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.

Details

The terms domain coloring or phase portrait refer to the type of visualization technique for functions of a complex variable used here, or some variation thereof. For a function
f
of a complex variable, each point
z
in the complex plane is colored according to the value of
f(z)
. The brightness represents the absolute value (or modulus) of
f(z)
, with light representing large absolute value and dark representing small absolute value. The hue represents the argument (or phase) of
f(z)
, with red representing positive real numbers, yellow-green representing positive imaginary numbers, cyan representing negative real numbers, and so forth.
These plots illustrate various phenomena in complex analysis. A dark spot contains a zero of
f
, while a white spot contains a pole. The order of a zero or pole is given by the number of rays of each color emanating from it. Where a single color forms a saddle-shaped region (such as near the point
π/2+0
for the function
sin(z)
), the derivative must have a zero. The graphs of
1/z

and
sin(1/z)
contain essential singularities, and in particular illustrate Picard's great theorem that in any neighborhood of an essential singularity
f
must contain every complex value with at most one exception. Observe that functions such as
log
and the inverse trigonometric functions exhibit branch cuts.
In addition to the standard polynomial, trigonometric, and exponential functions, we have included: the Möbius transformation
(1-z)/(1+z)
, the Joukowski map
z+1/z
, the Koebe function
z
2
(1-z)
, a few instances of the beta functions
Β
z
(a,b)
z
∫
0
a-1
t
b-1
(1-t)
dt
, the gamma functions
Γ(z)=
∞
∫
0
z-1
t
-t
e
dt
and
Γ(a,z)=
∞
∫
z
a-1
t
-t
e
dt
, and the Weierstrass ℘ functions
℘(u;
g
2
,
g
3
)
defined by
℘(z;
g
2
,
g
3
)=x
if
z
x
∫
∞
-1/2
4
3
t
-
g
2
t-
g
3

dt
, and finally the famous Riemann zeta function
ζ(z)
, defined as the analytic continuation of the Dirichlet series
∞
∑
k=1
-z
k
for
Re(z)>1
.

References

[1] E. Wegert, Visual Complex Functions: An Introduction with Phase Portraits, Basel: Birkhäuser/Springer Basel AG, 2012.
[2] A. Sandoval-Romero and A. Hernández-Garduño, "Domain Coloring on the Riemann Sphere," The Mathematica Journal, 2015. doi:10.3888/tmj.17-9.

External Links

Phase of a Complex Polynomial

Permanent Citation

Matthew Romney
​
​"Domain Coloring for Common Functions in Complex Analysis"​
​http://demonstrations.wolfram.com/DomainColoringForCommonFunctionsInComplexAnalysis/​
​Wolfram Demonstrations Project​
​Published: March 2, 2016