Gauss hypergeometric function
E596519
The Gauss hypergeometric function is a special function defined by a power series that generalizes many elementary and higher transcendental functions and plays a central role in mathematical analysis, differential equations, and mathematical physics.
Statements (52)
| Predicate | Object |
|---|---|
| instanceOf |
analytic function
ⓘ
hypergeometric function ⓘ special function ⓘ |
| alsoKnownAs |
Gaussian hypergeometric function
NERFINISHED
ⓘ
ordinary hypergeometric function ⓘ |
| appearsIn |
conformal field theory
ⓘ
general relativity ⓘ potential theory ⓘ probability theory ⓘ quantum mechanics ⓘ solutions of second-order linear ordinary differential equations with three regular singular points ⓘ statistics ⓘ |
| belongsTo | hypergeometric series family {}_pF_q ⓘ |
| convergesFor | |z|<1 for general complex parameters a,b,c with c not a nonpositive integer ⓘ |
| definedBySeries | {}_2F_1(a,b;c;z)=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!} ⓘ |
| dependsOnParameters |
a
ⓘ
b ⓘ c ⓘ |
| dependsOnVariable | z ⓘ |
| domain | complex variable z with analytic continuation to \mathbb{C}\setminus[1,\infty) ⓘ |
| generalizes |
Bessel functions
NERFINISHED
ⓘ
Chebyshev polynomials NERFINISHED ⓘ Gegenbauer polynomials NERFINISHED ⓘ Jacobi polynomials NERFINISHED ⓘ Legendre functions NERFINISHED ⓘ arcsin function ⓘ arctan function ⓘ binomial series (1-z)^{-a} ⓘ logarithm function ⓘ |
| hasBranchPoints |
z=1
ⓘ
z=\infty ⓘ |
| hasConnectionFormula | linear relations between values at z,1-z,1/z ⓘ |
| hasIntegralRepresentation | {}_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1 t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,dt for suitable parameters ⓘ |
| hasSymmetry | {}_2F_1(a,b;c;z)={}_2F_1(b,a;c;z) ⓘ |
| hasTransformationFormula |
Euler transformation {}_2F_1(a,b;c;z)=(1-z)^{c-a-b}{}_2F_1(c-a,c-b;c;z)
ⓘ
Pfaff transformation {}_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1\left(a,c-b;c;\frac{z}{z-1}\right) NERFINISHED ⓘ |
| isCaseOf | generalized hypergeometric function with p=2,q=1 ⓘ |
| namedAfter | Carl Friedrich Gauss NERFINISHED ⓘ |
| parameterDomain | complex parameters a,b,c with c not in \{0,-1,-2,\dots\} ⓘ |
| reducesTo | polynomial when a or b is a nonpositive integer ⓘ |
| relatedTo |
beta function
ⓘ
gamma function ⓘ |
| satisfiesDifferentialEquation | z(1-z)y''+[c-(a+b+1)z]y'-aby=0 ⓘ |
| specialValue |
{}_2F_1(a,b;a;z)=(1-z)^{-b}
ⓘ
{}_2F_1(a,b;b;z)=(1-z)^{-a} ⓘ {}_2F_1(a,b;c;0)=1 ⓘ |
| symbol | {}_2F_1(a,b;c;z) ⓘ |
| usedFor |
analytic continuation of many special functions
ⓘ
asymptotic analysis ⓘ evaluation of definite integrals ⓘ representation of orthogonal polynomials ⓘ |
| usesNotation | (q)_n for the Pochhammer symbol ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.