Graphest User Guide

Graphing a Circle

You can express a unit circle centered at origin in many ways:

Implicit relation
```
x^2 + y^2 = 1
```
Polar equation
```
r = 1
```

Combinations of explicit relations

y = sqrt(1 − x^2) ∨ y = −sqrt(1 − x^2)

x = sqrt(1 − y^2) ∨ x = −sqrt(1 − y^2)

Parametric equation
```
x = cos(t) ∧ y = sin(t)
```
Complex function
```
x + i y = exp(i t)
```

In Graphest, each of the above formulae is called a relation.

Entering a Relation

A relation must contain at least one of =, <, ≤, >, or ≥, as in y = sin(x) or sin(x) < cos(y).
- It can also be a compound formula, such as y = sin(x) ∨ y = cos(x).
Enter the product of x and y as x y, instead of xy.
- You don’t need a space around parentheses or in a expression like 2x.
Enter the sine of \(x\) as sin(x), instead of sinx or sin x.
Use parentheses ( ) to group a part of a relation, as in 1 / (x + 3).
- Other kinds of brackets such as [ ] or { } cannot be used for this purpose.
You can use x, y, r and θ as coordinate variables.
- Each point in Cartesian coordinates has infinitely many counterparts in polar coordinates. For example, \((x, y) = (1, 0)\) is identical to \((r, θ) = (1, 0), (1, ±2π), (1, ±4π), …; (−1, ±π), (−1, ±3π), …\)
- You can restrict the range of r or θ, as in r = θ ∧ 0 ≤ θ < 2π.

You can use t as a real parameter. Example:

x = t cos(t) ∧ y = t sin(t) ∧ 0 ≤ t < 2π

You can use m and n as independent integer parameters. Example:

(x − 0.3m)^2 + (y − 0.3n)^2 < 0.1^2 ∧ |n| ≤ 5 ∧ mod(m + n, 3) = 0

Reading a Graph

Each pixel of a graph is painted based on whether it contains a solution of the relation or not.

Pixel Color	Meaning
Opaque	The pixel contains a solution.
Translucent	The pixel may or may not contain a solution.
Transparent	The pixel does not contain a solution.

Pixel Color

Meaning

Opaque

The pixel contains a solution.

Translucent

The pixel may or may not contain a solution.

Transparent

The pixel does not contain a solution.

Built-in Definitions

Constants

Input Interpreted as Notes Links

Input	Interpreted as	Notes	Links
Real Numbers
`123` `123.5` `.5`	123 123.5 0.5
`e`	\(e ≈ 2.71828\)	The base of natural logarithms.	MW WFS
`pi` or `π`	\(π ≈ 3.14159\)		MW WFS
`gamma` or `γ`	\(γ ≈ 0.577216\)	The Euler–Mascheroni constant.	MW WFS
Complex Numbers
`i`	\(i = \sqrt{−1}\)	The imaginary unit.	MW WFS

Real Numbers

123
123.5
.5

123
123.5
0.5

e

\(e ≈ 2.71828\)

The base of natural logarithms.

MW WFS

pi or π

\(π ≈ 3.14159\)

MW WFS

gamma or γ

\(γ ≈ 0.577216\)

The Euler–Mascheroni constant.

MW WFS

Complex Numbers

i

\(i = \sqrt{−1}\)

The imaginary unit.

MW WFS

Variables

Input Interpreted as Notes

Input	Interpreted as	Notes
Cartesian Coordinate System
`x`	\(x\)	The horizontal coordinate.
`y`	\(y\)	The vertical coordinate.
Polar Coordinate System
`r`	\(r\)	The radial coordinate.
`theta` or `θ`	\(θ\)	The angular coordinate.
Integer Parameters
`m`	\(m\)	A parameter that spans all integers.
`n`	\(n\)	A parameter that spans all integers.
Real Parameter
`t`	\(t\)	A parameter that spans all real numbers.

Cartesian Coordinate System

x

\(x\)

The horizontal coordinate.

y

\(y\)

The vertical coordinate.

Polar Coordinate System

r

\(r\)

The radial coordinate.

theta or θ

\(θ\)

The angular coordinate.

Integer Parameters

m

\(m\)

A parameter that spans all integers.

n

\(n\)

A parameter that spans all integers.

Real Parameter

t

\(t\)

A parameter that spans all real numbers.

A relation \(R\) that contains parameters is interpreted as \(∃m, n ∈ ℤ, ∃t ∈ ℝ : R\).

Real Functions

Arithmetic

Input Interpreted as Notes

Input	Interpreted as	Notes
`−x`	\(−x\)
`x + y`	\(x + y\)
`x − y`	\(x − y\)
`x y` or `x * y`	\(xy = x ⋅ y\)
`x / y`	\(\dfrac{x}{y}\)	Undefined for \(y = 0\).

−x

\(−x\)

x + y

\(x + y\)

x − y

\(x − y\)

x y or x * y

\(xy = x ⋅ y\)

x / y

\(\dfrac{x}{y}\)

Undefined for \(y = 0\).

Number Parts

Input Interpreted as Notes Links

Input	Interpreted as	Notes	Links
`abs(x)` or `\|x\|`	\(\|x\|\)	The absolute value of \(x\).	MW WFS
`sign(x)` or `sgn(x)`	\(\sgn x = \begin{cases} -1 & \if x < 0, \\ 0 & \if x = 0, \\ 1 & \if x > 0 \end{cases}\)	The sign function.	MW WFS

abs(x) or |x|

\(|x|\)

The absolute value of \(x\).

MW WFS

sign(x) or sgn(x)

\(\sgn x = \begin{cases} -1 & \if x < 0, \\ 0 & \if x = 0, \\ 1 & \if x > 0 \end{cases}\)

The sign function.

MW WFS

Exponentiation and Logarithm

Input Interpreted as Notes Links

Input	Interpreted as	Notes	Links
`sqrt(x)`	\(\sqrt{x} = x^{1/2}\)	Undefined for \(x < 0\).	MW WFS
`x^y`	\(x^y\)	`x^y^z` is equivalent to `x^(y^z)`.	MW WFS
`x^^y`	\(x^y\) with some extension	The cube root of \(x\) can be entered as `x^^(1/3)`. `x^^y^^z` is equivalent to `x^^(y^^z)`.
`exp(x)`	\(\exp x = e^x\)		MW WFS
`ln(x)`	\(\ln x = \log_e x\)	Undefined for \(x ≤ 0\).	MW WFS
`log(b, x)`	\(\log_b x = \dfrac{\ln x}{\ln b}\)	Undefined for \(x ≤ 0\), \(b ≤ 0\), and \(b = 1\).	MW WFS

sqrt(x)

\(\sqrt{x} = x^{1/2}\)

Undefined for \(x < 0\).

MW WFS

x^y

\(x^y\)

x^y^z is equivalent to x^(y^z).

MW WFS

x^^y

\(x^y\) with some extension

The cube root of \(x\) can be entered as x^^(1/3).
x^^y^^z is equivalent to x^^(y^^z).

exp(x)

\(\exp x = e^x\)

MW WFS

ln(x)

\(\ln x = \log_e x\)

Undefined for \(x ≤ 0\).

MW WFS

log(b, x)

\(\log_b x = \dfrac{\ln x}{\ln b}\)

Undefined for \(x ≤ 0\), \(b ≤ 0\), and \(b = 1\).

MW WFS

Comparison of `x^y` and `x^^y`

For \(x ≥ 0\), both x^y and x^^y gives the same value, \(x^y\). For \(x < 0\), x^y is only defined for integer exponents, while x^^y is also defined for rational number exponents with odd denominators. The exact definition of these operators are as follows.

For an integer \(n\), both x^n and x^^n gives:

\(x^n = \begin{cases} \overbrace{x × ⋯ × x}^{(n \text{ copies})} & \if n > 0, \\ 1 & \if n = 0 ∧ x ≠ 0, \\ 1 / x^{-n} & \if n < 0. \end{cases}\)

\(0^0\) is left undefined.
For a non-integer \(y\), they can give different results:
- x^y gives the principal value of \(e^{y \ln x}\) or its limit as \(x → 0\):
  
  \(x^y = \begin{cases} 0 & \if x = 0 ∧ y > 0, \\ e^{y \ln x} & \otherwise. \end{cases}\)
- If \(y\) is a rational number \(p/q\) with odd \(q\), assuming \(p\) and \(q\) \((> 0)\) be coprime integers, x^^y gives \((\sqrt[q]{x})^p\), where \(\sqrt[q]{x}\) is the real-valued \(q\)th root of \(x\).
- Otherwise, x^^y gives the same result as x^y described above.

Trigonometry

Input Interpreted as Notes Links

Input	Interpreted as	Notes	Links
Trigonometric Functions
`sin(x)`	\(\sin x\)		MW WFS
`cos(x)`	\(\cos x\)		MW WFS
`tan(x)`	\(\tan x\)		MW WFS
Inverse Trigonometric Functions
`asin(x)`	\(\sin^{−1} x\)	The range is \([−π/2, π/2]\).	MW WFS
`acos(x)`	\(\cos^{−1} x\)	The range is \([0, π]\).	MW WFS
`atan(x)`	\(\tan^{−1} x\)	The range is \((−π/2, π/2)\).	MW WFS
`atan2(y, x)`		The two-argument arctangent. Undefined for \((x, y) = (0, 0)\). The range is \((−π, π]\).	MW WFS
Hyperbolic Functions
`sinh(x)`	\(\sinh x\)		MW WFS
`cosh(x)`	\(\cosh x\)		MW WFS
`tanh(x)`	\(\tanh x\)		MW WFS
Inverse Hyperbolic Functions
`asinh(x)`	\(\sinh^{−1} x\)		MW WFS
`acosh(x)`	\(\cosh^{−1} x\)		MW WFS
`atanh(x)`	\(\tanh^{−1} x\)		MW WFS

Trigonometric Functions

sin(x)

\(\sin x\)

MW WFS

cos(x)

\(\cos x\)

MW WFS

tan(x)

\(\tan x\)

MW WFS

Inverse Trigonometric Functions

asin(x)

\(\sin^{−1} x\)

The range is \([−π/2, π/2]\).

MW WFS

acos(x)

\(\cos^{−1} x\)

The range is \([0, π]\).

MW WFS

atan(x)

\(\tan^{−1} x\)

The range is \((−π/2, π/2)\).

MW WFS

atan2(y, x)

The two-argument arctangent.
Undefined for \((x, y) = (0, 0)\).
The range is \((−π, π]\).

MW WFS

Hyperbolic Functions

sinh(x)

\(\sinh x\)

MW WFS

cosh(x)

\(\cosh x\)

MW WFS

tanh(x)

\(\tanh x\)

MW WFS

Inverse Hyperbolic Functions

asinh(x)

\(\sinh^{−1} x\)

MW WFS

acosh(x)

\(\cosh^{−1} x\)

MW WFS

atanh(x)

\(\tanh^{−1} x\)

MW WFS

Divisibility

Input Interpreted as Notes Links

Input	Interpreted as	Notes	Links
`mod(x, y)`	\(x \bmod y = x - y \left⌊ \dfrac{x}{y} \right⌋\)	The remainder of \(x/y\) (modulo operation). Undefined for \(y = 0\). The range for a fixed \(y\) is \(\begin{cases} (y, 0] & \if y < 0, \\ [0, y) & \if y > 0. \end{cases}\)	MW WFS
`gcd(x₁, …, x_n)`	\(\gcd \set{x_1, …, x_n}\)	The greatest common divisor of the numbers in the set \(\set{x_1, …, x_n}\). \(\gcd \set{x, 0}\) is defined to be \(\|x\|\) for any rational number \(x\). Undefined if any of the numbers is irrational.	MW WFS
`lcm(x₁, …, x_n)`	\(\lcm \set{x_1, …, x_n}\)	The least common multiple of the numbers in the set \(\set{x_1, …, x_n}\). \(\lcm \set{x, 0}\) is defined to be 0 for any rational number \(x\). Undefined if any of the numbers is irrational.	MW WFS

mod(x, y)

\(x \bmod y = x - y \left⌊ \dfrac{x}{y} \right⌋\)

The remainder of \(x/y\) (modulo operation).
Undefined for \(y = 0\).
The range for a fixed \(y\) is \(\begin{cases} (y, 0] & \if y < 0, \\ [0, y) & \if y > 0. \end{cases}\)

MW WFS

gcd(x₁, …, x_n)

\(\gcd \set{x_1, …, x_n}\)

The greatest common divisor of the numbers in the set \(\set{x_1, …, x_n}\).
\(\gcd \set{x, 0}\) is defined to be \(|x|\) for any rational number \(x\).
Undefined if any of the numbers is irrational.

MW WFS

lcm(x₁, …, x_n)

\(\lcm \set{x_1, …, x_n}\)

The least common multiple of the numbers in the set \(\set{x_1, …, x_n}\).
\(\lcm \set{x, 0}\) is defined to be 0 for any rational number \(x\).
Undefined if any of the numbers is irrational.

MW WFS

Ordering

Input Interpreted as Notes Links

Input	Interpreted as	Notes	Links
`max(x₁, …, x_n)` `min(x₁, …, x_n)`	\(\max \set{x_1, …, x_n}\) \(\min \set{x_1, …, x_n}\)	The largest and the smallest elements of the set \(\set{x_1, …, x_n}\), respectively.	MW WFS MW WFS
`rankedMax([x₁, …, x_n], k)` `rankedMin([x₁, …, x_n], k)`		The \(k\)th largest and the \(k\)th smallest elements of the list \(\list{x_1, …, x_n}\), respectively.

max(x₁, …, x_n)
min(x₁, …, x_n)

\(\max \set{x_1, …, x_n}\)
\(\min \set{x_1, …, x_n}\)

The largest and the smallest elements of the set \(\set{x_1, …, x_n}\), respectively.

MW WFS
MW WFS

rankedMax([x₁, …, x_n], k) rankedMin([x₁, …, x_n], k)

The \(k\)th largest and the \(k\)th smallest elements of the list \(\list{x_1, …, x_n}\), respectively.

Rounding

Input Interpreted as Notes Links

Input	Interpreted as	Notes	Links
`floor(x)` or `⌊x⌋`	\(⌊x⌋\)	The floor function.	MW WFS
`ceil(x)` or `⌈x⌉`	\(⌈x⌉\)	The ceiling function.	MW WFS

floor(x) or ⌊x⌋

\(⌊x⌋\)

The floor function.

MW WFS

ceil(x) or ⌈x⌉

\(⌈x⌉\)

The ceiling function.

MW WFS

Complex Functions

To use a complex function when all arguments are real, add a dummy imaginary part to one of them as x + 0i.

Arithmetic

Input Interpreted as Notes

Input	Interpreted as	Notes
`−z`	\(−z\)
`z + w`	\(z + w\)
`z − w`	\(z − w\)
`z w` or `z * w`	\(zw = z ⋅ w\)
`z / w`	\(\dfrac{z}{w}\)	Undefined for \(w = 0\).

−z

\(−z\)

z + w

\(z + w\)

z − w

\(z − w\)

z w or z * w

\(zw = z ⋅ w\)

z / w

\(\dfrac{z}{w}\)

Undefined for \(w = 0\).

Number Parts

Input Interpreted as Notes Links

Input	Interpreted as	Notes	Links
Real-Valued
`Re(z)`	\(\Re z\)	The real part of \(z\).	MW WFS
`Im(z)`	\(\Im z\)	The imaginary part of \(z\).	MW WFS
`abs(z)` or `\|z\|`	\(\|z\|\)	The absolute value of \(z\).	MW WFS
`arg(z)`	\(\arg z\)	The argument of \(z\). Undefined for \(z = 0\).	MW WFS
Complex-Valued
`~z`	\(\bar z\)	The complex conjugate of \(z\).	MW WFS
`sgn(z)` or `sign(z)`	\(\sgn z = \begin{cases} 0 & \if z = 0, \\ \dfrac{z}{\|z\|} & \if z ≠ 0 \end{cases}\)	The complex sign of \(z\).	MW WFS

Real-Valued

Re(z)

\(\Re z\)

The real part of \(z\).

MW WFS

Im(z)

\(\Im z\)

The imaginary part of \(z\).

MW WFS

abs(z) or |z|

\(|z|\)

The absolute value of \(z\).

MW WFS

arg(z)

\(\arg z\)

The argument of \(z\).
Undefined for \(z = 0\).

MW WFS

Complex-Valued

~z

\(\bar z\)

The complex conjugate of \(z\).

MW WFS

sgn(z) or sign(z)

\(\sgn z = \begin{cases} 0 & \if z = 0, \\ \dfrac{z}{|z|} & \if z ≠ 0 \end{cases}\)

The complex sign of \(z\).

MW WFS

Exponentiation and Logarithm

Input Interpreted as Notes Links

Input	Interpreted as	Notes	Links
`sqrt(z)`	\(\sqrt{z} = z^{1/2}\)	Branch cuts: \((−∞, 0)\), continuous from above.	MW WFS
`z^w`	\(z^w = \begin{cases} 0 & \if z = 0 ∧ \Re w > 0, \\ e^{w \ln z} & \if z ≠ 0 \end{cases}\)	Undefined for \((z, w)\) if \(z = 0 ∧ \Re w ≤ 0\). Branch cuts for a fixed non-integer \(w\): \((−∞, 0)\), continuous from above.	MW WFS
`exp(z)`	\(\exp z = e^z\)		MW WFS
`ln(z)`	\(\ln z = \log_e z\)	Undefined for \(z = 0\). Branch cuts: (−∞, 0), continuous from above.	MW WFS
`log(b, z)`	\(\log_b z = \dfrac{\ln z}{\ln b}\)	Undefined for \(z = 0\), \(b = 0\), and \(b = 1\). Branch cuts for a fixed \(b\): \((−∞, 0)\), continuous from above. Branch cuts for a fixed \(z\): \((−∞, 0)\), continuous from above.	MW WFS

sqrt(z)

\(\sqrt{z} = z^{1/2}\)

Branch cuts: \((−∞, 0)\), continuous from above.

MW WFS

z^w

\(z^w = \begin{cases} 0 & \if z = 0 ∧ \Re w > 0, \\ e^{w \ln z} & \if z ≠ 0 \end{cases}\)

Undefined for \((z, w)\) if \(z = 0 ∧ \Re w ≤ 0\).
Branch cuts for a fixed non-integer \(w\): \((−∞, 0)\), continuous from above.

MW WFS

exp(z)

\(\exp z = e^z\)

MW WFS

ln(z)

\(\ln z = \log_e z\)

Undefined for \(z = 0\).
Branch cuts: (−∞, 0), continuous from above.

MW WFS

log(b, z)

\(\log_b z = \dfrac{\ln z}{\ln b}\)

Undefined for \(z = 0\), \(b = 0\), and \(b = 1\).
Branch cuts for a fixed \(b\): \((−∞, 0)\), continuous from above.
Branch cuts for a fixed \(z\): \((−∞, 0)\), continuous from above.

MW WFS

Trigonometry

Input Interpreted as Notes Links

Input	Interpreted as	Notes	Links
Trigonometric Functions
`sin(z)`	\(\sin z\)		MW WFS
`cos(z)`	\(\cos z\)		MW WFS
`tan(z)`	\(\tan z\)		MW WFS
Inverse Trigonometric Functions
`asin(z)`	\(\sin^{−1} z\)	Branch cuts: \((−∞, −1)\), continuous from above; \((1, ∞)\), continuous from below.	MW WFS
`acos(z)`	\(\cos^{−1} z\)	Branch cuts: \((−∞, −1)\), continuous from above; \((1, ∞)\), continuous from below.	MW WFS
`atan(z)`	\(\tan^{−1} z\)	Branch cuts: \((−i∞, −i)\), continuous from the left; \((i, i∞)\), continuous from the right.	MW WFS
Hyperbolic Functions
`sinh(z)`	\(\sinh z\)		MW WFS
`cosh(z)`	\(\cosh z\)		MW WFS
`tanh(z)`	\(\tanh z\)		MW WFS
Inverse Hyperbolic Functions
`asinh(z)`	\(\sinh^{−1} z\)	Branch cuts: \((−i∞, −i)\), continuous from the left; \((i, i∞)\), continuous from the right.	MW WFS
`acosh(z)`	\(\cosh^{−1} z\)	Branch cuts: \((−∞, 1)\), continuous from above.	MW WFS
`atanh(z)`	\(\tanh^{−1} z\)	Branch cuts: \((−∞, −1)\), continuous from above; \((1, ∞)\), continuous from below.	MW WFS

Trigonometric Functions

sin(z)

\(\sin z\)

MW WFS

cos(z)

\(\cos z\)

MW WFS

tan(z)

\(\tan z\)

MW WFS

Inverse Trigonometric Functions

asin(z)

\(\sin^{−1} z\)

Branch cuts: \((−∞, −1)\), continuous from above; \((1, ∞)\), continuous from below.

MW WFS

acos(z)

\(\cos^{−1} z\)

Branch cuts: \((−∞, −1)\), continuous from above; \((1, ∞)\), continuous from below.

MW WFS

atan(z)

\(\tan^{−1} z\)

Branch cuts: \((−i∞, −i)\), continuous from the left; \((i, i∞)\), continuous from the right.

MW WFS

Hyperbolic Functions

sinh(z)

\(\sinh z\)

MW WFS

cosh(z)

\(\cosh z\)

MW WFS

tanh(z)

\(\tanh z\)

MW WFS

Inverse Hyperbolic Functions

asinh(z)

\(\sinh^{−1} z\)

Branch cuts: \((−i∞, −i)\), continuous from the left; \((i, i∞)\), continuous from the right.

MW WFS

acosh(z)

\(\cosh^{−1} z\)

Branch cuts: \((−∞, 1)\), continuous from above.

MW WFS

atanh(z)

\(\tanh^{−1} z\)

Branch cuts: \((−∞, −1)\), continuous from above; \((1, ∞)\), continuous from below.

MW WFS

Formulae

Equations and Inequalities

Input Interpreted as Notes

Input	Interpreted as	Notes
`x = y`	\(x = y\)	\(x\) and/or \(y\) can be either real or complex.
`x < y`	\(x < y\)	\(x\) and \(y\) must be real.
`x <= y` or `x ≤ y`	\(x ≤ y\)	\(x\) and \(y\) must be real.
`x > y`	\(x > y\)	\(x\) and \(y\) must be real.
`x >= y` or `x ≥ y`	\(x ≥ y\)	\(x\) and \(y\) must be real.

x = y

\(x = y\)

\(x\) and/or \(y\) can be either real or complex.

x < y

\(x < y\)

\(x\) and \(y\) must be real.

x <= y or x ≤ y

\(x ≤ y\)

\(x\) and \(y\) must be real.

x > y

\(x > y\)

\(x\) and \(y\) must be real.

x >= y or x ≥ y

\(x ≥ y\)

\(x\) and \(y\) must be real.

These operators can be chained. For example, 0 ≤ θ < 2π is equivalent to 0 ≤ θ ∧ θ < 2π.

Logical Connectives

Input Interpreted as Notes

Input	Interpreted as	Notes
`P && Q` or `P ∧ Q`	\(P ∧ Q\)	Logical conjunction (logical AND).
`P \|\| Q` or `P ∨ Q`	\(P ∨ Q\)	Logical disjunction (logical OR).
`!P` or `¬P`	\(¬P\)	Negation (logical NOT).

P && Q or P ∧ Q

\(P ∧ Q\)

Logical conjunction (logical AND).

P || Q or P ∨ Q

\(P ∨ Q\)

Logical disjunction (logical OR).

!P or ¬P

\(¬P\)

Negation (logical NOT).

\(P\) and \(Q\) must be formulae.

Conditional Expressions

Input Interpreted as Notes

Input	Interpreted as	Notes
`if(P, x, y)`	\(\begin{cases} x & \if P, \\ y & \otherwise \end{cases}\)	\(P\) must be a formula. \(x\) and/or \(y\) can be either real or complex.

if(P, x, y)

\(\begin{cases} x & \if P, \\ y & \otherwise \end{cases}\)

\(P\) must be a formula.
\(x\) and/or \(y\) can be either real or complex.

Special Functions

All functions accept only real inputs at the moment.

Input Interpreted as Notes Links

Input	Interpreted as	Notes	Links
`W(x)` `W(k, x)`	\(W(x) = W_0(x)\) \(W_k(x)\)	The Lambert W function. \(k\) must be either 0 or −1.	MW WFS MW WFS
`Gamma(x)` or `Γ(x)`	\(Γ(x)\)	The gamma function.	MW WFS
`Gamma(a, x)` or `Γ(a, x)`	\(Γ(a, x)\)	The upper incomplete gamma function. \(a\) must be an exact number^[1].	MW WFS
`lnGamma(x)` or `lnΓ(x)`	\(\ln Γ(x)\)	The log-gamma function.	MW WFS
`psi(x)` or `ψ(x)`	\(ψ(x)\)	The digamma function.	MW WFS
`erf(x)`	\(\operatorname{erf}(x)\)	The error function.	MW WFS
`erfc(x)`	\(\operatorname{erfc}(x)\)	The complementary error function.	MW WFS
`erfi(x)`	\(\operatorname{erfi}(x)\)	The imaginary error function.	MW WFS
`erfinv(x)`	\(\operatorname{erf}^{-1}(x)\)	The inverse error function.	MW WFS
`erfcinv(x)`	\(\operatorname{erfc}^{-1}(x)\)	The inverse complementary error function.	MW WFS
`Ei(x)`	\(\operatorname{Ei}(x)\)	The exponential integral.	MW WFS
`li(x)`	\(\operatorname{li}(x)\)	The logarithmic integral.	MW WFS
`Si(x)`	\(\operatorname{Si}(x)\)	The sine integral.	MW WFS
`Ci(x)`	\(\operatorname{Ci}(x)\)	The cosine integral.	MW WFS
`Shi(x)`	\(\operatorname{Shi}(x)\)	The hyperbolic sine integral.	MW WFS
`Chi(x)`	\(\operatorname{Chi}(x)\)	The hyperbolic cosine integral.	MW WFS
`S(x)` `C(x)`	\(S(x)\) \(C(x)\)	The Fresnel integrals.	MW WFS MW WFS
`J(n, x)` `Y(n, x)`	\(J_n(x)\) \(Y_n(x)\)	The Bessel functions. \(n\) must be an integer or a half-integer.	MW WFS MW WFS
`I(n, x)` `K(n, x)`	\(I_n(x)\) \(K_n(x)\)	The modified Bessel functions. \(n\) must be an integer or a half-integer.	MW WFS MW WFS
`Ai(x)` `Bi(x)` `Ai'(x)` `Bi'(x)`	\(\operatorname{Ai}(x)\) \(\operatorname{Bi}(x)\) \(\operatorname{Ai'}(x)\) \(\operatorname{Bi'}(x)\)	The Airy functions and their derivatives.	MW WFS MW WFS MW WFS MW WFS
`sinc(x)`	\(\operatorname{sinc}(x) = \begin{cases} 1 & \if x = 0, \\ \dfrac{\sin x}{x} & \if x ≠ 0 \end{cases}\)	The (unnormalized) sinc function.	MW WFS
`K(m)`	\(K(m)\)	The complete elliptic integral of the first kind.	MW WFS
`E(m)`	\(E(m)\)	The complete elliptic integral of the second kind.	MW WFS
`zeta(s)` or `ζ(s)`	\(ζ(s)\)	The Riemann zeta function.	MW WFS