Confluent Hypergeometric Function of the Second Kind U(a;c;z)

Compute the Tricomi function U(a;c;z) from the connection formula using Kummer's function M. Enter a, c, and z (z>0) to get U and its related M values.

Input

Enter a, c, and z (z＞0) to compute the confluent hypergeometric function of the second kind U(a;c;z), the Tricomi function, via the connection formula.

Parameter a

Real number. Numerator parameter of the series.

Parameter c

Real number. Values near integers are unstable.

Argument z

Enter a positive real number (z＞0).

Result

U(0.5; 1.5; 2)

0.7071067812

M(a; c; z)

2.36445389

M(a-c+1; 2-c; z)

First term

-0

Second term

0.70710678

Calculation breakdown

Values of each Kummer function M, each term, and the number of series terms used.

Quantity	Value
M(a; c; z)	2.36445389
M(a-c+1; 2-c; z)	1
First term G(1-c)/G(a-c+1)·M	-0
Second term G(c-1)/G(a)·z^(1-c)·M	0.70710678
Terms for M(a; c; z)	21
Terms for M(a-c+1; 2-c; z)	1

How it works

For non integer c, the second kind confluent hypergeometric function is U(a;c;z)=G(1-c)/G(a-c+1)·M(a;c;z)+G(c-1)/G(a)·z^(1-c)·M(a-c+1;2-c;z), where G is the gamma function and M is Kummer's function of the first kind.
Kummer's function M(a;c;z)=sum of (a)n/(c)n·z^n/n! is evaluated as a power series, where (q)n is the Pochhammer symbol and (a)0 equals 1.
Input is restricted to z>0. Smaller z converges faster; larger z needs more terms and may lose precision through cancellation.
When c is close to an integer the gamma factors in each term have poles that cancel in the limit, so this method becomes numerically unstable. Treat near integer c results as approximate.
A warning appears when the series does not settle within the term limit. The table lists each M value, each term, and the number of terms used.

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