sage: real(1+i)
1
sage: real(exp(i))
cos(1)
sage: imaginary(1+i)
1

sage: abs(1+i)
sqrt(2)

sage: arg(1+i)
1/4*pi

sage: pts = [e^(I*(theta)) for theta in srange(0, 2*pi, pi/8)]
sage: list_plot([(real(p),imag(p)) for p in pts])

sage: log(-i)
-1/2*I*pi
sage: log(-1)
I*pi
sage: log(-1+0.001*I)
4.99999750058881e-7 + 3.14059265392313*I
sage: log(-1-0.001*I)
4.99999750058881e-7 - 3.14059265392313*I

sage: complex_plot(1/(1+z^2), (-5,5), (-5, 5))

sage: f(z) = sin(z)/z
sage: f.limit(z=0)
z |--> 1

sage: f(z)=z^2
sage: diff(f)
z |--> 2*z

sage: f(z) = conjugate(z)
sage: diff(f)
z |--> diff(conjugate(z), z)

sage: conj(x,y) = (x,-y)
sage: diff(conj)
[ (x, y) |--> 1  (x, y) |--> 0]
[ (x, y) |--> 0 (x, y) |--> -1]