91Ó°ÊÓ

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 $\frac{\left({\mathit{z}}_{\mathbf{1}}\mathbf{,}{\mathit{z}}_{\mathbf{2}}\mathbf{,}{\mathit{z}}_{\mathbf{3}}\right)}{\mathbf{3}}$.

Find each of the following in the x+iy form and check your answers by computer.

$\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left(2\mathrm{Ï€}i\right)$

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

${\left(1+2\mathit{i}\right)}^{\mathbf{3}}$.

Verify the formulas in Problems 17 to 24.

${\mathbf{tanh}}^{\mathbf{-}\mathbf{1}}\left(z\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{In}\left(\frac{1+z}{1-z}\right)$

Evaluate each of the following in $\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathit{i}\mathit{y}$form, and compare with a computer solution.

$\mathbf{ln}\mathbf{(}\mathbf{-}\mathit{i}\mathbf{)}$

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

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.${\mathit{e}}^{\mathbf{i}\mathbf{Ï€}\mathbf{/}\mathbf{2}}$

In each of the following problems, z represents the displacement of a particle from the origin. Find (as functions of t) its speed and the magnitude of its acceleration, and describe the motion.

$\mathbf{z}\mathbf{=}\mathbf{-}\mathbf{5}{\mathbf{e}}^{\mathbf{¾\pm Ó\neg ³Ù}}\mathbf{,}\mathbf{Ó\neg }\mathbf{=}\mathbf{const}$

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

arctan2i

For each of the following numbers, first visualize where it is in the complex plane. With a little practice, you can quickly find $\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{r}\mathbf{,}\mathit{θ}\mathbf{}$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 i-1.