NAME

y0, y1, yn — Bessel functions of the second kind

SYNOPSIS

[XSI] #include <math.h>

double y0(double x);
double y1(double x);
double yn(int n, double x);

DESCRIPTION

The y0(), y1(), and yn() functions shall compute Bessel functions of x of the second kind of orders 0, 1, and n, respectively. y0(x) shall be equivalent to yn(0, x), and y1(x) shall be equivalent to yn(1, x).

An application wishing to check for error situations should set errno to zero and call feclearexcept(FE_ALL_EXCEPT) before calling these functions. On return, if errno is non-zero or fetestexcept(FE_INVALID | FE_DIVBYZERO | FE_OVERFLOW | FE_UNDERFLOW) is non-zero, an error has occurred.

RETURN VALUE

Upon successful completion, these functions shall return the relevant Bessel value of x of the second kind.

^[MXX] If x is NaN, NaN shall be returned.

If the x argument to these functions is negative, ^[MXX]  either NaN (if supported) or the same return value as when x is 0.0 (see below) shall be returned, and a domain error may occur.

If x is 0.0, y0() and y1() shall return -HUGE_VAL and a pole error may occur. If x is 0.0 and n is not both negative and odd, yn() shall return -HUGE_VAL and a pole error may occur. If x is 0.0 and n is negative and odd, yn() shall return +HUGE_VAL and a pole error may occur.

^[MXX] If x is +Inf, +0 shall be returned.

If the correct result would cause underflow ^[MXX]  and is not representable, a range error may occur, and the function shall return ^[MXX]  0.0, or (if the IEC 60559 Floating-Point option is not supported) an implementation-defined value no greater in magnitude than DBL_MIN.
^[MXX] If the correct result would cause underflow, and is representable, a range error may occur and the correct value shall be returned.

If the correct result of calling y1() would cause overflow, -HUGE_VAL shall be returned and a range error may occur. If n is not both negative and odd, and the correct result of calling yn() would cause overflow, -HUGE_VAL shall be returned and a range error may occur. If n is negative and odd, and the correct result of calling yn() would cause overflow, +HUGE_VAL shall be returned and a range error may occur.

ERRORS

These functions may fail if:

Domain Error

The value of x is negative.

If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [EDOM]. If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, then the invalid floating-point exception shall be raised.

Pole Error

The value of x is zero.

If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, then the divide-by-zero floating-point exception shall be raised.

Range Error

The value of x is too large in magnitude, or the correct result would cause underflow.

If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, then the underflow floating-point exception shall be raised.

The y1() and yn() functions may fail if:

Range Error

The correct result would cause overflow.

If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, then the overflow floating-point exception shall be raised.

EXAMPLES

None.

APPLICATION USAGE

On error, the expressions (math_errhandling & MATH_ERRNO) and (math_errhandling & MATH_ERREXCEPT) are independent of each other, but at least one of them must be non-zero.

RATIONALE

None.

FUTURE DIRECTIONS

None.

SEE ALSO

feclearexcept, fetestexcept, isnan, j0

XBD 4.23 Treatment of Error Conditions for Mathematical Functions, <math.h>

CHANGE HISTORY

First released in Issue 1. Derived from Issue 1 of the SVID.

Issue 5

The DESCRIPTION is updated to indicate how an application should check for an error. This text was previously published in the APPLICATION USAGE section.

Issue 6

The normative text is updated to avoid use of the term "must" for application requirements.

The RETURN VALUE and ERRORS sections are reworked for alignment of the error handling with the ISO/IEC 9899:1999 standard.

IEEE Std 1003.1-2001/Cor 2-2004, item XSH/TC2/D6/148 is applied, updating the RETURN VALUE and ERRORS sections. The changes are made for consistency with the general rules stated in "Treatment of Error Conditions for Mathematical Functions" in the Base Definitions volume of POSIX.1-2024.

Issue 7

POSIX.1-2008, Technical Corrigendum 1, XSH/TC1-2008/0746 [68] is applied.

Issue 8

Austin Group Defect 714 is applied, changing the behavior of these functions for special cases to be a better match for their mathematical behavior.