# zgraph help

This is the documentation for zgraph, an interactive program for domain coloring of complex functions.

This program was created in December 2016 to extend the complex polynomial toy.

## 1. Basic usage

In the text box in the upper left corner of the window, enter an expression that is a function of the variable $z$ using standard computer math notation, and then press Enter on your keyboard. The program will then interpret the function, and, if there are no errors, update the left and right views.

The left view is a domain coloring of the complex plane, and the right view is a domain coloring of the complex plane mapped onto the Riemann sphere (the complex plane closed up with an additional point at infinity, forming a sphere). In this particular implementation, the color represents the argument of the value of the complex function at that particular point, and the gray contours indicate curves where the function has constant magnitude. The contours are logarithmically spaced, meaning the magnitude of the function changes by a constant multiplicative factor between consecutive contours.

Click and drag
Rotate the Riemann sphere.
Left arrow and Right arrow
Adjust the animation speed of the argument coloring. Right arrow makes the rainbow rotate counterclockwise around zeros and clockwise around poles.
Down arrow and Up arrow
Adjust the animation speed of the contour lines. Up arrow makes the contours move away from zeros and towards poles.

Known quirk: If these commands do not work, click on the graph to ensure the focus is not on the expression editor or the Load/Save buttons.

Click the Load button in the upper right to see examples of functions that people have saved. If you find an interesting function, click the Save button to upload the function for others to see. (The author reserves the right to prune this list of functions!)

## 3. Families of functions

The complex function may additionally be functions of the real-valued parameters $s$ and $t$.

Shift + Left arrow and Shift + Right arrow
Adjust the rate that the $t$ variable changes as a function of time.
Shift + Down arrow and Shift + Up arrow
Adjust the rate that the $s$ variable changes as a function of time.

## 4. Language reference

The program accepts a number of standard complex functions and operators. There is also some non-standard notation for function composition. In the following, let a and b represent valid expressions. The usual order of operations holds.

z
The variable $z$ that expressions are a function of.
s and t
Real-valued variables $s$ and $t$; see Families of functions.
a + b
a - b
Subtraction.
-a
Negation.
+a
Unary plus (no effect).
a b or a * b
Multiplication.
a / b
Division.
a ** b or a ^ b
Exponentiation. The implementation is ${e}^{b\mathrm{ln}a}$, where the logarithm has a branch cut on the negative real ray.
(a)
Parenthetical grouping.
exp(a)
The analytic function ${e}^{a}$.
sin(a)
The analytic function $\mathrm{sin}a$.
cos(a)
The analytic function $\mathrm{cos}a$.
tan(a)
The function $\frac{\mathrm{sin}a}{\mathrm{cos}a}$.
ln(a)
The function $\mathrm{ln}a$, with a branch cut along the negative real ray.
conj(a)
The complex conjugate $\overline{a}$.
re(a)
The real part of $a$.
im(a)
The imaginary part of $a$.
abs(a)
The absolute value $|a|$, which for complex numbers we define to be the magnitude of $a$.
arg(a)
The argument of $a$, as a number between $-\pi$ and $\pi$.
i
The imaginary unit $i=\sqrt{-1}$, which we imagine to be ninety degrees counterclockwise from $1$.
pi
The circular circumference-to-diameter ratio $\pi$.
e
The natural exponential base $e$.
a sequence of digits 09 with at most one decimal point
Interpreted as the corresponding floating-point number.
a then b
If $f\left(z\right)=a$ and $g\left(z\right)=b$, this expression represents the composition $g\circ f$. In particular, the value of z in b is the value of a.
a iterate n
If $f\left(z\right)=a$, then this represents ${f}^{n}\left(z\right)=f\left(f\left(\cdots f\left(z\right)\cdots \right)\right)$. It must be the case that n is a positive integer constant. This has higher precedence than then.
a iterateprod n or a iteratesum n
This is like iterate, but the result is the product or sum of the iterates rather than the last iterate.
c
The original value of z, even in the face of function composition. Useful for iterate.

Formal grammar:

compose = iterate {"then" iterate}.
iterate = sum {("iterate" | "iterateprod" | "iteratesum") "num"}.
sum = ["+"|"-"] prod {("+"|"-") prod}.
prod = exp {["*"|"/"] exp}.
exp = term {("^"|"**") ["+"|"-"] term}.
term = fun | "(" compose ")" | "num" | "z" | "s" | "t" | "e" | "i" | "pi" | "const".
fun = elem "(" compose {"," compose} ")".
elem = "exp" | "sin" | "cos" | "tan" | "ln" | "conj" | "re" | "im" | "abs" | "arg".


## 5. Implementation details

Expressions are compiled to the OpenGL Shader Language (GLSL), and this program is executed by GPU. The process begins by converting the complex functions to a list of real-valued functions of two real variables, and these are somewhat optimized using global value numbering. The program logs the resulting shader program to the console in your browser’s developer tools, for the curious.

A consequence of this is that the accuracy of the graph is up to the graphics driver and its particular implementation of mediump floating-point numbers for your particular graphics card. For instance, floating-point numbers with too great of a magnitude might yield any number of colors, but in practice it seems most drivers default to black.

It would be nice to have the circle from the complex polynomials toy, however this would involve needing to sample the points on the circle properly. There, I cheated by adding enough points that most people wouldn’t notice this limitation.