This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Hypergeometric2F1

 Hypergeometric2F1is the hypergeometric function .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• The function has the series expansion .
• For certain special arguments, Hypergeometric2F1 automatically evaluates to exact values.
• Hypergeometric2F1 has a branch cut discontinuity in the complex plane running from to .
Evaluate numerically:
Evaluate symbolically:
Plot :
Expand Hypergeometric2F1 in a Taylor series at the origin:
Evaluate numerically:
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Evaluate symbolically:
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Plot :
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Expand Hypergeometric2F1 in a Taylor series at the origin:
 Out[1]=
 Scope   (8)
Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Hypergeometric2F1 automatically evaluates to simpler functions for certain parameters:
Hypergeometric series terminates if either of the first two parameters is a negative integer:
Exact value of Hypergeometric2F1 at unity:
Expand Hypergeometric2F1 in a series around :
Give the result for an arbitrary symbolic direction :
Expand Hypergeometric2F1 in a series near :
 Applications   (1)
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:
Force at a large distance from the sphere:
Use FunctionExpand to expand Hypergeometric2F1 into other functions:
Find limits of Hypergeometric2F1 from below and above the branch cut:
Mathematica implicitly assumes variables to be generic complex numbers:
When is a positive integer, the above reduces to a polynomial, yielding a finite result:
is equivalent to for generic :
However, if is a negative integer, Hypergeometric2F1 returns a polynomial:
The discrete Kepler problem with initial conditions and can be solved as a hypergeometric function:
The energy depends on :
Finite norm states exist for an attractive potential with and :