Setup:

Clear["Global`*"];

#1:

With Mathematica, plot the Re and Im part of z = x + I y in the complex plane .
                        Plot3D[Im[Sqrt[x + I y]], {x, -4, 4}, {y, -4, 4}]
Plot3D[Re[Sqrt[x + I y]], {x, -4, 4}, {y, -4, 4}]
Comment on your answer .

Clear["Global`*"];

Plot3D[Im[Sqrt[x+I y]],{x,-4,4},{y,-4,4}]

Plot3D[Re[Sqrt[x+I y]],{x,-4,4},{y,-4,4}]

#3:

For each of the following numbers, plot

the number and label it in ﬁve ways as

in Figure 3.3. Also plot the complex

conjugate of the number.

list={1+I, I-1,1-I (Sqrt[3]),I-(Sqrt[3])}

clist=list //Conjugate

ResourceFunction["ComplexToPolar"][1-2 I]

do[point_]:=Module[{},
re=Re[point];
im=Im[point];
Return[{re,im}]
]

list2=do /@ list

clist2=do /@ clist

p1= ListPlot[list2,PlotStyle->{Red,PointSize->0.03},AspectRatio->1,PlotRange->All];

p2= ListPlot[clist2,PlotStyle->{Blue,PointSize->0.03},AspectRatio->1,PlotRange->All];

Show[p1,p2,AspectRatio->1,PlotRange->All]

?CoordinateTransform

CoordinateTransform[ "Polar" -> "Cartesian",{r,θ}]

CoordinateTransform[  "Cartesian"->"Polar" ,{x,y}]

{list2,clist2} //MatrixForm

doPolar[vec_]:=Module[{x,y,r,θ},
{x,y}=vec;
r= (Sqrt[(x^2)+(y^2)]);
θ=ArcTan[x,y];
Return[{r,θ}];
]

plist2=doPolar /@ list2

cplist2=doPolar /@ clist2

doExp[vec_]:=Module[{},
{r,θ}=vec;
Return[r Exp[I θ]]
]

doExp /@ plist2

doExp /@ cplist2

doTrig[vec_]:=Module[{},
{r,θ}=vec;
Return[r (cos[θ]+I sin[θ])]
]

doTrig /@ plist2

doTrig /@ plist2 /.{cos->Cos,sin->Sin} //Simplify

list

doTrig /@ cplist2 /.{cos->Cos,sin->Sin} //Simplify

clist

#4:

Compute the Re and Im part of:

a) (3+I)/(2+I)

b) Convert this into polar form.

c) Compute the absolute value

tmp= ((3+I)/(2+I))

((3+I)/(2+I))*((2-I)/(2-I))

do[tmp]

do[tmp] //doPolar

ctmp=tmp //Conjugate

tmp ctmp

tmp ctmp //Sqrt

#6:

Clear["Global`*"];

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

a) x + Iy = 3 I − 4

b) (2x − 3y − 5) + I (x + 2y + 1) = 0

Part a

eq1= x + i y == 3 i − 4

eq1= x + i y -( 3 i − 4)

eq2= eq1 /.{i->0}

eq3= eq1 -  eq2 /.{i->1}

sol= Solve[{eq2==0,eq3==0},{x,y}]

eq1 /.sol

Part b

eq1= (2x−3y−5)+i (x+2y+1)

eq2= eq1 /.{i->0}

eq3= eq1 -  eq2 /.{i->1}

sol= Solve[{eq2==0,eq3==0},{x,y}]

eq1 /.sol