Special function

Elementary functions

• Polynomials: can be generated by addition and multiplication alone.
• Square root: yields a number whose square is the given one.
• Exponential function: raise a fixed number to a variable power.
• Logarithm: the inverses of exponential functions; useful to solve equations involving exponentials.
• Trigonometric functions: sine, cosine, etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.
• Hyperbolic functions: formally similar to the trigonometric functions.
• Absolute value: drops the sign of a given number.
• Floor function: largest integer ≤ a given number.
• Identity function: maps a given element to itself
• Constant function: a fixed value regardless of arguments, if any.

Special functions

• Gamma function: A generalization of the factorial function.
  • Beta function: Corresponding binomial coefficient analogue.
  • Digamma function, Polygamma function
• Riemann zeta function: A special case of Dirichlet series.
  • Dirichlet eta function is an allied function.
• Elliptic integrals: Arising from the path length of ellipses; important in many applications.
• Elliptic functions: The inverses of elliptic integrals; used to model double-periodic phenomena. Particular types are Weierstrass's elliptic functions and Jacobi's elliptic functions.
• Hypergeometric functions: Versatile family of power series
• Legendre function: From the theory of spherical harmonics
• Bessel functions: Defined by a differential equation; useful in astronomy, electromagnetism, and mechanics.
• Logarithmic integral: Integral of the reciprocal of the logarithm, important in the prime number theorem.
• Lambert's W function: inverse of f(w) = w exp(w).
• Error function: an integral important for normal random variables.

Number theoretic functions

• Sigma function: Sums of powers of factors of a given natural number.
• Euler's phi function: Number of numbers relatively prime to 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.

Other standard special functions

• Carlson symmetric form
• Clausen function
• Dawson function
• Dedekind eta function
• Incomplete gamma function

Miscellaneous

• Ackermann function: in the theory of computation, a recursive 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.
• Heaviside step function: 0 for negative arguments and 1 for positive arguments. The integral of the Dirac delta distribution.