1. 计算出下列复数的实部，虚部，模和幅角

（1）$1+i\sqrt{3}$

x = 1 + I*Sqrt[3]
Re[x]
Im[x]
Abs[x]
Arg[x]

（2）$1-cos\alpha +isin\alpha ,0\le \alpha <2\pi$

f = Refine[1 - Cos[a] + I*Sin[a], 0 <= a < 2*Pi]

Refine[Re[f], 0 <= a < 2*Pi]

Refine[Im[f], 0 <= a < 2*Pi]

Refine[Norm[f], 0 <= a < 2*Pi]

Refine[ArcTan[f], 0 <= a < 2*Pi]

（3）$e^{isinx},x为实数$

f = Refine[E^{I*Sin[x]}, x \[Element] Reals]

ComplexExpand[Re[f]]

ComplexExpand[Im[f]]

ComplexExpand[Abs[f]]

ComplexExpand[Arg[f]]

（4）$e^{iz}$

Clear[z];
f = E^{I*z}
ComplexExpand[Re[f], z]
ComplexExpand[Im[f], z]
ComplexExpand[Abs[f], z]
ComplexExpand[Arg[f], z]

2. 计算以上复数的和、差、积、商（任意组合、各两个）

f1 = 1 + I*Sqrt[3]
f2 = Refine[1 - Cos[a] + I*Sin[a], 0 <= a < 2*Pi]
f3 = ComplexExpand[E^{I*Sin[x]}]
f4 = E^{I*z}

ComplexExpand[f1 + f2]
ComplexExpand[f3 + f4, z]
ComplexExpand[f1 - f2]
ComplexExpand[f3 - f4, z]
ComplexExpand[f1 * f2]
ComplexExpand[f3 * f4, z]
ComplexExpand[f1 / f2]
ComplexExpand[f3 / f4, z]

3. 计算下列复数方程的解，并画出它们的位置

（1）$z^2+1=0$

Solve[z^2 + 1 == 0, z];
temp1 = z /. {{z -> -I}, {z -> I}}
ListPlot[(Tooltip[{Re[#1], Im[#1]}] &) /@ temp1]

（2）$z^3+8=0$

sol2 = Solve[z^3 + 8 == 0, z];
temp2 = z /. sol2
ListPlot[(Tooltip[{Re[#1], Im[#1]}] &) /@ temp2]

（3）$z^4-1=0$

sol3 = Solve[z^4 - 1 == 0, z];
temp3 = z /. sol3
ListPlot[(Tooltip[{Re[#1], Im[#1]}] &) /@ temp3]

（4）$z^4+1=0$

sol4 = Solve[z^4 + 1 == 0, z];
temp4 = z /. sol4
ListPlot[(Tooltip[{Re[#1], Im[#1]}] &) /@ temp4]

（5）$z^{2n}+1=0$

i = 1;
temp5 = {};

While[i < 10, sol5 = Solve[z^{2*i} + 1 == 0, z]; p = z /. sol5; 
 temp5 = Join[temp5, p]; i++]
Print[temp5]
ListPlot[(Tooltip[{Re[#1], Im[#1]}] &) /@ temp5]

4. 把下列关系用几何图形表示出来

（1）$|z|<2,|z|=2,|z|>2$

RegionPlot[x^2 + y^2 < 4, {x, -3, 3}, {y, -3, 3}]

ContourPlot[x^2 + y^2 == 4, {x, -3, 3}, {y, -3, 3}]

RegionPlot[x^2 + y^2 > 4, {x, -3, 3}, {y, -3, 3}]

（2）$Rez>\frac{1}{2},1<Imz<2$

RegionPlot[x > 1/2 && 1 < y < 2, {x, 0, 3}, {y, 0, 3}]