public static double s21ba(
	double x,
	double y,
	out int ifail
)

Public Shared Function s21ba ( _
	x As Double, _
	y As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double s21ba(
	double x, 
	double y, 
	[OutAttribute] int% ifail
)

static member s21ba : 
        x:float * 
        y:float * 
        ifail:int byref -> float 

s21ba returns a value of an elementary integral, which occurs as a degenerate case of an elliptic integral of the first kind.

s21ba calculates an approximate value for the integral where x≥0 and y≠0.

This function, which is related to the logarithm or inverse hyperbolic functions for y<x and to inverse circular functions if x<y, arises as a degenerate form of the elliptic integral of the first kind. If y<0, the result computed is the Cauchy principal value of the integral.

The basic algorithm, which is due to Carlson (1979) and Carlson (1988), is to reduce the arguments recursively towards their mean by the system:

The quantity Sn for n=0,1,2,3,… decreases with increasing n, eventually Sn∼1/4n. For small enough Sn the required function value can be approximated by the first few terms of the Taylor series about the mean. That is The truncation error involved in using this approximation is bounded by 16Sn6/1-2Sn and the recursive process is stopped when Sn is small enough for this truncation error to be negligible compared to the machine precision.

Within the domain of definition, the function value is itself representable for all representable values of its arguments. However, for values of the arguments near the extremes the above algorithm must be modified so as to avoid causing underflows or overflows in intermediate steps. In extreme regions arguments are prescaled away from the extremes and compensating scaling of the result is done before returning to the calling program.

You should consult the S class which shows the relationship of this function to the classical definitions of the elliptic integrals.