InverseBetaRegularized
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1 InverseBetaRegularized Notations Traditional name Inverse of the regularized incomplete beta function Traditional notation I z 1 a, b Mathematica StandardForm notation InverseBetaRegularizedz, a, b Primary definition z I w a, b ; w I z 1 a, b Specific values Specialized values I 1 0 a, b 0 ; a I 1 1 a, b 1 ; a 0 General characteristics Domain and analyticity I z 1 a, b is an analytical function of z, a, b which is defined in z a bi z 1 a, b Symmetries and periodicities Symmetry No symmetry
2 2 Periodicity No periodicity Series representations Generalized power series Expansions at generic point z z 0 For the function itself I z 1 a, b I z0 1 a, b 1 w 1b a, bw 1a z z w12 b a a b 2 w 1 a, b 2 w 12 a z z w13 b 2 a 2 w 1 2 a 4 b 7 w 3 w 1 2 b 3 w b 2 w 2 1 a, b 3 w 13 a z z w14 b 6 a 3 11 a 2 6 a a b 2 2 a 2 b 3 3 a 3 b 4 w 3 3 a 1 2 a 2 b a 3 b 4 w 2 a 1 11 b 3 a 6 a 6 b w 1 a, b 4 w 14 a z z w15 b 24 a 4 50 a 3 35 a 2 10 a a b 2 2 a 2 b 3 3 a 3 b 4 4 a 4 b 5 w a 1 2 a 2 b 3 3 a 3 b 4 4 a 4 b 5 w 3 2 a 1 72 a b 37 a 2 b 72 b a b b 75 w 2 2 a 1 2 a 1 13 b 2 a 12 a 12 b w 1 a, b 5 w 15 a z z w16 b a b 2 2 a 2 b 3 3 a 3 b 4 4 a 4 b 5 5 a 5 b 6 w 5 5 a 1 2 a 2 b 3 3 a 3 b 4 4 a 4 b 5 5 a 5 b 6 w 4 2 a a 437 b 3 12 a 150 a b 2 a 3 a 600 a b 1923 b a 2 a 5 a 60 a w 3 2 a a a 212 b 2 a a 1200 a b 2 4 a 3 5 a 3 a 15 a w 2 a a a 10 a 60 a a 1 2 a 1 a 100 a b w 62 w a a a 2 a 60 a a, b 6 w 16 a z z 0 6 ; z z 0 w I z0 1 a, b I 1 z a, b I 1 z0 a, b1 Oz z 0 Expansions at z I 1 z a, b a z a, b 1a b 1 b 1 a 2 3 b a a 5 b 4 a 1 a z a, b2a a z a, b 3a Oz 4a ; a 0 2 a 1 2 a 2
3
4
5
6 b w 10 Oz 11a ; w a z a, b 1a a 0 Expansions at generic point a a 0 For the function itself I z 1 a, b I z 1 a 0, b w 1 w 1b w a 0 w a 0, b logw Ψb a 0 Ψa 0 1 a 0 2 3F 2 a 0, a 0, 1 b; a 0 1, a 0 1; w a a w 1 w12 b 2 a 0 3 4F 3a 0, a 0, a 0, 1 b; a 0 1, a 0 1, a 0 1; w 1 w b w a 0 a 0, b I w a 0, b Ψa 0 Ψb a 0 logw Ψb a 0 Ψa 0 1 w b w a 0 a 0, b I w a 0, b logw Ψb a 0 Ψa 0 1 a 0 1 w 1b w a 0 w a 0 a 0 2 3F 2a 0, a 0, 1 b; a 0 1, a 0 1; w a 0, b I w a 0, b logw Ψb a 0 Ψa 0 logw 1 w b b 1 w 1a 0 a 0 2 3F 2a 0, a 0, 1 b; a 0 1, a 0 1; w w a 0 a 0 2 3F 2a 0, a 0, 1 b; a 0 1, a 0 1; w a 0, b I w a 0, b logw Ψb a 0 Ψa 0 w 1 w a 0 a 0 2 a 2 0 b 1 w 3 F 2a 0 1, a 0 1, 2 b; a 0 2, a 0 2; w 3 F 2a 0, a 0, 1 b; a 0 1, a 0 1; w w a 0 a 0 2 3F 2a 0, a 0, 1 b; a 0 1, a 0 1; w a 0, b I w a 0, b logw Ψb a 0 Ψa 0 1 b w 12 a 0 a 0, b I w a 0, b logw Ψb a 0 Ψa 0 w a 0 a 0 2 3F 2a 0, a 0, 1 b; a 0 1, a 0 1; w a 0, b I w a 0, b logw Ψb a 0 Ψa 0 w 2 a 0 a 0, b I w a 0, b w a 0 Ψ 1 a 0 Ψ 1 b a 0 1 w b w 1 w a 0 a 0 2 3F 2a 0, a 0, 1 b; a 0 1, a 0 1; w w 1 a 0, b I w a 0, b logw Ψb a 0 Ψa 0 a a 0 2 ; a a 0 w I 1 z a 0, b Q 1 a, z Q 1 a 0, z1 Oa a 0 Expansions at generic point b b 0 For the function itself
7 I 1 z a, b I 1 z a, b 0 w 1a 1 w 1b 0 1 w b 0 3 F 2 1 a, b 0, b 0 ; b 0 1, b 0 1; 1 w 1w b 0, a log1 w Ψb 0 Ψa b 0 b b 0 b w12 a 2 w 1 b 0 3 4F 3b 0, b 0, b 0, 1 a; b 0 1, b 0 1, b 0 1; 1 w w a 1 w 1b 0 1w b 0, a Ψb 0 Ψa b 0 log1 w Ψb 0 Ψa b 0 w a 1 w 2b 0 b 0 2 3F 2b 0, b 0, 1 a; b 0 1, b 0 1; 1 w 1 w b 0 b 0 2 3F 2b 0, b 0, 1 a; b 0 1, b 0 1; 1 w 1w b 0, a log1 w Ψb 0 Ψa b 0 w 1 a 1 w 2b 0 b 0 2 3F 2b 0, b 0, 1 a; b 0 1, b 0 1; 1 w 1 w b 0 b 0 2 3F 2b 0, b 0, 1 a; b 0 1, b 0 1; 1 w 1w b 0, a log1 w Ψb 0 Ψa b 0 a 1 1 w 22 b 0 1w b 0, a log1 w Ψb 0 Ψa b 0 1 w b 0 b 0 2 3F 2b 0, b 0, 1 a; b 0 1, b 0 1; 1 w 1w b 0, a log1 w Ψb 0 Ψa b 0 1 w 2 b 0 w 1 1w b 0, a log1 w Ψb 0 Ψa b 0 1 w b 0 log1 w w a b 0 1 1w b 0, a log1 w Ψb 0 Ψa b 0 w 1 w b 0 b 0 2 3F 2b 0, b 0, 1 a; b 0 1, b 0 1; 1 w b 0 1 w 1 w 2 b 0 w 1 1w b 0, a 1 w b 0 Ψ 1 b 0 Ψ 1 a b 0 w a 1 w b 0 b 0 2 3F 2b 0, b 0, 1 a; b 0 1, b 0 1; 1 w w 1w b 0, a log1 w Ψb 0 Ψa b 0 w b b 0 2 ; b b 0 w I 1 z a, b I 1 z a, b I 1 z a, b 0 1 Ob b 0 Differential equations Ordinary nonlinear differential equations wz 1 wz w z 1 a a b 2 wz w z 2 0 ; wz I 1 z a, b Differentiation Low-order differentiation With respect to z I 1 z a, b 1 w 1b w 1a a, b ; w I 1 z a, b z 2 I z 1 a, b z w 12 b w 12 a w 1 a b 2 w 1 a, b 2 ; w I 1 z a, b I 1 z a, b 1 w 13 b w 13 a 2 a 2 w 1 2 a 4 b 7 w 3 w 1 2 b 2 7 b 6 w 2 4 b 6 w 1 a, b 3 ; z 3 w I z 1 a, b
8 I 1 z a, b 1 w 14 b w 14 a 6 a 3 11 a 2 6 a a b 2 2 a 2 b 3 3 a 3 b 4 w 3 z 4 3 a 1 2 a 2 b 3 3 a 3 b 4 w 2 a 1 11 b 3 a 6 a 6 b w 1 a, b 4 ; w I 1 z a, b I 1 z a, b 1 w 15 b w 15 a 24 a 4 50 a 3 35 a 2 10 a z 5 a b 2 2 a 2 b 3 3 a 3 b 4 4 a 4 b 5 w 4 4 a 1 2 a 2 b 3 3 a 3 b 4 4 a 4 b 5 w 3 2 a 1 72 a b 37 a 2 b 72 b a b b 75 w 2 2 a 1 2 a 1 13 b 2 a 12 a 12 b w 1 a, b 5 ; w I 1 z a, b I 1 z a, b z 6 1 w 16 b w 16 a a b 2 2 a 2 b 3 3 a 3 b 4 4 a 4 b 5 5 a 5 b 6 w 5 5 a 1 2 a 2 b 3 3 a 3 b 4 4 a 4 b 5 5 a 5 b 6 w 4 2 a a 437 b 3 12 a 150 a b 2 a 3 a 600 a b 1923 b a 2 a 5 a 60 a w 3 2 a a a 212 b 2 a a 1200 a b 2 4 a 3 5 a 3 a 15 a w 2 a a a 10 a 60 a a 1 2 a 1 a 100 a b w 62 w a a a 2 a 60 a a, b 6 ; w I 1 z a, b I 1 z a, b 1 w 17 b w 17 a a b 2 2 a 2 b 3 3 a 3 b 4 4 a 4 b 5 5 a 5 b 6 6 a 6 b 7 w 6 z 7 6 a 1 2 a 2 b 3 3 a 3 b 4 4 a 4 b 5 5 a 5 b 6 6 a 6 b 7 w 5 3 a a 229 b a 200 a b 3 4 a 72 a 75 a b 2 8 a a 3 a 600 a b a a 4 a 45 a 20 a b 140 w 4 4 a a 100 a b a a a 335 b 2 2 a a 18 a 300 a b a a 4 a 45 a 20 a b 70 w 3 3 a 1 2 a a 25 a b 2 a 18 a 200 a b 1135 b a a 90 a 20 a w 2 6 a a a 4 a 9 a 20 a a 1 2 a 1 3 a 1 6 a 5 a 4 5 b w 126 w a a a 4 a 9 a 20 a a, b 7 ; w I 1 z a, b
9 I 1 z a, b 1 w 18 b w 18 a 5040 a 7 w a b 1343 w 363 w 1 6 z 8 4 a 5 w w 9 b b 1343 w w 1 5 a 4 w 20 b 9 b 980 b w 2 4 b b w b w 1 4 a 3 w 4 b 10 b 18 b 245 b w 3 b 24 b b w 2 12 b b w b w 1 3 a 2 w 2 b b 10 b 27 b 196 b w 4 b b 4 b b w 3 b b b w 2 7 b b w b w 1 2 a w w 5 b 6 36 w w 5263 b 5 4 w 3 w w b 4 w 2 w w w b 3 w w w w w b 2 2 w 1 44 w 1 w w 1 w b w 24 w 2 w 3 w 2958 w w 1 w b 2 2 b 3 3 b 4 4 b 5 5 b 6 6 b 7 7 b 8 w b 3 3 b 4 4 b 5 5 b 6 6 b 7 7 b 8 w 5 3 b b b 12 b 2323 b w 4 5 b b 20 b 835 b w 3 b b b w 2 3 b 1894 b w 247 b a, b 8 ; w I z 1 a, b
10 I 1 z a, b 1 w 19 b w 19 a a 8 w a b 3001 w 761 w 1 7 z 9 4 a 6 w w 36 b b 3001 w w a 5 w 3 2 b 42 b 2240 b w b b w b w 1 5 a 4 w 3 20 b b 84 b 560 b w b 5 b b w 2 4 b b w w b w a 3 w 2 b 10 b 2 b 63 b 448 b w 4 2 b 2 b 10 b b w 3 2 b b b w 2 b b w w b w a 2 w 3 b b 2 b b 84 b 1120 b w 5 2 b b 2 b b b w 4 b b 2 b b w 3 b 8 b b w 2 b b w b w a w w 6 b 7 72 w w b 6 4 w 4 w w b 5 2 w 3 3 w w w b 4 w 2 w 2 w w w b 3 w w w w w w b 2 w 1 w 4 w 1 w w 1 w b 3 w w 8 w w w w w 1 w b 2 2 b 3 3 b 4 4 b 5 5 b 6 6 b 7 7 b 8 8 b 9 w b 3 3 b 4 4 b 5 5 b 6 6 b 7 7 b 8 8 b 9 w 6 4 b b b b 36 b 6361 b w 5 2 b b b 4 b b w 4 2 b b b b w 3 4 b b b w 2 2 b 9511 b w 502 b a, b 9
11 I 1 z a, b 1 w 110 b w 110 a a 9 w a b w 7129 w 1 8 z a 7 w w b b w w a 6 w 63 b 16 b 7560 b w b b w b w 1 6 a 5 w 3 4 b 21 b 32 b 5670 b w 3 12 b 6 b b w 2 8 b b w w b w 1 5 a 4 w 3 5 b 4 b 21 b 8 b 4536 b w b 5 b 12 b b w 3 10 b 4 b b w 2 2 b b w b w a 3 w b b 2 b 63 b 32 b 3780 b w b b b 9 b b w 4 b b 90 b b w 3 b b b w b b w b w a 2 w w 6 b w w b w 4 3 w w b 5 2 w w w w b 4 w 2 w w 3 w w b 3 w 2 w 1 w 1 w w 1 w b 2 w w 2 w 3 w w w b w w w w 9 w w w 1 2 a w w 7 b w w b 7 36 w 5 9 w w b 6 4 w 4 w w w b 5 w 3 w 3 w w w b 4 2 w 2 3 w w 4 w w w b 3 2 w w w 2 w 3 w w w b w 1 2 w 1 w 2 w 1 w w 1 w b w 6 w 2 2 w 20 w w 3 w w w 1 w b 2 2 b 3 3 b 4 4 b 5 5 b 6 6 b 7 7 b 8 8 b 9 9 b 10 w b 3 3 b 4 4 b 5 5 b 6 6 b 7 7 b 8 8 b 9 9 b 10 w 7 4 b b b b b 36 b b w 6 2 b b b b b b w 5 6 b b b b b w 4 2 b b b b w 3 2 b b b w b b 4665 w 1013 b a, b 10 ; w I z 1 a, b With respect to a
12 I 1 z a, b 1 w 1b w 1a w a a 2 3F 2a, a, 1 b; a 1, a 1; w a, b I w a, b logw Ψa Ψa b ; a w I 1 z a, b I 1 z a, b 1 w 12 b w b 1 a 2 3F 2a, a, 1 b; a 1, a 1; w a 2 w a a 2 3F 2a, a, 1 b; a 1, a 1; w a, b I w a, b logw Ψa Ψa b w 1a a, b I w a, b w 1 a 2 3F 2a, a, 1 b; a 1, a 1; w w a 1 w b Ψ 1 a Ψ 1 a b w a w 1 a, b I w a, b logw Ψa Ψa b w 2 a 1 w b a, b I w a, b Ψa Ψa b logw Ψa Ψa b w a w 1 a 2 a 2 b 1 w 3 F 2a 1, a 1, 2 b; a 2, a 2; w 3 F 2a, a, 1 b; a 1, a 1; w w a a 2 3F 2a, a, 1 b; a 1, a 1; w a, b I w a, b logw Ψa Ψa b w a 1 w b a, b I w a, b logw Ψa Ψa b 1 a 1 w 1b w a w a a 2 3F 2a, a, 1 b; a 1, a 1; w a, b I w a, b logw Ψa Ψa b logw w a 1 b a, b I w a, b logw Ψa Ψa b w a a 2 3F 2a, a, 1 b; a 1, a 1; w a, b I w a, b logw Ψa Ψa b w 12 a 2 1 w b a 3 4F 3a, a, a, 1 b; a 1, a 1, a 1; w ; w I 1 z a, b With respect to b I 1 z a, b 1 w b w 1 w 1a b 1 w b b 2 3F 2b, b, 1 a; b 1, b 1; 1 w 1w b, a log1 w Ψb Ψa b ; w I 1 z a, b I 1 z a, b w 12 a 2 w 1 b 3 4F 3b, b, b, 1 a; b 1, b 1, b 1; 1 w w a 1 w 1b 1w b, a b 2 Ψb Ψa b log1 w Ψb Ψa b w a 1 w 2b b 2 3F 2b, b, 1 a; b 1, b 1; 1 w 1 w b b 2 3F 2b, b, 1 a; b 1, b 1; 1 w 1w b, a log1 w Ψb Ψa b w 1 a 1 w 2b b 2 3F 2b, b, 1 a; b 1, b 1; 1 w 1 w b b 2 3F 2b, b, 1 a; b 1, b 1; 1 w 1w b, a log1 w Ψb Ψa b a 1 1 w 22 b 1w b, a log1 w Ψb Ψa b 1 w b b 2 3F 2b, b, 1 a; b 1, b 1; 1 w 1w b, a log1 w Ψb Ψa b 1 w 2 b w 1 1w b, a log1 w Ψb Ψa b 1 w b log1 w w a b 1 1 w b b 2 3F 2b, b, 1 a; b 1, b 1; 1 w w b 1 1w b, a log1 w Ψb Ψa b w 1 w 2 b w 1 1w b, a 1 w b Ψ 1 b Ψ 1 a b w a 1 w b b 2 3F 2b, b, 1 a; b 1, b 1; 1 w w 1w b, a log1 w Ψb Ψa b w ; w I 1 z a, b Integration Indefinite integration
13 13 Involving only one direct function I 1 z a, b z a 1 a, b 2F 1 a 1, 1 b; a 2; I 1 z a, b I 1 z a, b a1 Representations through equivalent functions With inverse function I 1 Iz a, b z a,b Iz a, b a, b z a,b 1 I Iz1 a,bz a, b I z1,z 2 a, b
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