Chapter 2: Complex Numbers

If$\mathit{x}{\mathit{y}}^{\mathbf{3}}\mathbf{-}\mathit{y}{\mathit{x}}^{\mathbf{3}}\mathbf{=}\mathbf{6}$is the equation of a curve, find the slope and the equation of the tangent line at the point$\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{)}$. Computer plot the curve and the tangent line on the same axes.

Test each of the following series for convergence.

$\mathbf{\sum }\frac{\mathbf{1}}{{\mathbf{n}}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{i}}{\mathbf{n}}$

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

sin 2z=2 sin z cos z

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

$\mathbf{arccos}\left(i\sqrt{8}\right)$

Describe geometrically the set of points in the complex plane satisfying the following equations. $\left|z-1+i\right|\mathbf{=}\mathbf{2}$.

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

Describe geometrically the set of points in the complex plane satisfying the following equations.$\mathbf{|}\mathit{z}\mathbf{+}\mathbf{1}\mathbf{|}\mathbf{+}\mathbf{|}\mathit{z}\mathbf{-}\mathbf{1}\mathbf{|}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{}$

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

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

${\mathit{z}}^{\mathbf{2}}\mathbf{=}\mathbf{-}{\overline{\mathbf{z}}}^{\mathbf{2}}\mathbf{.}$.

Findand as functions of for the example above, and verify for this case that $\mathit{v}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{a}$are correctly given by the method of the example.