Complex Numbers

Solve for all possible values of the real numbers $\mathit{x}$ and $\mathit{y}$in the following equations.

$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}\mathbf{=}\mathit{y}\mathbf{+}\mathit{i}\mathit{x}$

Solve for all possible values of the real numbers $\mathit{x}$ and $\mathit{y}$ in the following equations.

$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}\mathbf{=}\mathbf{3}\mathit{i}\mathbf{-}\mathit{i}\mathit{x}$

Solve for all possible values of the real numbers $\mathit{x}$ and $\mathit{y}$ in the following equations. $\left(2x-3y-5\right)\mathbf{+}\mathit{i}\left(x+2y+1\right)\mathbf{=}\mathbf{0}$

Solve for all possible values of the real numbers $\mathit{x}$ and $\mathit{y}$ in the following equations.$\left(x+2y+3\right)\mathbf{+}\mathit{i}\left(3x-y-1\right)\mathbf{=}\mathbf{0}$

Solve for all possible values of the real numbers  and in the following equations ${\left(x-iy\right)}^{\mathbf{2}}\mathbf{=}\mathbf{2}\mathit{i}\mathit{x}$.

Solve for all possible values of the real numbers  and in the following equations ${\left(x+iy\right)}^{\mathbf{2}}\mathbf{=}{\left(x-iy\right)}^{\mathbf{2}}$.

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

Solve for all possible values of the real numbers  x and y in the following equations.

${\left(x+iy\right)}^{\mathbf{3}}\mathbf{=}\mathbf{-}\mathbf{1}$

Solve for all possible values of the real numbers x and y in the following equations. $\left|1-\left(x+iy\right)\right|\mathbf{=}\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$

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.

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

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

 $\mathit{z}\mathbf{-}\overline{\mathbf{z}}\mathbf{=}\mathbf{5}\mathit{i}$. 

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

The angle $\mathit{z}\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{2}}$. 

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

 $\mathit{R}\mathit{e}\mathbf{\left(}{\mathit{z}}^{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{4}$. 

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

 $\mathbf{R}\mathit{e}\mathbf{\left(}\mathit{z}\mathbf{\right)}\mathbf{>}\mathbf{2}$. 

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

 $\left|z+3i\right|\mathbf{=}\mathbf{4}$. 

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.

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

Find  and  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.

Find  and  if $\mathit{z}\mathbf{=}\frac{\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{i}\mathbf{t}\mathbf{)}}{\mathbf{2}\mathbf{t}\mathbf{+}\mathbf{i}}$ .

Question: Find v and a if $\mathit{z}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathbf{2}\mathit{t}\mathbf{+}\mathit{i}\mathbf{ }\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{t}$, can you describe the motion.

Prove that an absolutely convergent series of complex numbers converges. This means to prove that $\mathbf{\sum }\mathbf{(}{\mathbf{a}}_{\mathbf{n}}\mathbf{+}{\mathbf{ib}}_{\mathbf{n}}\mathbf{)}\mathbf{ }$converges ($\mathbf{ }{\mathbf{a}}_{\mathbf{n}}$and ${\mathbf{b}}_{\mathbf{n}}$real) if $\mathbf{\sum }\sqrt{{{\mathbf{a}}_{\mathbf{n}}}^{\mathbf{2}}\mathbf{+}{{\mathbf{b}}_{\mathbf{n}}}^{\mathbf{2}}}\mathbf{ }$converges. Hint: Convergence of $\mathbf{\sum }\mathbf{(}{\mathbf{a}}_{\mathbf{n}}\mathbf{+}{\mathbf{ib}}_{\mathbf{n}}\mathbf{)}$means that $\mathbf{\sum }{\mathbf{a}}_{\mathbf{n}}\mathbf{ }$and $\mathbf{\sum }{\mathbf{b}}_{\mathbf{n}}$ both converge. Compare $\mathbf{\sum }\mathbf{\left|}{\mathbf{a}}_{\mathbf{n}}\mathbf{\right|}$ and$\mathbf{ }\mathbf{\sum }\mathbf{\left|}{\mathbf{b}}_{\mathbf{n}}\mathbf{\right|}$ with $\mathbf{\sum }\sqrt{{{\mathbf{a}}_{\mathbf{n}}}^{\mathbf{2}}\mathbf{+}{{\mathbf{b}}_{\mathbf{n}}}^{\mathbf{2}}}$  , and use Problem 7.9 of Chapter 1 .

Test each of the following series for convergence.

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

Test each of the following series for convergence.

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

Test each of the following series for convergence.

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

Test each of the following series for convergence.

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

Test each of the following series for convergence.

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

Test each of the following series for convergence.

$\underset{\mathbf{ }}{\mathbf{\sum }\frac{{\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{i}\mathbf{)}}^{\mathbf{n}}}{\mathbf{n}}}$

Test each of the following series for convergence.

$\mathbf{ }\mathbf{\sum }_{\mathbf{ }}{\mathit{e}}^{\mathbf{i}\mathbf{n}\frac{\mathbf{\pi }}{\mathbf{6}}}$

Test each of the following series for convergence.

$\mathbf{\sum }_{\mathbf{ }}\frac{{\mathbf{i}}^{\mathbf{n}}}{\mathbf{n}}$

Test each of the following series for convergence.

$\underset{\mathbf{ }}{\mathbf{\sum }{\left(\frac{1+i}{1-i\sqrt{3}}\right)}^{\mathbf{n}}}$

Test each of the following series for convergence.

 $\mathbf{\sum }_{\mathbf{ }}{\left(\frac{2+i}{3-4i}\right)}^{\mathbf{2}\mathbf{n}}$

Test each of the following series for convergence.

 $\mathbf{\sum }_{\mathbf{ }}\frac{{\left(3+2\mathit{i}\right)}^{\mathbf{n}}}{\mathbf{n}\mathbf{!}}$

Test each of the following series for convergence.

$\mathbf{\sum }_{\mathbf{ }}\frac{{\mathbf{(}\mathbf{1}\mathbf{+}\mathit{i}\mathbf{)}}^{\mathbf{n}}}{{\mathbf{(}\mathbf{2}\mathbf{=}\mathbf{i}\mathbf{)}}^{\mathbf{n}}}$

Prove that a series of complex terms diverge if $\mathit{\rho }\mathbf{>}\mathbf{1}$  (  = ratio test limit). Hint: The ${\mathit{n}}^{\mathbf{t}\mathbf{h}}$ term of a convergent series tends to zero.

Find the disk of convergence for each of the following complex power series.a

${\mathbf{e}}^{\mathbf{z}}\mathbf{=}\mathbf{1}\mathbf{+}\mathbf{z}\mathbf{+}\frac{{\mathbf{z}}^{\mathbf{2}}}{\mathbf{2}\mathbf{!}}\mathbf{+}\frac{{\mathbf{z}}^{\mathbf{3}}}{\mathbf{3}\mathbf{!}}\mathbf{·}\mathbf{·}\mathbf{·}\left[\mathrm{equation} (8.1)\right]$

Find the disk of convergence for each of the following complex power series.

 $\mathit{z}\mathbf{-}\frac{{\mathbf{z}}^{\mathbf{2}}}{\mathbf{2}}\mathbf{+}\frac{{\mathbf{z}}^{\mathbf{3}}}{\mathbf{3}}\mathbf{-}\frac{{\mathbf{z}}^{\mathbf{4}}}{\mathbf{4}}\mathbf{+}\mathbf{\dots }$

Find the disk of convergence for each of the following complex power series.

$\mathbf{1}\mathbf{-}\frac{{\mathbf{z}}^{\mathbf{2}}}{\mathbf{3}\mathbf{!}}\mathbf{+}\frac{{\mathbf{z}}^{\mathbf{4}}}{\mathbf{5}\mathbf{!}}\mathbf{-}\mathbf{·}\mathbf{·}\mathbf{·}$

Find the disk of convergence for each of the following complex power series.

 $\underset{\mathbf{ }}{\mathbf{\sum }{\mathbf{z}}^{\mathbf{n}}}$

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.

Find the disk of convergence for each of the following complex power series.

$\mathbf{\sum }_{\mathbf{n}\mathbf{-}\mathbf{1}}^{\mathbf{\infty }}{\mathit{n}}^{\mathbf{2}}{\mathbf{\left(}\mathbf{3}\mathit{i}\mathit{z}\mathbf{\right)}}^{\mathbf{n}}\mathbf{.}$ 

: Find the disk of convergence for each of the followingcomplex power series.

 $\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{n}}{\mathbf{z}}^{\mathbf{2}\mathbf{n}}}{\mathbf{\left(}\mathbf{2}\mathbf{n}\mathbf{\right)}\mathbf{!}}$

Find the disk of convergence for each of the following complex power series.

 $\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{z}}^{\mathbf{2}\mathbf{n}}}{\mathbf{(}\mathbf{2}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{!}}$

Find the disk of convergence for each of the following complex power series.

 $\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{z}}^{\mathbf{n}}}{\sqrt{\mathbf{n}}}$

Find the disk of convergence for each of the following complex power series.

 $\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{{\mathbf{\left(}\mathbf{i}\mathbf{z}\mathbf{\right)}}^{\mathbf{n}}}{{\mathbf{n}}^{\mathbf{2}}}$