图形根据教材解释绘制，贴出MATLAB代码来，供大家参考。欢迎提供宝贵意见！

     设 z = x + i y,  w = u + i v, 则 w = z^2 = (x^2-y^2) + i*2*x*y 反映了两对变量 u,v 和x,y 之间的对应关系，因为涉及到 4 个变量，无法用一个平面内的几何图形表示出来， 必须把它看成两个复平面上的点集之间的对应关系。下面的图形反映了复平面z 上的三角形、矩形、双曲线和直线点集与其对应到w上的映射点集。

clear,clc,close all

figure('color',[1,1,1]);alpha = pi/6;

x = 0:0.1:1; y = tan(alpha)*x; x=[x,x(end)*ones(1,10),x(1)];

y = [y,linspace(y(end),0,10),y(1)];

subplot(2,2,1);fill(x,y,'m');axis equal;axis([0,1.2,0,1.2]);

text(0.5,0.8,'$$z$$','interpreter','latex');

xlabel('$$x$$','interpreter','latex');ylabel('$$y$$','interpreter','latex');

%%% uov 映射点解

z = x + i*y; w = z.^2; u = real(w); v = imag(w);

u = [u,u(end),u(1)];v=[v,0,v(1)];

subplot(2,2,2);fill(u,v,'c');axis equal;axis([0,1.2,0,1.2]);

text(0.8,0.8,'$$w=z^2$$','interpreter','latex');

xlabel('$$u$$','interpreter','latex');ylabel('$$v$$','interpreter','latex');

%%%%%%%%%%%%%%%%%% 矩形区域

n = 10; x1 = -1; x2 = 1; y1 = 0.5; y2 = 1;

x = [linspace(x1,x2,n),x2*ones(1,n),linspace(x2,x1,n),x1*ones(1,n)];

y = [y1*ones(1,n),linspace(y1,y2,n),y2*ones(1,n),linspace(y2,y1,n)];

subplot(2,2,3);fill(x,y,'m');axis equal;axis([-1,1.2,0,1.2]);

text(0.5,0.8,'$$z$$','interpreter','latex');

xlabel('$$x$$','interpreter','latex');ylabel('$$y$$','interpreter','latex');

%%% uov 映射点解

z = x + i*y; w = z.^2; u = real(w); v = imag(w);

u = [u,u(end),u(1)];v=[v,0,v(1)];

subplot(2,2,4);fill(u,v,'c');axis equal;axis([-2,2,-2.2,2.2]);

text(0.8,0.8,'$$w=z^2$$','interpreter','latex');

xlabel('$$u$$','interpreter','latex');ylabel('$$v$$','interpreter','latex');

%%%%%%%%%%%%%

figure('color',[1,1,1]); x1 = -2; x2 = 2; y1 = -2; y2 = 2;

h = 0.01;[x,y] = meshgrid(x1:h:x2,y1:h:y2);

u = x.^2 - y.^2; v = 2*x.*y;

subplot(2,2,1);contour(x,y,u,'b'), hold on; contour(x,y,v,'r--'),

xlabel('$$x$$','interpreter','latex');ylabel('$$y$$','interpreter','latex');

h = 0.5; n = (x2-x1)/h;

for i = 1:n+1

   subplot(2,2,2);hold on;plot([x1+(i-1)*h,x1+(i-1)*h],[y1,y2],'b');

   plot([x1,x2],[y1+(i-1)*h,y1+(i-1)*h],'r--');

end

box;xlabel('$$u$$','interpreter','latex');ylabel('$$v$$','interpreter','latex');

%%%%

x1 = -2; x2 = 2; y1 = -2; y2 = 2; h = 1.0; n = (x2-x1)/h;

for i = 1:n+1 % 固定z平面直线

   subplot(2,2,3);hold on;plot([x1+(i-1)*h,x1+(i-1)*h],[y1,y2],'r--');

   plot([x1,x2],[y1+(i-1)*h,y1+(i-1)*h],'b');

end

box;xlabel('$$x$$','interpreter','latex');ylabel('$$y$$','interpreter','latex');

for i = 1:n

   subplot(2,2,4); hold on; lam = i*0.5; u = -8:0.1:8;

   u1 = (lam^2-u).*((lam^2-u)>=0);v1 = 2*lam*sqrt(u1);plot(u,v1,'b--');%x为常数

   v2 = -2*lam*sqrt(u1);plot(u,v2,'b--');

   u2 = (lam^2+u).*((lam^2+u)>=0);v3 = 2*lam*sqrt(u2);plot(u,v3,'r-');%y为常数

   v4 = -2*lam*sqrt(u2);plot(u,v4,'r-');    

end

axis([-8,8,-8,8]);grid on;box

xlabel('$$u$$','interpreter','latex');ylabel(''$$v$$','interpreter','latex');