Chapter 2: Complex Numbers

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 .

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.

Express the following complex numbers in the$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$form. Try to visualize each complex number, using sketches as in the examples if necessary${\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{\pi }\mathbf{/}\mathbf{4}}$.

Show from the power series (8.1) that ${\mathit{e}}^{{\mathbf{z}}_{\mathbf{1}}}\mathbf{·}{\mathit{e}}^{{\mathbf{z}}_{\mathbf{2}}}\mathbf{=}{\mathit{e}}^{{\mathbf{z}}_{\mathbf{1}}\mathbf{+}{\mathbf{z}}_{\mathbf{2}}}$.

Show that if the line through the origin and the point z is rotated $\mathbf{90}\mathbf{°}$about the origin, it becomes the line through the origin and the point iz. This fact is sometimes expressed by saying that multiplying a complex number byrotates it through$\mathbf{90}\mathbf{°}$. Use this idea in the following problem. Let$\mathit{z}\mathbf{=}\mathit{a}{\mathit{e}}^{\mathbf{i}\mathbf{\omega }\mathbf{t}}$be the displacement of a particle from the origin at time t. Show that the particle travels in a circle of radius a at velocity $\mathit{v}\mathbf{=}\mathit{a}\mathit{\omega }$and with acceleration of magnitude directed toward the centre${\mathit{v}}^{\mathbf{2}}\mathbf{/}\mathit{a}$of the circle.

Follow steps (a), (b), (c) above to find all the values of the indicated roots.

$\sqrt[\mathbf{3}]{\mathbf{-}\mathbf{8}\mathit{i}}$

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.

$\mathbf{7}\mathbf{(}\mathit{c}\mathit{o}\mathit{s}{\mathbf{110}}^{\mathbf{°}}\mathbf{-}\mathit{i}\mathit{s}\mathit{i}\mathit{n}{\mathbf{110}}^{\mathbf{°}}\mathbf{)}$.

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.

20.${\left(\frac{\sqrt{2}}{i-1}\right)}^{10}$

Verify the formulas in Problems 17 to 24.

$\mathit{s}\mathit{i}\mathit{n}{\mathit{h}}^{\mathbf{-}\mathbf{1}}\mathit{z}\mathbf{=}\mathit{l}\mathit{n}\mathbf{(}\mathit{z}\mathbf{\pm }\sqrt{{\mathbf{z}}^{\mathbf{2}}\mathbf{+}\mathbf{1}}\mathbf{)}$

Show that ${\mathbf{e}}^{\mathbf{nz}}\mathbf{=}\mathbf{(}\mathbf{coshz}\mathbf{+}\mathbf{sinhz}{\mathbf{)}}^{\mathbf{n}}\mathbf{=}\mathbf{cosh}\mathbf{}\mathbf{nz}\mathbf{+}\mathbf{sinh}\mathbf{}\mathbf{nz}\mathbf{.}$Use this and a similar equation for ${\mathit{e}}^{\mathbf{-}\mathbf{n}\mathbf{z}}$to find formulas for $\mathit{c}\mathit{o}\mathit{s}\mathit{h}\mathbf{3}\mathit{z}$and $\mathit{s}\mathit{i}\mathit{n}\mathit{h}\mathbf{3}\mathit{z}$in terms of $\mathbf{sinhz}$ and $\mathbf{coshz}$.