Algebraic functions

Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.

• Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer.
  • Constant function: polynomial of degree zero, graph is a horizontal straight line
  • Linear function: First degree polynomial, graph is a straight line.
  • Quadratic function: Second degree polynomial, graph is a parabola.
  • Cubic function: Third degree polynomial.
  • Quartic function: Fourth degree polynomial.
  • Quintic function: Fifth degree polynomial.
  • Sextic function: Sixth degree polynomial.
• Rational functions: A ratio of two polynomials.
• nth root
  • Square root: Yields a number whose square is the given one.
  • Cube root: Yields a number whose cube is the given one.

Elementary transcendental functions

Transcendental functions are functions that are not algebraic.

• Exponential function: raises a fixed number to a variable power.
• Hyperbolic functions: formally similar to the trigonometric functions.
  • Inverse hyperbolic functions: inverses of the hyperbolic functions, analogous to the inverse circular functions.

• Logarithms: the inverses of exponential functions; useful to solve equations involving exponentials.
  • Natural logarithm
  • Common logarithm
  • Binary logarithm
• Power functions: raise a variable number to a fixed power; also known as Allometric functions; note: if the power is a rational number it is not strictly a transcendental function.
• Periodic functions
  • Trigonometric functions: sine, cosine, tangent, cotangent, secant, cosecant, exsecant, excosecant, versine, coversine, vercosine, covercosine, haversine, hacoversine, havercosine, hacovercosine, Inverse trigonometric functions etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.

Piecewise special functions

Arithmetic functions

• Sigma function: Sums of powers of divisors of a given natural number.
• Euler's totient function: Number of numbers coprime to (and not bigger than) a given one.
• Prime-counting function: Number of primes less than or equal to a given number.
• Partition function: Order-independent count of ways to write a given positive integer as a sum of positive integers.
• Möbius μ function: Sum of the nth primitive roots of unity, it depends on the prime factorization of n.
• Prime omega functions
• Chebyshev functions
• Liouville function, λ(n) = (–1)^Ω(n)
• Von Mangoldt function, Λ(n) = log p if n is a positive power of the prime p
• Carmichael function

Antiderivatives of elementary functions

• Logarithmic integral function: Integral of the reciprocal of the logarithm, important in the prime number theorem.
• Exponential integral
• Trigonometric integral: Including Sine Integral and Cosine Integral
• Inverse tangent integral
• Error function: An integral important for normal random variables.
  • Fresnel integral: related to the error function; used in optics.
  • Dawson function: occurs in probability.
  • Faddeeva function

Gamma and related functions

• Gamma function: A generalization of the factorial function.
• Barnes G-function
• Beta function: Corresponding binomial coefficient analogue.
• Digamma function, Polygamma function
• Incomplete beta function
• Incomplete gamma function
• K-function
• Multivariate gamma function: A generalization of the Gamma function useful in multivariate statistics.
• Student's t-distribution
• Pi function ${\displaystyle \Pi (z)=z\Gamma (z)=(z)!}$

Elliptic and related functions

Bessel and related functions

Riemann zeta and related functions

Hypergeometric and related functions

• Hypergeometric functions: Versatile family of power series.
• Confluent hypergeometric function
• Associated Legendre functions
• Meijer G-function
• Fox H-function

Iterated exponential and related functions

• Hyper operators
• Iterated logarithm
• Pentation
• Super-logarithms
• Super-roots
• Tetration

Other standard special functions

• Lambert W function: Inverse of f(w) = w exp(w).
• Lamé function
• Mathieu function
• Mittag-Leffler function
• Painlevé transcendents
• Parabolic cylinder function
• Arithmetic–geometric mean

Miscellaneous functions

• Ackermann function: in the theory of computation, a computable function that is not primitive recursive.
• Dirac delta function: everywhere zero except for x = 0; total integral is 1. Not a function but a distribution, but sometimes informally referred to as a function, particularly by physicists and engineers.
• Dirichlet function: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous.
• Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function.
• Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
• Minkowski's question mark function: Derivatives vanish on the rationals.
• Weierstrass function: is an example of continuous function that is nowhere differentiable