Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9c7)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a<0

For fixed z and a=-41/8, b>=a

For fixed z and a=-41/8, b=29/8

http://functions.wolfram.com/07.23.03.b9c7.01

Input Form

Hypergeometric2F1[-(41/8), 29/8, 6, z] == (1/(3744803476138725 Pi z^5)) (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-4 (114196480 - 392104320 z + 210114135 z^2 + 196428540 z^3 + 565134570 z^4 - 18594795996 z^5 + 51528344679 z^6 - 65217168792 z^7 + 44549814672 z^8 - 16041689664 z^9 + 2403346176 z^10) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 40 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1784320 - 5833890 z + 2415105 z^2 + 3220140 z^3 - 347775120 z^4 + 1012254516 z^5 - 1311410457 z^6 + 909537714 z^7 - 331221528 z^8 + 50069712 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (-14274560 + 38641680 z + 805035 z^2 - 21735945 z^3 - 4847115735 z^4 + 14847876477 z^5 - 19813711176 z^6 + 14052514608 z^7 - 5213710656 z^8 + 801115392 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 (114196480 - 392104320 z + 210114135 z^2 + 196428540 z^3 + 565134570 z^4 - 18594795996 z^5 + 51528344679 z^6 - 65217168792 z^7 + 44549814672 z^8 - 16041689664 z^9 + 2403346176 z^10) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

Standard Form

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MathML Form

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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]