Special function/Catalogs/Catalog

Special functions are mathematical functions that turn up so often that they have been named. This page lists the most common special functions by category, along with some of the properties that are important to functions belonging to each category. It must be stressed that there is no single way to categorize functions; any practical classification will contain overlapping categories.

Algebraic functions

• Polynomials
  • Linear functions
  • Quadratic functions
  • Cubic functions
• Rational functions
• Power functions
  • Square root
  • Cube root

Complex parts

• Absolute value
• Real part
• Imaginary part
• Complex conjugate
• Complex argument
• Sign function

Elementary transcendental functions

Name	Notation
Exponential function	$\exp(x)$, $e^{x}$
Natural logarithm	$\log(x)$, $\ln(x)$

Trigonometric functions:

Name	Notation	Triangle formula	Exponential formula
Sine	$\sin(x)$	Opposite / Hypotenuse	$(e^{ix}-e^{-ix})/2i$
Cosine	$\cos(x)$	Adjacent / Hypotenuse	$(e^{ix}+e^{-ix})/2$
Tangent	$\tan(x)$	Opposite / Adjacent	$-{\mathit {i}}(e^{{\mathit {i}}x}-e^{-{\mathit {i}}x})/(e^{{\mathit {i}}x}+e^{-{\mathit {i}}x})$
Cosecant	$\csc(x)$	Hypotenuse / Opposite
Secant	$\sec(x)$	Hypotenuse / Adjacent
Cotangent	$\cot(x)$	Adjacent / Opposite

Hyperbolic functions:

Name	Notation	Exponential formula
Hyperbolic sine	$\sinh(x)$	$(e^{x}-e^{-x})/2$
Hyperbolic cosine	$\cosh(x)$	$(e^{x}+e^{-x})/2$
Hyperbolic tangent	$\tanh(x)$	$(e^{x}-e^{-x})/(e^{x}+e^{-x})$
Hyperbolic cosecant	$\mathrm {csch} (x)$	$2/(e^{x}-e^{-x})$
Hyperbolic secant	$\mathrm {sech} (x)$	$2/(e^{x}+e^{-x})$
Hyperbolic cotangent	$\coth(x)$	$(e^{x}+e^{-x})/(e^{x}-e^{-x})$

Inverse trigonometric functions:

Name	Notation	Triangle formula	Exponential formula
Arcsine	$\arcsin(x)$
Arccosine	$\arccos(x)$
Arctangent	$\arctan(x)$
Arccosecant	$\operatorname {arccsc}(x)$
Arcsecant	$\operatorname {arcsec}(x)$
Arccotangent	$\operatorname {arccot}(x)$

Inverse hyperbolic functions:

Name	Notation	Logarithmic formula
Inverse hyperbolic sine	$\mathrm {arcsinh} (x)$	$\ln {(x+{\sqrt {x^{2}+1)}}}$
Inverse hyperbolic cosine	$\mathrm {arccosh} (x)$	$\ln {(x+{\sqrt {x^{2}-1}})}$
Inverse hyperbolic tangent	$\mathrm {arctanh} (x)$	${\frac {1}{2}}\ln {\frac {1+x}{1-x}}$
Inverse hyperbolic cosecant	$\mathrm {arccsch} (x)$
Inverse hyperbolic secant	$\mathrm {arcsech} (x)$
Inverse hyperbolic cotangent	$\mathrm {arccoth} (x)$

Other:

• Sinc function
• Lambert W-function

Exponential integral related

Function	Notation	Definition
Exponential integral	$\mathrm {Ei} (x)$	$\textstyle -\int _{-x}^{\infty }{\frac {e^{-t}}{t}}\,dt$
Logarithmic integral	$\mathrm {li} (x)$	$\textstyle \int _{0}^{x}{\frac {1}{\ln t}}\,dt$

Trigonometric integrals:

Function	Notation	Definition
Sine integral	$\mathrm {Si} (x)$	$\textstyle \int _{0}^{x}{\frac {\sin t}{t}}\,dt$
Hyperbolic sine integral	$\mathrm {Shi} (x)$	$\textstyle \int _{0}^{x}{\frac {\sinh t}{t}}\,dt$
Cosine integral	$\mathrm {Ci} (x)$	$\textstyle \gamma +\ln x+\int _{0}^{x}{\frac {\cos t-1}{t}}\,dt$
Hyperbolic cosine integral	$\mathrm {Chi} (x)$	$\textstyle \gamma +\ln x+\int _{0}^{x}{\frac {\cosh t-1}{t}}\,dt$

Note: $\gamma$ is Euler's constant

Related to the normal distribution:

Name	Notation	Definition
Gaussian function	none standardized	$f(x)=ae^{-(x-b)^{2}/c^{2}}$
Error function	$\mathrm {erf} (x)$	$\textstyle {\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}dt$
Complementary error function	$\mathrm {erfc} (x)$	$1-\mathrm {erf} (x)$

See also gamma related functions below; in particular, the incomplete gamma functions.

Bessel function related

• Airy functions
• Bessel functions
• Fresnel integrals

Orthogonal polynomials

See catalog of orthogonal polynomials for a more detailed listing.

Name	Notation	Interval	Weight function	$f_{0}$, $f_{1}$, $f_{2}$, $f_{3}$, ...
Chebyshev (first kind)	$T_{n}$	$-1,1$	$(1-x^{2})^{-1/2}$	$1$, $x$, $2x^{2}-1$, $4x^{3}-3x$, ...
Chebyshev (second kind)	$U_{n}$	$-1,1$	$(1-x^{2})^{1/2}$	$1$, $2x$, $4x^{2}-1$, $8x^{3}-4x$, ...
Legendre	$P_{n}$	$-1,1$	$1$	$1$, $x$, ${\textstyle {\frac {1}{2}}}$$(3x^{2}-1)$, ${\textstyle {\frac {1}{2}}}$$(5x^{3}-3x)$, …
Hermite	$H_{n}$	$-\infty ,\infty$	$e^{-x^{2}}$
Laguerre	$L_{n}$	$0,\infty$	$e^{-x}$
Associated Laguerre	$L_{n}^{(\alpha )}$	$0,\infty$	$x^{\alpha }e^{-x}$

Factorial and gamma related

Name	Notation	Discrete formula	Continuous formula
Factorial	$x!$	$1\cdot 2\cdot 3\cdots x$	$\Gamma (x+1)$
Gamma function	$\Gamma (x)$	$(x-1)!$	$\Gamma (x)$
Double factorial	$x!!$	$1\cdot 3\cdot 5\cdots x\;\;(x\;\mathrm {odd} )$ $2\cdot 4\cdot 6\cdots x\;\;(x\;\mathrm {even} )$	${\frac {\Gamma (x+1)}{2^{\frac {x-1}{2}}*\Gamma ({\frac {x+1}{2}})}}\;\;(x\;\mathrm {odd} )$ $2^{\frac {x-1}{2}}*\Gamma ({\frac {x+1}{2}})\;\;(x\;\mathrm {even} )$
Binomial coefficient	$n \choose k$	${\frac {n!}{k!(n-k)!}}$	${\frac {\Gamma (n+1)}{\Gamma (k+1)\Gamma (n-k+1)}}$
Rising factorial	$(x)^{(n)}$	${\frac {(x+n-1)!}{(x-1)!}}$	${\frac {\Gamma (x+n)}{\Gamma (x)}}$
Falling factorial	$(x)_{(n)}$	${\frac {x!}{(x-n)!}}$	${\frac {\Gamma (x+1)}{\Gamma (x-n+1)}}$
Beta function	$\mathrm {\mathrm {B} } (x,y)$		${\frac {\Gamma (x)\Gamma (y)}{\Gamma (x+y)}}$
Harmonic number	$H_{n}$	$\textstyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}$	$\gamma +\psi (n+1)$
Digamma function	$\psi (x),\psi ^{(0)}(x)$	$H_{x-1}-\gamma$	${\begin{matrix}{\frac {d}{dx}}\end{matrix}}\ln \Gamma (x)$
Polygamma function (of order m)	$\psi ^{(m)}(x)$		$\left({\begin{matrix}{\frac {d}{dx}}\end{matrix}}\right)^{m+1}\ln \Gamma (x)$

• Incomplete gamma function
• Incomplete beta function
• Regularized gamma function
• Regularized beta function
• Barnes G-function

Notes:

• $\gamma$ is Euler's constant
• The polygamma functions are generalized to continuous m by the Hurwitz zeta function

Zeta function related

• Riemann zeta function
• Hurwitz zeta function
• Polylogarithm

Hypergeometric functions

Note: many of the preceding functions are special cases of the following:

• Hypergeometric functions
• Meijer G-function

See also

• Catalog of mathematical constants
• Catalog of probability distributions
• Catalog of number sequences

Further reading

• Introductory material: N. N. Lebedev (1972). Special Functions and their applications. Dover. 

References

• Milton Abramowitz and Irene A. Stegun (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.  (available online)
• I. S. Gradstein and I. M. Ryzhik (2000). Table of integrals, series and products. Academic Press. 
• A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi (1953). Higher Transcendental Functions (Vol I and II). McGraw-Hill Book Company.