# Fungrim entry: c60679

$\left|\,{}_2F_1\!\left(a, b, c, z\right) - \sum_{k=0}^{N - 1} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\left(c\right)_{k}} \frac{{z}^{k}}{k !}\right| \le \left|\frac{\left(a\right)_{N} \left(b\right)_{N}}{\left(c\right)_{N}} \frac{{z}^{N}}{N !}\right| \frac{1}{1 - D}\; \text{ where } D = \left|z\right| \left(1 + \frac{\left|a - c\right|}{\left|c + N\right|}\right) \left(1 + \frac{\left|b - 1\right|}{\left|1 + N\right|}\right)$
Assumptions:$a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, c \in \mathbb{C} \setminus \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|z\right| \lt 1 \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(c\right) + N \gt 0 \,\mathbin{\operatorname{and}}\, D \lt 1$
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| \lt 1 \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(c\right) + N \gt 0 \,\mathbin{\operatorname{and}}\, D \lt 1
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
Hypergeometric2F1$\,{}_2F_1\!\left(a, b, c, z\right)$ Gauss hypergeometric function
RisingFactorial$\left(z\right)_{k}$ Rising factorial
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
CC$\mathbb{C}$ Complex numbers
ZZLessEqual$\mathbb{Z}_{\le n}$ Integers less than or equal to n
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Re$\operatorname{Re}\!\left(z\right)$ 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))), Tuple(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))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC