Complex Numbers

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

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

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

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

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

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

$\mathbf{2}\mathbf{+}\mathbf{2}\mathbf{i}\sqrt{\mathbf{3}}$  

Find the values of the indicated roots.

 $\sqrt[\mathbf{3}]{\mathbf{2}\mathbf{i}\mathbf{-}\mathbf{2}}$

Find the values of the indicated roots:

$\sqrt[\mathbf{4}]{\mathbf{8}\mathbf{i}\sqrt{\mathbf{3}}\mathbf{-}\mathbf{8}}$

Find the values of the indicated roots.

$\sqrt[\mathbf{8}]{\frac{\mathbf{-}\mathbf{1}\mathbf{-}\mathbf{I}\sqrt{\mathbf{3}}}{\mathbf{2}}}$

Find the values of the indicated roots:

$\sqrt[\mathbf{5}]{\mathbf{-}\mathbf{1}\mathbf{-}\mathbf{i}}$

Find the values of the indicated roots.

 $\sqrt[\mathbf{5}]{\mathbf{i}}$

Using the fact that a complex equation is really two real equations, find the double angle formulas $(for sin 2\theta , cos2\theta )$by using equation 10.2.

As in Problem 27, find the formulas for $\mathbf{(}\mathit{f}\mathit{o}\mathit{r}\mathbf{ }\mathbf{sin}\mathbf{ }\mathbf{3}\mathit{\theta }\mathbf{,}\mathbf{ }\mathbf{cos}\mathbf{3}\mathit{\theta }\mathbf{)}$.

Show that the center of mass of three identical particles situated at the point ${\mathit{z}}_{\mathbf{1}}\mathbf{,}{\mathit{z}}_{\mathbf{2}}\mathbf{,}{\mathit{z}}_{\mathbf{3}}$ is $\frac{\left({\mathit{z}}_{\mathbf{1}}\mathbf{,}{\mathit{z}}_{\mathbf{2}}\mathbf{,}{\mathit{z}}_{\mathbf{3}}\right)}{\mathbf{3}}$ .

Show that the sum of the three cube roots of 8 is zero.

Show that the sum of the n nth roots of any complex number is zero.

The three cube roots of +1 are often called 1,$\mathbf{\omega }$ , and ${\mathbf{\omega }}^{\mathbf{2}}$. Show that this is reasonable, that is, show that the cube roots of +1 are +1 and two other numbers, each of which is the square of the other.

Verify the results given for the roots in Example 4. You can find the exact values in terms of $\sqrt{\mathbf{3}}$  by using trigonometric addition formulas or more easily by using a computer to solve ${\mathit{z}}^{\mathbf{6}}\mathbf{=}\mathbf{-}\mathbf{8}\mathit{i}$. (You still may have to do a little work by hand to put the computer’s solution into the given form.)

Solve the equations ${\mathit{e}}^{\mathbf{i}\mathbf{\theta }}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{+}\mathit{i}\mathbf{ }\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathbf{,}{\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{\theta }}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{-}\mathit{i}\mathbf{ }\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }$, for cos θ and sin θ and so obtain equations (11.3).

Find each of the following in rectangular form $\mathbf{x}\mathbf{+}\mathbf{iy}$ and check your results by computer. Remember to save time by doing as much as you can in your head.

${\mathbf{e}}^{\mathbf{-}\left(i\pi /4\right)\mathbf{+}\mathbf{ln}\mathbf{3}}$.

Find each of the following in rectangular form $\mathbf{x}\mathbf{+}\mathbf{iy}$ and check your results by computer. Remember to save time by doing as much as you can in your head.

${\mathit{e}}^{\mathbf{3}\mathbf{l}\mathbf{n}\mathbf{-}\mathit{i}\mathit{\pi }}$.

Find each of the following in rectangular form x+iy and check your results by computer. Remember to save time by doing as much as you can in your head.

${\mathit{e}}^{\left(i\pi /4\right)\mathbf{+}\left(ln2\right)\mathbf{/}\mathbf{2}}$.

Find each of the following in rectangular form x+iy and check your results by computer. Remember to save time by doing as much as you can in your head.

$\mathit{c}\mathit{o}\mathit{s}\left(i \mathrm{In}5\right)$.

Find each of the following in rectangular form $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ and check your results by computer. Remember to save time by doing as much as you can in your head.

$\mathbf{tan}\mathbf{\left(}\mathit{i}\mathbf{ln}\mathbf{2}\mathbf{\right)}$

Find each of the following in rectangular form x+iy and check your results by computer. Remember to save time by doing as much as you can in your head.

$\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathbf{ }\mathit{\pi }\mathbf{-}\mathbf{2}\mathit{i}\mathbf{ }\mathit{l}\mathit{n}\mathbf{3}\mathbf{)}\mathbf{.}$

Find each of the following in rectangular form x + iy and check your results by computer. Remember to save time by doing as much as you can in your head.

$\mathit{s}\mathit{i}\mathit{n}\left(\pi -i \mathrm{In}3\right)$.

Find each of the following in rectangular form $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ and check your results by computer. Remember to save time by doing as much as you can in your head.

$\mathit{s}\mathit{i}\mathit{n}\left(i\mathrm{In}i\right)$ .

In the following integrals express the sines and cosines in exponential form and then integrate to show that ${\mathbf{\int }}_{\mathbf{-}\mathbf{\pi }}^{\mathbf{\pi }}\mathit{c}\mathit{o}\mathit{s}\mathbf{2}\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathbf{3}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}\mathbf{0}$

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

${\mathbf{\int }}_{\mathbf{-}\mathbf{\pi }}^{\mathbf{\pi }}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathbf{3}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}\mathit{\pi }$

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

${\mathbf{\int }}_{\mathbf{-}\mathbf{\pi }}^{\mathbf{\pi }}\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathbf{3}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}\mathbf{0}$

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

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{2}\mathbf{x}}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathbf{4}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}\mathit{\pi }$

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

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

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

${\mathbf{\int }}_{\mathbf{-}\mathbf{\pi }}^{\mathbf{\pi }}\mathit{s}\mathit{i}\mathit{n}\mathbf{3}\mathit{x}\mathbf{ }\mathit{c}\mathit{o}\mathit{s}\mathbf{ }\mathbf{4}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}\mathbf{0}$

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:

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

Evaluate$\mathbf{\int }{\mathit{e}}^{\left(a+ib\right)\mathbf{x}}\mathbf{ }\mathit{d}\mathit{x}$and take real and imaginary parts to show that:

$\mathbf{\int }{\mathit{e}}^{\mathbf{a}\mathbf{x}}\mathbf{ }\mathit{s}\mathit{i}\mathit{n}\mathbf{ }\mathit{b}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{ax}}\mathbf{ }\left(a \sin \mathrm{bx}-b \mathrm{cod} \mathrm{bx}\right)}{{\mathbf{a}}^{\mathbf{2}}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}}$

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

 $\mathbf{sinh}\mathbf{ }\mathbf{z}\mathbf{=}\mathbf{sinh}\mathbf{ }\mathbf{x}\mathbf{ }\mathbf{cos}\mathbf{ }\mathbf{y}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{cosh}\mathbf{ }\mathbf{x}\mathbf{ }\mathbf{sin}\mathbf{ }\mathbf{y}$

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

cosh z=cosh x cos y+i sinh x sin y

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

 sin 2z=2 sin z cos z

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

$\mathbf{cos}\mathbf{ }\mathbf{2}\mathit{z}\mathbf{=}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{z}\mathbf{-}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathbf{ }\mathit{z}$

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

sinh 2z=2 sinh z cosh z

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

 $\mathbf{cos}\mathit{h}\mathbf{ }\mathbf{2}\mathit{z}\mathbf{=}{\mathbf{cosh}}^{\mathbf{2}}\mathbf{ }\mathbf{z}\mathbf{+}{\mathbf{sinh}}^{\mathbf{2}}\mathbf{z}$

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

$\frac{\mathbf{d}}{\mathbf{dz}}\mathbf{cos}\mathbf{ }\mathbf{z}\mathbf{=}\mathbf{-}\mathbf{sin}\mathbf{ }\mathbf{z}$

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

$\frac{\mathbf{d}}{\mathbf{dz}}\mathbf{cosh}\mathbf{ }\mathbf{z}\mathbf{=}\mathbf{sinhz}$

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

$\mathit{c}\mathit{o}\mathit{s}{\mathit{h}}^{\mathbf{2}}\mathit{z}\mathbf{-}\mathit{s}\mathit{i}\mathit{n}{\mathit{h}}^{\mathbf{2}}\mathit{z}\mathbf{=}\mathbf{1}$

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

 $\mathbf{cos}\mathbf{ }\mathbf{z}\mathbf{=}\mathbf{cos}\mathbf{ }\mathbf{x}\mathbf{ }\mathbf{cosh}\mathbf{ }\mathbf{y}\mathbf{-}\mathbf{i}\mathbf{ }\mathbf{sin}\mathbf{ }\mathbf{x}\mathbf{ }\mathbf{sinh}\mathbf{ }\mathbf{y}$

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

${\mathbf{cos}}^{\mathbf{4}}\mathbf{ }\mathbf{z}\mathbf{+}{\mathbf{sin}}^{\mathbf{4}}\mathbf{z}\mathbf{=}\mathbf{1}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{sin}}^{\mathbf{2}}\mathbf{2}\mathbf{z}$

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

 $\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathbf{3}\mathit{z}\mathbf{\right)}\mathbf{=}\mathbf{4}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{3}}\mathbf{\left(}\mathit{z}\mathbf{\right)}\mathbf{-}\mathbf{3}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathit{z}\mathbf{\right)}$

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

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

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

siniz = i sin z

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

$\mathbf{ }\mathbf{tani}\mathbf{ }\mathbf{z}\mathbf{=}\mathbf{i}\mathbf{ }\mathbf{tanhz}$

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

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