Chapter 2: Complex Numbers

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

${\mathit{i}}^{\mathit{2}\mathit{i}\mathit{/}\mathit{\pi }}$

Follow steps (a), (b), (c) above to find all the values of the indicated roots $\sqrt[\mathbf{4}]{\mathbf{-}\mathbf{4}}$.

Find the disk of convergence of the series$\mathbf{\sum }_{\mathbf{}}{\left(z-2i\right)}^{\mathbf{n}}\mathbf{/}\mathit{n}$.

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

$\mathbf{sin}\mathit{i}\mathit{z}\mathbf{=}\mathit{i}\mathbf{}\mathbf{sinh}\mathbf{}\mathit{z}$

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

$\mathbf{2}\mathbf{.}\mathbf{8}{\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{)}}$.

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.

14.${(1+i\sqrt{3})}^6$

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.

Follow steps (a), (b), (c) above to find all the values of the indicated roots $\sqrt[\mathbf{4}]{\mathbf{-}\mathbf{1}}$.

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

$\mathbf{arcsin}\left(3i/4\right)$