Setup:

Clear["Global`*"];

#1:

Clear["Global`*"];

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`*"];

p1=Plot3D[+Im[Sqrt[x+I y]],{x,-4,4},{y,-4,4}]
p2=Plot3D[-Im[Sqrt[x+I y]],{x,-4,4},{y,-4,4},ColorFunction->"Pastel"]
Show[p1,p2]

p3=Plot3D[+Re[Sqrt[x+I y]],{x,-4,4},{y,-4,4}]
p4=Plot3D[-Re[Sqrt[x+I y]],{x,-4,4},{y,-4,4},ColorFunction->"Pastel"]
Show[p3,p4,PlotRange->All]

ParametricPlot3D[{Log[r] Cos[ϕ],Log[r] Sin[ϕ],ϕ/(π)},
{ϕ,-2.5 π,2.5 π},
{r,1/10,10},Mesh->{30,10},
PlotRange->All,
ColorFunction->(Hue[16 #4/π,16 #4/π,0.8]&),
AxesLabel->{"Re z","Im z","Arg z"},
Ticks->{Automatic,Automatic,Evaluate[({#,#*"π"}&)/@Range[-3,3]]},
PlotLabel->Style[" z |-> {Re log z, Im log z, arg z}",16]]

#3:

Clear["Global`*"];

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

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

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

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

list2

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

list //Abs

#4:

Clear["Global`*"];

Compute the Re and Im part of:

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

b) Convert this into polar form.

c) Compute the absolute value

tmp3=((( (((1+I))/((1-I)))))^3)

tmp2= (((3I-7))/((I+4)))

tmp2//Abs

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

tmp2=((3+I)/(2+I))*((2-I)/(2-I))

tmp2 //Re

tmp2 //Im

tmp2 //Abs

f[x_,y_]=CoordinateTransform[  "Cartesian"->"Polar" ,{x,y}]

f[Re[tmp2],Im[tmp2]]

ctmp=tmp //Conjugate

tmp ctmp

tmp ctmp //Sqrt

#5:

x=2

Clear["Global`*"];

5) 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

Solve[x + I y == 3 I - 4, {x,y},Assumptions->{x,y}∈Reals]

Solve[x + I y == 3 I - 4, {x,y},Assumptions->{x,y}∈Reals]

tmp1= x + I y -( 3 I - 4)

tmp2=Im[tmp1] /.{Im[y]->0,Re[x]->x}

Solve[tmp2==0,x]

#6:

Clear["Global`*"];

6) Show that multiplying two complex numbers z1=x1+I y1, z2=x2+I y2 can be represented as a
rotation and rescaling of z1 by z2. Hint: convert to polar coordinates to show the
property, but express the final result in terms of {x1,x2,y1,y2}

z1= x1+ I y1;
z2= x2+ I y2;

z1 z2 //Expand

Solve[(2x−3y−5)+I (x+2y+1)==0, {x,y},Assumptions->{x,y}∈Reals]

tx[x_,y_]=CoordinateTransform[  "Cartesian"->"Polar" ,{x,y}]

z1=tx[x1,y1]

z2=tx[x2,y2]

output = (Sqrt[(x1^2)+(y1^2)])(Sqrt[(x2^2)+(y2^2)])Exp[ArcTan[x1,y1]+ArcTan[x2,y2]]

output //FullSimplify

#7:

Clear["Global`*"];

) Compute the Re and Im parts of ln[z] where a) z1= r Exp[Iθ] and b) z2= z1* Exp[2πI]

tmp1= Log[r (E^(I θ))* 1] //PowerExpand

r (E^(I (θ)))*(E^(I (2π)))

tmp2= Log[r (E^(I (θ+4π )))]  //PowerExpand

tmp2= Log[r (E^(I (θ+2π I)))] //PowerExpand

tmp1-tmp2 //Expand

#Gamow Question:

Clear["Global`*"];

{a,b}={0.4,0.5}

pine={1,0};
oak = {-1,0};
gallow:={a,b};
vpine=pine-gallow;
voak=oak - gallow;

doVec[pt2_]:=Arrow[{{0,0},pt2}] (* draw from origin *)
doVec[pt1_,pt2_]:=Arrow[{pt1,pt2}] (* draw difference *)
doVec[pt1_,pt2_,pt3_]:=Arrow[{pt1+pt3,pt2+pt3}] (* shift from pt3 *)

rotL[{a_,b_}]:={-b,a}
rotR[{a_,b_}]:={b,-a}

Graphics[{Thickness[0.01],
Black,doVec[{1,2}],
Red,doVec[rotR[{1,2}]],
Blue,doVec[rotL[{1,2}]]
},Axes->True,PlotRange->All,AspectRatio->Automatic]

Graphics[{Thickness[0.01],
Green,doVec[oak],
Purple,doVec[pine],
Black,doVec[gallow],
Red,doVec[vpine],
Blue,doVec[voak]} ,
Axes->True,PlotRange->All,AspectRatio->Automatic]

Graphics[{Thickness[0.01],
Green,doVec[oak],
Purple,doVec[pine],
Black,doVec[gallow],
Red,doVec[vpine],
Blue,doVec[voak]},Axes->True,PlotRange->All,AspectRatio->Automatic]

Graphics[{Thickness[0.01],
Green,doVec[oak],
Purple,doVec[pine],
Black,doVec[gallow],
Red,doVec[{0,0},vpine,gallow],doVec[vpine],
Blue,doVec[{0,0},voak,gallow],doVec[voak],
},Axes->True,PlotRange->All,AspectRatio->Automatic]

Graphics[{Thickness[0.01],
Green,doVec[oak],
Purple,doVec[pine],
Black,doVec[gallow],
Red,doVec[{0,0},vpine,gallow],doVec[vpine],
Blue,doVec[{0,0},voak,gallow],doVec[voak],
},Axes->True,PlotRange->All,AspectRatio->Automatic]

Graphics[{Thickness[0.01],
Green,doVec[oak],
Purple,doVec[pine],
Black,doVec[gallow],
Red,doVec[{0,0},vpine,gallow],doVec[{0,0},rotL[vpine],pine],
Blue,doVec[{0,0},voak,gallow],doVec[{0,0},rotR[voak],oak],
},Axes->True,PlotRange->All,AspectRatio->Automatic]

Graphics[{Thickness[0.01],
Green,doVec[oak],
Purple,doVec[pine],
Black,doVec[gallow],
Red,doVec[{0,0},vpine,gallow],doVec[{0,0},rotL[vpine],pine],
Blue,doVec[{0,0},voak,gallow],doVec[{0,0},rotR[voak],oak],
Green, doVec[ (rotL[vpine]+pine),(rotR[voak]+oak) ],
Thickness[0.02],Dotted,Red, doVec[ (rotR[voak]+oak),(rotL[vpine]+pine) ]
},Axes->True,PlotRange->All,AspectRatio->Automatic]