Chapter 2: Complex Numbers

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.

19. ${(1-i)}^8$

Follow steps (a), (b), (c) above to find all the values of the indicate droots $\sqrt[\mathbf{3}]{\mathbf{i}}$.

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

$\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{\pi }\mathbf{+}\mathbf{i}\mathbf{}\mathbf{I}\mathbf{n}\left(2\right)\mathbf{)}$

Verify the formulas in Problems 17 to 24.

$\mathit{a}\mathit{r}\mathit{c}\mathit{t}\mathit{a}\mathit{n}\mathbf{}\mathit{z}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}\mathbf{i}}\mathbf{}\frac{\mathbf{l}\mathbf{n}\mathbf{1}\mathbf{+}\mathbf{i}\mathbf{z}}{\mathbf{1}\mathbf{-}\mathbf{i}\mathbf{z}}$

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.

$\mathbf{5}\mathbf{(}\mathit{c}\mathit{o}\mathit{s}{\mathbf{20}}^{\mathbf{°}}\mathbf{+}\mathit{i}\mathit{s}\mathit{i}\mathit{n}{\mathbf{20}}^{\mathbf{°}}\mathbf{)}$.

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

$\mathit{t}\mathit{a}\mathit{n}\mathit{h}\mathbf{}\mathit{z}\mathbf{=}\frac{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{h}\mathbf{}\mathbf{x}\mathbf{+}\mathbf{i}\mathbf{}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{}\mathbf{y}}{\mathbf{1}\mathbf{+}\mathbf{i}\mathbf{}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{h}\mathbf{}\mathbf{x}\mathbf{}\mathbf{}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{}\mathbf{y}\mathbf{}}$

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

${\left(\frac{1+i}{1-i}\right)}^{\mathbf{2718}}$

For each of the following numbers, first, visualize where it is in the complete 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 numbers 1+i .

Find each of the following in the x+ iyform and compare a computer solution$\mathit{a}\mathit{r}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathbf{2}\mathbf{\right)}$.

Prove that an absolutely convergent series of complex numbers converges. This means to prove that $\mathbf{\sum }\mathbf{(}{\mathbf{a}}_{\mathbf{n}}\mathbf{+}{\mathbf{ib}}_{\mathbf{n}}\mathbf{)}\mathbf{}$converges ($\mathbf{}{\mathbf{a}}_{\mathbf{n}}$and ${\mathbf{b}}_{\mathbf{n}}$real) if $\mathbf{\sum }\sqrt{{{\mathbf{a}}_{\mathbf{n}}}^{\mathbf{2}}\mathbf{+}{{\mathbf{b}}_{\mathbf{n}}}^{\mathbf{2}}}\mathbf{}$converges. Hint: Convergence of $\mathbf{\sum }\mathbf{(}{\mathbf{a}}_{\mathbf{n}}\mathbf{+}{\mathbf{ib}}_{\mathbf{n}}\mathbf{)}$means that localid="1658823032447" $\mathbf{\sum }{\mathbf{a}}_{\mathbf{n}}\mathbf{}$and $\mathbf{\sum }{\mathbf{b}}_{\mathbf{n}}$ both converge. Compare $\mathbf{\sum }\mathbf{\left|}{\mathbf{a}}_{\mathbf{n}}\mathbf{\right|}$ and$\mathbf{}\mathbf{\sum }\mathbf{\left|}{\mathbf{b}}_{\mathbf{n}}\mathbf{\right|}$with $\mathbf{\sum }\sqrt{{{\mathbf{a}}_{\mathbf{n}}}^{\mathbf{2}}\mathbf{+}{{\mathbf{b}}_{\mathbf{n}}}^{\mathbf{2}}}$ , and use Problem 7.9of Chapter 1.