Gauss Hypergeometric Function 2F1 Calculator

Compute the Gauss hypergeometric function 2F1(a,b;c;z) by power series. Enter a, b, c, and z to see the value, convergence status, term count, and how the partial sums settle.

Input

Enter the parameters a, b, c and the variable z to evaluate the Gauss hypergeometric function 2F1(a,b;c;z) by its power series.

Parameter a

Parameter b

Parameter c

Not zero or a negative integer

Variable z

Converges when the absolute value is below 1

Result

Value of 2F1(a,b;c;z)

1.1787360798

at a = 0.5, b = 1, c = 1.5, z = 0.4

Convergence

In range

Terms summed

Absolute value of z

0.4

Last term size

1.101298e-15

How the partial sums settle

The horizontal axis is the term index and the vertical axis is the partial sum. The dashed line marks the final approximation.

Terms and partial sums

Shows each term value and the running partial sum for up to the first 12 terms.

Index k	Term value	Partial sum
0	1	1
1	0.13333333	1.13333333
2	0.032	1.16533333
3	0.00914286	1.17447619
4	0.00284444	1.17732063
5	0.00093091	1.17825154
6	0.00031508	1.17856662
7	0.00010923	1.17867585
8	3.855059e-5	1.1787144
9	1.379705e-5	1.1787282
10	4.993219e-6	1.17873319
11	1.823610e-6	1.17873501

How it works

The Gauss hypergeometric function is defined by 2F1(a,b;c;z) = sum over k of (a)_k (b)_k / ((c)_k k!) z^k. Here (q)_k is the rising Pochhammer symbol with (q)_0 = 1 and (q)_k = q(q+1)...(q+k-1).
The partial sums are built by multiplying the term ratio term(k+1)/term(k) = (a+k)(b+k) / ((c+k)(k+1)) z step by step.
The power series converges when the absolute value of z is less than 1. For absolute value of z at or above 1 it generally diverges, so out of range only a few reference terms are shown.
When c is zero or a negative integer the factor (c)_k contains a zero, so the function is undefined and the input is rejected.
The sum is truncated once a term falls below the tolerance or the term limit is reached, so the result is an approximation.

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