naginterfaces.library.specfun.ellipint_​symm_​1_​degen

naginterfaces.library.specfun.ellipint_symm_1_degen(x, y)

ellipint_symm_1_degen returns a value of an elementary integral, which occurs as a degenerate case of an elliptic integral of the first kind.

For full information please refer to the NAG Library document for s21ba

https://www.nag.com/numeric/nl/nagdoc_30/flhtml/s/s21baf.html

Parameters

xfloat

The argument $x$ of the function.

yfloat

The argument $y$ of the function.

Returns

rcfloat

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

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$.

Constraint: $x\ge 0.0$.

The function is undefined.

(errno $2$)

On entry, $y=0.0$.

Constraint: $y\ne 0.0$.

The function is undefined and returns zero.

Notes

ellipint_symm_1_degen calculates an approximate value for the integral

$$R_C(x,y)=\frac{1}{2}{\int }_0^{\infty }\frac{dt}{(t+y).\sqrt{t+x}}$$

where $x\ge 0$ and $y\ne 0$.

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:

$$\begin{array}{c}\begin{array}{cc}{x}_0=x& y_0=y\\ {\mu }_n=(x_n+2y_n)/3\text{,}& S_n=(y_n-x_n)/3{\mu }_n\\ & {\lambda }_n=y_n+2\sqrt{x_n{y}_n}\\ x_{n+1}=(x_n+{\lambda }_n)/4\text{,}& y_{n+1}=(y_n+{\lambda }_n)/4\text{.}\end{array}\end{array}$$

The quantity $\left|S_n\right|$ for $n=0,1,2,3,\dots$ decreases with increasing $n$, eventually $\left|S_n\right|\sim 1/4^n$. For small enough $S_n$ the required function value can be approximated by the first few terms of the Taylor series about the mean. That is

$$R_C(x,y)=(1+\frac{3S_n^2}{10}+\frac{S_n^3}{7}+\frac{3S_n^4}{8}+\frac{9S_n^5}{22})/\sqrt{{\mu }_n}\text{.}$$

The truncation error involved in using this approximation is bounded by $16{\left|S_n\right|}^6/(1-2\left|S_n\right|)$ and the recursive process is stopped when $S_n$ 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.

References

NIST Digital Library of Mathematical Functions

Carlson, B C, 1979, Computing elliptic integrals by duplication, Numerische Mathematik (33), 1–16

Carlson, B C, 1988, A table of elliptic integrals of the third kind, Math. Comput. (51), 267–280