Hypergeometric2F1

Hypergeometric2F1[a,b,c,z]

is the hypergeometric function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
The function has the series expansion $TemplateBox[{a, b, c, z}, Hypergeometric2F1]=sum_(k=0)^(infty)TemplateBox[{a, k}, Pochhammer]TemplateBox[{b, k}, Pochhammer]/TemplateBox[{c, k}, Pochhammer] z^k/k!$ , where is the Pochhammer symbol.
For certain special arguments, Hypergeometric2F1 automatically evaluates to exact values.
Hypergeometric2F1 can be evaluated to arbitrary numerical precision.
Hypergeometric2F1 automatically threads over lists.
Hypergeometric2F1[a,b,c,z] has a branch cut discontinuity in the complex plane running from to .
FullSimplify and FunctionExpand include transformation rules for Hypergeometric2F1.
Hypergeometric2F1 can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples (7)

Evaluate numerically:

Evaluate symbolically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Expand Hypergeometric2F1 in a Taylor series at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

Scope (44)

Numerical Evaluation (5)

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate Hypergeometric2F1 efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix Hypergeometric2F1 function using MatrixFunction:

Specific Values (6)

Hypergeometric2F1 automatically evaluates to simpler functions for certain parameters:

Exact value of Hypergeometric2F1 at unity:

Hypergeometric series terminates if either of the first two parameters is a negative integer:

Find a value of satisfying the equation :

Permutation symmetry:

Heun functions can simplify to hypergeometric functions:

Visualization (3)

Plot the Hypergeometric2F1 function:

Plot Hypergeometric2F1 as a function of its third parameter :

Plot the real part of :

Plot the imaginary part of :

Function Properties (9)

Real domain of Hypergeometric2F1:

Complex domain of Hypergeometric2F1:

is an analytic function on its real domain:

It is neither analytic nor meromorphic in the complex plane:

is non-decreasing on its real domain:

is injective:

is not surjective:

is non-negative on its real domain:

has both singularity and discontinuity for :

is convex on its real domain:

TraditionalForm formatting:

Differentiation (3)

First derivative:

Higher derivatives:

Plot higher derivatives for , and :

Formula for the derivative:

Integration (3)

Indefinite integral of Hypergeometric2F1:

Definite integral of Hypergeometric2F1:

Integral involving a power function:

Series Expansions (6)

Taylor expansion for Hypergeometric2F1:

Plot the first three approximations for around :

General term in the series expansion of Hypergeometric2F1:

Expand Hypergeometric2F1 in a series near :

Expand Hypergeometric2F1 in a series around :

Give the result for an arbitrary symbolic direction :

Apply Hypergeometric2F1 to a power series:

Integral Transforms (2)

Compute the Laplace transform using LaplaceTransform:

HankelTransform:

Function Identities and Simplifications (2)

Argument simplification:

Recurrence identities:

Function Representations (5)

Basic definition:

Relation to the JacobiP polynomial:

Hypergeometric2F1 can be represented as a DifferentialRoot:

Hypergeometric2F1 can be represented in terms of MeijerG:

TraditionalForm formatting:

Applications (3)

An expression for the force acting on an electric point charge outside a neutral dielectric sphere of radius :

The limit of infinite dielectric constant, corresponding to an uncharged insulated conducting sphere:

An approximation for the force at a large distance from the sphere:

Two players roll dice. If the total of both numbers is less than 10, the second player is paid 4 cents; otherwise, the first player is paid 9 cents. Is the game fair? Compute the probability that the first player gets paid:

The game is not fair, since mean scores per game are not equal:

Find the probability that after n games the player at the disadvantage scores more:

The probability exhibits oscillations:

The maximum probability is attained at :