WOLFRAM|DEMONSTRATIONS PROJECT

Domain Coloring for Common Functions in Complex Analysis

​
f(x)=
z
z

1-z
1+z
1/z

log(z)
Γ(z)
2
z
Γ(2,z)
z
Γ(3,z)
z
2
(1-z)
z+
1
z
Β
z
(1,2)
℘(z;1,2)
Β
z
(1,3)
℘(z;1,)
Β
z
(1,4)
sec(z)
cos(z)
sin(z)
sec(z)
sin
1
z
cot(z)
csc(z)
tan(z)
tan(z)
csc(z)
cot(z)
sin(z)
ζ(z)
detail
low
medium
high
range
π/8
2π/3
2π
π/4
π
4π
π/2
3π/2
8π
A domain coloring or phase portrait is a popular and attractive way to visualize functions of a complex variable. The absolute value (modulus) of the function at a point is represented by brightness (dark for small modulus, light for large modulus), while the argument (phase) is represented by hue (red for positive real values, yellow-green for positive imaginary values, etc.). This Demonstration creates domain colorings for many well-known functions in complex analysis: trigonometric and exponential functions, Möbius transformations, the Joukowski map, the Koebe function, Weierstrass
℘
functions, gamma and beta functions, and the Riemann zeta function.