf{1}$.

Express the following complex numbers in the 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.

17.$\frac{1}{{\left(1+i\right)}^3}$

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly findin 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.

$\sqrt{\mathbf{2}}{\mathit{e}}^{\mathbf{-}\mathbf{i}\frac{\mathbf{\pi }}{\mathbf{4}}}$.

Follow steps (a), (b), (c) above to find all the values of the indicate droots $\sqrt[\mathbf{5}]{\mathbf{-}\mathbf{1}}$

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.

17.$\frac{1}{{(1+i)}^3}$

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

$\mathbf{}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\mathit{i}\mathbf{}\mathit{z}\mathbf{=}\mathit{i}\mathbf{}\mathit{t}\mathit{a}\mathit{n}\mathit{z}$

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

${\left(i-1\right)}^{\mathbf{l}\mathbf{+}\mathbf{1}}$

Question:First simplify each of the following numbers to the x+iyform or to the$\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$form. Then plot the number in the complex plane.

${\left(1.7-3.2i\right)}^{\mathbf{2}}$ .

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:

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