Chapter 2: Complex Numbers

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

ln ( i - 1 )

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.

role="math" localid="1658739754131" ${\mathit{e}}^{\mathbf{3}\mathbf{l}\mathbf{n}\mathbf{-}\mathit{i}\mathit{\pi }}$.

Test each of the following series for convergence.

$\mathbf{\sum }{\left(\frac{1-i}{1+i}\right)}^{\mathbf{n}}$

Question: Find each of the following in the x+iy form and compare a computer solution.

${\mathbf{sinh}}^{\mathbf{-}\mathbf{1}}\mathbf{(}\mathbf{i}\mathbf{/}\mathbf{2}\mathbf{)}$

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.

$\mathit{z}\mathbf{=}\left(1+i\right)\mathit{t}\mathbf{-}\left(2+i\right)\left(1-t\right)$Hint: Show that the particle moves along a straight line through the points$\mathbf{1}\mathbf{+}\mathit{i}$and$\mathbf{-}\mathbf{2}\mathbf{-}\mathit{i}$.

Solve for all possible values of the real numbers x and y in the following equations.$\mathbf{|}\mathbf{x}\mathbf{+}\mathbf{i}\mathbf{y}\mathbf{|}\mathbf{=}\mathit{y}\mathbf{-}\mathit{i}\mathit{x}\mathbf{.}$ .

Describe geometrically the set of points in the complex plane satisfying the following equations. $\left|z\right|\mathbf{=}\mathbf{2}$.

Describe geometrically the set of points in the complex plane satisfying the following equations.

$\mathit{R}\mathit{e}\mathbf{}\mathit{z}\mathbf{=}\mathbf{0}$.

Describe geometrically the set of points in the complex plane satisfying the following equations.

$\left|z-1\right|\mathbf{=}\mathbf{1}$.

Describe geometrically the set of points in the complex plane satisfying the following equations.

$\left|z-1\right|\mathbf{<}\mathbf{1}$.