Confluent Hypergeometric Function (First Kind) Calculator

Enter the parameters a and c with the argument z to compute the confluent hypergeometric function of the first kind (Kummer function) M(a;c;z)=₁F₁(a;c;z) from its power series, with term count and convergence shown.

Input

Enter the parameters a and c with the argument z to evaluate the confluent hypergeometric function M(a;c;z)=₁F₁(a;c;z) by its series.

Parameter a

Numerator parameter. Any real number is allowed.

Parameter c

Denominator parameter. Zero or negative integers are not allowed (poles).

Argument z

The point at which to evaluate the function. Any real number is allowed.

Result

Value of M(a=1; c=2; z=1)

1.7182818285

The series converged within the target tolerance.

Terms summed

Convergence

Converged

Last term magnitude

1.561921e-16

Series terms and partial sums

The value of each leading term and the running partial sum up to that term.

Index k	Term value	Partial sum
0	1	1
1	0.5	1.5
2	0.16666667	1.66666667
3	0.04166667	1.70833333
4	0.00833333	1.71666667
5	0.00138889	1.71805556
6	0.00019841	1.71825397
7	2.480159e-5	1.71827877
8	2.755732e-6	1.71828153
9	2.755732e-7	1.7182818
10	2.505211e-8	1.71828183
11	2.087676e-9	1.71828183

How it works

The confluent hypergeometric function of the first kind (Kummer function) is M(a;c;z)=₁F₁(a;c;z)=Σ (a)_k/((c)_k k!) z^k. Here (q)_k is the Pochhammer symbol (rising factorial), with (q)_0=1 and (q)_k=q(q+1)…(q+k−1).
This tool builds the terms iteratively using the ratio term_(k+1)=term_k×(a+k)/((c+k)(k+1))×z and accumulates the partial sum.
The series converges for every value of z (the radius of convergence is infinite), but when the magnitude of z or of a is large the intermediate terms can become very large and cancellation may reduce accuracy.
When c is zero or a negative integer, (c)_k becomes zero at some step and the function is undefined (a pole). In that case the tool reports that it cannot be evaluated.
The summation stops once a term is small enough relative to the partial sum, which is treated as convergence. If the term limit is reached without convergence, the value is approximate, so check the convergence status shown.
Special cases match known closed forms, for example M(a;a;z)=exp(z) when a equals c, which is useful for cross-checking results.

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