Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.adox)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 4 and fixed z

For fixed z and a=-11/4, b>=a

For fixed z and a=-11/4, b=21/4

http://functions.wolfram.com/07.23.03.adox.01

Input Form

Hypergeometric2F1[-(11/4), 21/4, 5, -z] == (4096 Sqrt[2] (2 Sqrt[1 + z] (-4928 + 16632 z - 54747 z^2 + 263648 z^3 + 5333760 z^4 + 9584640 z^5 + 4587520 z^6) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + 2 (-4928 + 11704 z - 38115 z^2 + 208901 z^3 + 5597408 z^4 + 14918400 z^5 + 14172160 z^6 + 4587520 z^7) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - (-9856 + 25872 z - 83853 z^2 + 442519 z^3 + 3719040 z^4 + 5437440 z^5 + 2293760 z^6) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - 2 Sqrt[1 + z] (-4928 + 16632 z - 54747 z^2 + 263648 z^3 + 5333760 z^4 + 9584640 z^5 + 4587520 z^6) EllipticK[(-1 + Sqrt[1 + z])/ (1 + Sqrt[1 + z])]))/(15058768725 Pi z^4 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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</apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 15058768725 </cn> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]