Chapter 2: Complex Numbers

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.

$\mathit{c}\mathit{o}\mathit{s}\mathit{h}\left(2\pi i\right)$

Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.

$\left|e^{\mathrm{i\pi }/2}\right|$

Verify the formulas in Problems 17 to 24.

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

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).

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

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

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

arctan2i

Test each of the following series for convergence.

$\mathbf{\sum }\mathbf{(}\mathbf{1}\mathbf{+}\mathit{i}{\mathbf{)}}^{\mathbf{n}}$

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

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