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Title: Values of the Gauss hypergeometric function ${}_2F_1(a,b;c;z)$-Definition: Real principal values of Gauss' hypergeometric function CITE{Wiki} ${}_2F_1(a,b;c;z)=F(a,b;c;z)$,- with rational parameters $a,b,c,z$, ordered upper parameters $b\geq a$, $c\notin\mathbb{Z}_{\leq0}$,- and $z<1$. DLMF CITE{DLMFDefinitions} defines the Gauss series and the principal- branch.+Definition: ${}_2F_1(a,b;c;z)=F(a,b;c;z)$ is Gauss' hypergeometric function CITE{Wiki},+ and this table holds its real principal values at rational $a$, $b$, $c$ and $z$+ with $c\notin\mathbb{Z}_{\leq0}$ and $z<1$. DLMF CITE{DLMFDefinitions} defines the+ Gauss series and the principal branch. Keywords: - Gauss hypergeometric function
comment-unregularised: This is the unregularised Gauss function $F(a,b;c;z)$, not Olver's scaled function $\mathbf{F}(a,b;c;z)$ from CITE{DLMFDefinitions}.- comment-branch: The argument is kept off the branch cut $[1,\infty)$. Each stored- row is a real principal value, and the generator refuses a row if its computed- imaginary part does not contain zero.+ comment-branch: The argument is kept off the branch cut $[1,\infty)$, so every stored+ value is a real principal value. comment-symmetry: The upper parameters are ordered with $b\geq a$ because ${}_2F_1(a,b;c;z)={}_2F_1(b,a;c;z)$, as the series in CITE{formula-series} shows.- comment-rational-rows: Rows known from the stated identities to be rational are- omitted. This includes $z=0$, non-positive integer upper parameters, elementary- rows from CITE{formula-elementary} with integer remaining exponent, Kummer and- half-argument rows from CITE{formula-kummer-minus-one} and CITE{formula-half-argument}- where a denominator gamma factor has a pole, degree-one Pfaff transforms from- CITE{formula-pfaff-transformation} whose terminating polynomial vanishes, and- integer-parameter rows that a Pfaff or Euler transformation reduces to a rational- function. Formulas: formula-series: With $(q)_n$ denoting the Pochhammer symbol, ${}_2F_1(a,b;c;z)=\sum_{n=0}^{\infty}(a)_n(b)_n
- table: HREF{Complete_elliptic_integral_of_the_first_kind_K}[Complete elliptic integral of the first kind $K(m)$]- relation: is the specialisation in CITE{formula-elliptic}+ relation: is $\pi\,{}_2F_1(1/2,1/2;1;m)/2$ from CITE{formula-elliptic} - table: HREF{Complete_elliptic_integral_of_the_second_kind_E}[Complete elliptic integral of the second kind $E(m)$]- relation: is the other specialisation in CITE{formula-elliptic}+ relation: is $\pi\,{}_2F_1(-1/2,1/2;1;m)/2$ from CITE{formula-elliptic} - table: HREF{Values_of_Kummer's_confluent_hypergeometric_function}[Values of Kummer's confluent hypergeometric function $M(a;b;z)$]
title: 'DLMF 19.5: Maclaurin and Related Expansions' url: https://dlmf.nist.gov/19.5- Requested129:- title: Requested in numberdb-data#129- url: https://github.com/numberdb/numberdb-data/issues/129 Tags: - special values
rigour: proven rigour details: Each value was computed as a Sage complex ball with arb at `numberdb.bits(digits,- losing=64)` bits, with the imaginary part checked to contain zero. The 761 entries+ losing=64)` bits, with the imaginary part checked to contain zero. The 730 entries were checked against mpmath at 130 decimal digits, using CITE{formula-pfaff-transformation} before the mpmath call when its direct continuation failed; against CITE{formula-euler-transformation} on 279 stored rows; against CITE{formula-elementary} on 153 stored rows; and against CITE{formula-elliptic} on 4 stored rows. The widest returned ball had radius less- than $1.2\cdot 10^{-115}$, and the write-path formatter found no stored row that- shortened to an exact rational.+ than $1.2\cdot 10^{-115}$, and after the rational rows described in the completeness+ note were omitted, the write-path formatter found no stored row that shortened+ to an exact rational. complete: 'no'- complete-note: it holds every non-rational entry in the screened grid with $a,b\in\{-1/2,1/2,1,3/2,2,3\}$,- $b\geq a$, $c\in\{1/2,1,3/2,2,3\}$, and $z\in\{-4,-2,-3/2,-1,-3/4,-1/2,-1/4,1/4,1/2\}$+ complete-note: it holds every non-rational entry in the grid $a,b\in\{-1/2,1/2,1,3/2,2,3\}$,+ $b\geq a$, $c\in\{1/2,1,3/2,2,3\}$, and $z\in\{-4,-2,-3/2,-1,-3/4,-1/2,-1/4,1/4,1/2\}$;+ the omitted rational rows are elementary cases, Kummer and half-argument cases+ where a denominator gamma factor has a pole, terminating Pfaff or Euler transformations+ to rational functions, and five isolated rows verified by symbolic simplification Display properties: number-header: ${}_2F_1(a,b;c;z)$
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