Chapter 2: Complex Numbers

Prove that a series of complex terms diverge if $\mathit{\rho }\mathbf{>}\mathbf{1}$( = ratio test limit). Hint: The${\mathit{n}}^{\mathbf{t}\mathbf{h}}$term of a convergent series tends to zero.

Find the disk on convergence for each of the following complex power series.

$\mathbf{\sum }_{\mathbf{n}\mathbf{-}\mathbf{0}}^{\mathbf{\infty }}\mathit{n}\mathbf{(}\mathit{n}\mathbf{+}\mathbf{1}\mathbf{)}{\mathbf{(}\mathit{z}\mathbf{-}\mathbf{2}\mathit{i}\mathbf{)}}^{\mathbf{n}}$

Find the disk on convergence for each of the following complex power series.

$\mathbf{\sum }_{\mathbf{n}\mathbf{-}\mathbf{0}}^{\mathbf{m}}\frac{{\left(z-2+i\right)}^{\mathbf{n}}}{{\mathbf{2}}^{\mathbf{n}}}$

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.

$\left(cos\mathit{\pi }\mathit{-}\mathit{i}\mathit{}\sin \mathit{\pi }\right)$.

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

siniz = i sin z

In the following integrals express the sines and cosines in exponential form and then integrate to show that

${\mathbf{\int }}_{\mathbf{-}\mathbf{x}}^{\mathbf{x}}\mathbf{sin}\mathbf{2}\mathbf{xcos}\mathbf{3}\mathbf{xdx}\mathbf{=}\mathbf{0}$

Evaluate each of the following in$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form, and compare with a computer solution.${\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{i}}$

Find each of the following in the x + iyform and compare a computer solution.

arctan(2 + i )

Follow steps (a), (b), and (c) above to find all the values of the indicate droots $\sqrt[\mathbf{6}]{\mathbf{-}\mathbf{64}}$.

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.

$\frac{\mathbf{5}\mathbf{-}\mathbf{2}\mathbf{i}}{\mathbf{5}\mathbf{+}\mathbf{2}\mathbf{i}}$.