Aesthetic plots in Complex PlaneAmir H. KhanshanIslamic Azad University(IAU), Irankhanshan@yahoo.com

Complex functions can create beautiful and wonderful graphs in the complex plane. In this worksheet we look at some of them. Also, some interesting animations will be presented. The results not only are some mathematical fun but also are of interest in educational field. Similar graphics or animations can help students to visualize practical phenomenons in some engineering fields such as fluid mechanics or electrical engineering.Noting the following items can be helpful in undrestanding and creating new graphs. - adding the function by NiMmJSJaRzYjIiIh will move the figure by Re(NiMmJSJaRzYjIiIh) to the right and by Im(NiMmJSJaRzYjIiIh) up.- multiplying by real factor k will scale the figure.- multiplying by NiMtJSRleHBHNiMqJiUiaUciIiIlJnRoZXRhR0Yo rotates the figure by NiMlJnRoZXRhRw== .- multiplying by complex number Z will scale the shape by the factor of abs(Z) and rotate it by argument(Z).

restart;with(plots):f1:=(x+I*x^n)/(1+(x-x^m)*I);n:=2:m:=3:complexplot(f1,x=-6..6,thickness=2);n:=3:m:=3:complexplot(f1,x=-6..6,thickness=2,color=blue);n:=4:m:=6:complexplot(f1,x=-6..6,thickness=2,color=blue);Change m and n and have fun with different wonderful plots!n:='n':f2:=(1-I*5*x^1)/(1+x^n*I);n:=5:complexplot(f2,x=-7..7,thickness=2,color=blue);(Increasing n while it remains odd will not change the shape of figure, just flattening the right section.)n:='n':f3:=(cos(n*x)+I*x^3)/(1+(x-x^3)*I);n:=1:complexplot(f3,x=-6..6,thickness=2,color=blue);n:=2:complexplot(f3,x=-6..6,thickness=2,color=blue);n:=3:complexplot(f3,x=-6..6,thickness=2,color=blue);n:=4:complexplot(f3,x=-6..6,thickness=2,color=blue);n:='n':animate(complexplot,[f3,x=-6..6,color=COLOR(RGB,.5,.2,n/6)],n=1..6,thickness=2);complexplot((sin(1*x)-I*x^3)/(cos(3*x)+(-x^2)*I),x=-6..6,thickness=2,color=blue);n:='n':m:='m':p:='p':f4:=(cos(n*x)-I*p*x)/(cos(m*x)-x*I);

It's not hard to show that for m=n the graph is a circle centered at (1+p)/2. m:=n:abs(f4-(1+p)/2) assuming real;and the radius is:simplify(%) assuming(p,real);

n:=3:m:=2:p:=4:complexplot(f4,x=-6..6,thickness=2,color=blue);n:='n':f5:=(sin(x)+I*cos(x))/(cos(x)+(-x^n)*I);n:=1:plots[complexplot](f5,x=-11..11,thickness=2,color=blue,axes=none);n:=2:plots[complexplot](f5,x=-6..6,thickness=2,color=blue,axes=none);n:=3:plots[complexplot](f5,x=-6..6,thickness=2,color=blue,axes=none);Rotation around Z0:Z0:=0:n:=3:animate(complexplot,[(f5+Z0)*exp(-I*theta),x=-6..6,color=green],theta=0..2*Pi,thickness=2,axes=none,frames=31);animating as a pendulum:Z0:=0:n:=3:animate(complexplot,[-(f5+Z0)*exp(I*sin(theta)),x=-6..6,color=green],theta=0..2*Pi,thickness=2,axes=none,frames=31);(Note to the role of NiMtJSRzaW5HNiMlJnRoZXRhRw== in the exponent which oscillates the shape.)n:='n':f6:=(sin(x)-I*cos(n*x))/(1+x*I);For n=1 we have a leaf maybe,n:=1:plots[complexplot](f6,x=-11..11,thickness=2,color=blue,axes=none);n:=1:plots[complexplot](f6,x=-4.5..4.5,thickness=2,color=blue,axes=none);It looks like an onion. To change the direction it suffices to insert a minus before the function(since NiMvLSUkZXhwRzYjKiYlI1BpRyIiIiUiSUdGKSwkRikhIiI=)plots[complexplot](-f6,x=-4.5..4.5,thickness=2,color=blue,axes=none);Now, it resembles a dome. Scaling and transfering the graph can be done easily.k:=0.2:Z0:=1+I:plots[complexplot](-k*f6+Z0,x=-4.5..4.5,thickness=2,color=blue);k:=0.5:Z0:=I:complexplot([seq((-k*f6+Z0)*exp(I*i/3),i=1..19)],x=-4.5..4.5,thickness=2,color=green,axes=none);k:=0.5:Z0:=1+I:complexplot([seq((-k*f6+Z0)*exp(I*i/3),i=1..19)],x=-4.5..4.5,thickness=2,color=green,axes=none);The same function for another n:n:=3:plots[complexplot](f6,x=-4..4,thickness=2,color=blue,axes=none);It may be difficult to find the path from begining to end on this complicated curve. The following animation will help.F:=plots[complexplot]((sin(x)-I*cos(3*x))/(1+x*I),x=-4..4,thickness=2,color=cyan,axes=none,style=line):animate(complexplot,[(sin(a)-I*cos(3*a))/(1+a*I),x=-1..1],a=-4..4,frames=91,style=point,symbolsize=20,background=F);m:='m':p:='p':q:='q':f7:=(sin(p*t)+cos(q*t)*I)*exp(I*m*sin(t));For p=q the shape is a perfect circle. For m=0 there is a symmetrical shape. Here are some other values.q:=5:p:=4:m:=0:complexplot(f7,t=0..6.3,thickness=2);p:=5:q:=1:m:=1:complexplot(f7,t=0..6.3,thickness=2);m:=3:p:=4:q:=3:complexplot(f7,t=0..7,thickness=2);m:=3:p:=2:q:=3:complexplot(f7,t=0..7,thickness=2);m:='m':p:='p':q:='q':f8:=(sin(p*t)+cos(q*t)*I)*exp(I*m*t);m:=2:p:=3:q:=5:complexplot(f8,t=0..7,thickness=2);Dance of complex points:f8u:=unapply(f8,t):F:=complexplot(f8,t=-4..4,thickness=2,color=cyan,axes=none,style=line):animate(complexplot,[[f8u(t-2),f8u(-t+2)],x=-1.5..1.5],t=-3..3.28,frames=111,style=point,symbolsize=20,background=F);The following command animates moving of the complex point on the curve. (The begining and end value of t has been such selected to have a continous move when continous playing is active.)animate(complexplot,[[f8u(t),f8u(t-.05),f8u(t-0.1),f8u(t-0.15)],x=-1.5..1.5,color=[blue,green,black,yellow]],t=-2.2..4,frames=91,style=point,symbolsize=20,background=F);p:=4:q:=8:m:=1:complexplot(f8,t=0..6.3,thickness=2,color=blue);p:=6:q:=8:m:=1:complexplot(f8,t=0..6.3,thickness=2,color=blue);p:=6:q:=18:m:=1:complexplot(f8,t=0..6.3,thickness=2,color=blue);p:=7:q:=35:m:=3:complexplot(f8,t=0..6.3,thickness=2,color=blue,axes=none);p:=6:q:=12:m:=1:complexplot(f8,t=0..6.3,thickness=2,color=blue,axes=none);Rotating counterclockwise:p:=6:q:=12:m:=1:animate(complexplot,[f8*exp(I*theta),t=0..6.3,color=green],theta=0..Pi/3,thickness=2,frames=21,axes=none);...and much more!m:=3:p:=10:q:=3:complexplot(f8,t=0..7,thickness=2,color=violet);animating gradual evolution of the graph:m:=3:p:=10:q:=3:animate(complexplot,[f8,t=0..a,color=green],a=0..7,frames=91,thickness=2);m:=1:p:=18:q:=1:complexplot(f8,t=0..7,thickness=2,color=violet);m:=1:p:=12:q:=1:complexplot((sin(p*t)+cos(q*t)*I+2)*exp(I*m*t),t=0..7,thickness=2,color=violet);m:=1:p:=12:q:=1:complexplot((sin(p*t)+cos(q*t)*I+2+2*I)*exp(I*m*t),t=0..7,thickness=2,color=violet);p:=5:q:=1:m:=1:complexplot((sin(p*t)+cos(q*t)*I)*exp(I*m*sin(4*t)),t=0..6.3,thickness=2);Tiling the graph with appropriate distances can create interesting figures.f9:=(sin(p*t)+cos(q*t)*I)*exp(I*m*sin(4*t)):p:=5:q:=1:m:=1:complexplot([f9,f9+2,f9-2,f9+2*I,f9-2*I],t=0..6.3,thickness=2);

[1] Maplesoft website: http:\\www.maplesoft.com[2] Maple Applications Center: http:\\www.mapleapps.com[3] J. Mathews and R. Howell. Complex Analysis for Mathematics and Engineering. Jones and Bartlett, 1997.