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Fungrim entry: c60679

2F1 ⁣(a,b,c,z)k=0N1(a)k(b)k(c)kzkk!(a)N(b)N(c)NzNN!11D   where D=z(1+acc+N)(1+b11+N)\left|\,{}_2F_1\!\left(a, b, c, z\right) - \sum_{k=0}^{N - 1} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\left(c\right)_{k}} \frac{{z}^{k}}{k !}\right| \le \left|\frac{\left(a\right)_{N} \left(b\right)_{N}}{\left(c\right)_{N}} \frac{{z}^{N}}{N !}\right| \frac{1}{1 - D}\; \text{ where } D = \left|z\right| \left(1 + \frac{\left|a - c\right|}{\left|c + N\right|}\right) \left(1 + \frac{\left|b - 1\right|}{\left|1 + N\right|}\right)
Assumptions:aCandbCandcC{0,1,}andzCandz<1andNZ0andRe(c)+N>0andD<1a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, c \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| < 1 \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(c) + N > 0 \,\mathbin{\operatorname{and}}\, D < 1
TeX:
\left|\,{}_2F_1\!\left(a, b, c, z\right) - \sum_{k=0}^{N - 1} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\left(c\right)_{k}} \frac{{z}^{k}}{k !}\right| \le \left|\frac{\left(a\right)_{N} \left(b\right)_{N}}{\left(c\right)_{N}} \frac{{z}^{N}}{N !}\right| \frac{1}{1 - D}\; \text{ where } D = \left|z\right| \left(1 + \frac{\left|a - c\right|}{\left|c + N\right|}\right) \left(1 + \frac{\left|b - 1\right|}{\left|1 + N\right|}\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, c \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| < 1 \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(c) + N > 0 \,\mathbin{\operatorname{and}}\, D < 1
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
Sumnf(n)\sum_{n} f(n) Sum
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
Powab{a}^{b} Power
Factorialn!n ! Factorial
CCC\mathbb{C} Complex numbers
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
Entry(ID("c60679"),
    Formula(Where(LessEqual(Abs(Sub(Hypergeometric2F1(a, b, c, z), Sum(Mul(Div(Mul(RisingFactorial(a, k), RisingFactorial(b, k)), RisingFactorial(c, k)), Div(Pow(z, k), Factorial(k))), For(k, 0, Sub(N, 1))))), Mul(Abs(Mul(Div(Mul(RisingFactorial(a, N), RisingFactorial(b, N)), RisingFactorial(c, N)), Div(Pow(z, N), Factorial(N)))), Div(1, Sub(1, D)))), Equal(D, Mul(Mul(Abs(z), Add(1, Div(Abs(Sub(a, c)), Abs(Add(c, N))))), Add(1, Div(Abs(Sub(b, 1)), Abs(Add(1, N)))))))),
    Variables(a, b, c, z, N),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, SetMinus(CC, ZZLessEqual(0))), Element(z, CC), Less(Abs(z), 1), Element(N, ZZGreaterEqual(0)), Greater(Add(Re(c), N), 0), Less(D, 1))))

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2019-10-05 13:11:19.856591 UTC