'\" t
.\" Copyright 1993 David Metcalfe (david@prism.demon.co.uk)
.\" and Copyright 2008, Linux Foundation, written by Michael Kerrisk
.\"     <mtk.manpages@gmail.com>
.\"
.\" SPDX-License-Identifier: Linux-man-pages-copyleft
.\"
.\" References consulted:
.\"     Linux libc source code
.\"     Lewine's _POSIX Programmer's Guide_ (O'Reilly & Associates, 1991)
.\"     386BSD man pages
.\" Modified Sat Jul 24 19:08:17 1993 by Rik Faith (faith@cs.unc.edu)
.\" Modified 2002-08-25, aeb
.\" Modified 2004-11-12 as per suggestion by Fabian Kreutz/AEB
.\" 2008-07-24, mtk, created this page, based on material from j0.3.
.\"
.TH y0 3 2023-07-20 "Linux man-pages 6.05.01"
.SH NAME
y0, y0f, y0l, y1, y1f, y1l, yn, ynf, ynl \-
Bessel functions of the second kind
.SH LIBRARY
Math library
.RI ( libm ", " \-lm )
.SH SYNOPSIS
.nf
.B #include <math.h>
.PP
.BI "double y0(double " x );
.BI "double y1(double " x );
.BI "double yn(int " n ", double " x );
.PP
.BI "float y0f(float " x );
.BI "float y1f(float " x );
.BI "float ynf(int " n ", float " x );
.PP
.BI "long double y0l(long double " x );
.BI "long double y1l(long double " x );
.BI "long double ynl(int " n ", long double " x );
.fi
.PP
.RS -4
Feature Test Macro Requirements for glibc (see
.BR feature_test_macros (7)):
.RE
.PP
.BR y0 (),
.BR y1 (),
.BR yn ():
.nf
    _XOPEN_SOURCE
        || /* Since glibc 2.19: */ _DEFAULT_SOURCE
        || /* glibc <= 2.19: */ _SVID_SOURCE || _BSD_SOURCE
.fi
.PP
.BR y0f (),
.BR y0l (),
.BR y1f (),
.BR y1l (),
.BR ynf (),
.BR ynl ():
.nf
    _XOPEN_SOURCE >= 600
        || (_ISOC99_SOURCE && _XOPEN_SOURCE)
        || /* Since glibc 2.19: */ _DEFAULT_SOURCE
        || /* glibc <= 2.19: */ _SVID_SOURCE || _BSD_SOURCE
.fi
.SH DESCRIPTION
The
.BR y0 ()
and
.BR y1 ()
functions return Bessel functions of
.I x
of the second kind of orders 0 and 1, respectively.
The
.BR yn ()
function
returns the Bessel function of
.I x
of the second kind of order
.IR n .
.PP
The value of
.I x
must be positive.
.PP
The
.BR y0f (),
.BR y1f (),
and
.BR ynf ()
functions are versions that take and return
.I float
values.
The
.BR y0l (),
.BR y1l (),
and
.BR ynl ()
functions are versions that take and return
.I "long double"
values.
.SH RETURN VALUE
On success, these functions return the appropriate
Bessel value of the second kind for
.IR x .
.PP
If
.I x
is a NaN, a NaN is returned.
.PP
If
.I x
is negative,
a domain error occurs,
and the functions return
.RB \- HUGE_VAL ,
.RB \- HUGE_VALF ,
or
.RB \- HUGE_VALL ,
respectively.
(POSIX.1-2001 also allows a NaN return for this case.)
.PP
If
.I x
is 0.0,
a pole error occurs,
and the functions return
.RB \- HUGE_VAL ,
.RB \- HUGE_VALF ,
or
.RB \- HUGE_VALL ,
respectively.
.PP
If the result underflows,
a range error occurs,
and the functions return 0.0
.PP
If the result overflows,
a range error occurs,
and the functions return
.RB \- HUGE_VAL ,
.RB \- HUGE_VALF ,
or
.RB \- HUGE_VALL ,
respectively.
(POSIX.1-2001 also allows a 0.0 return for this case.)
.SH ERRORS
See
.BR math_error (7)
for information on how to determine whether an error has occurred
when calling these functions.
.PP
The following errors can occur:
.TP
Domain error: \fIx\fP is negative
.I errno
is set to
.BR EDOM .
An invalid floating-point exception
.RB ( FE_INVALID )
is raised.
.TP
Pole error: \fIx\fP is 0.0
.\" Before POSIX.1-2001 TC2, this was (inconsistently) specified
.\" as a range error.
.I errno
is set to
.B ERANGE
and an
.B FE_DIVBYZERO
exception is raised
(but see BUGS).
.TP
Range error: result underflow
.\" e.g., y0(1e33) on glibc 2.8/x86-32
.I errno
is set to
.BR ERANGE .
No
.B FE_UNDERFLOW
exception is returned by
.\" This is intended behavior
.\" See https://www.sourceware.org/bugzilla/show_bug.cgi?id=6806
.BR fetestexcept (3)
for this case.
.TP
Range error: result overflow
.\" e.g., yn(10, 1e-40) on glibc 2.8/x86-32
.I errno
is set to
.B ERANGE
(but see BUGS).
An overflow floating-point exception
.RB ( FE_OVERFLOW )
is raised.
.SH ATTRIBUTES
For an explanation of the terms used in this section, see
.BR attributes (7).
.TS
allbox;
lbx lb lb
l l l.
Interface	Attribute	Value
T{
.na
.nh
.BR y0 (),
.BR y0f (),
.BR y0l ()
T}	Thread safety	MT-Safe
T{
.na
.nh
.BR y1 (),
.BR y1f (),
.BR y1l ()
T}	Thread safety	MT-Safe
T{
.na
.nh
.BR yn (),
.BR ynf (),
.BR ynl ()
T}	Thread safety	MT-Safe
.TE
.sp 1
.SH STANDARDS
.TP
.BR y0 ()
.TQ
.BR y1 ()
.TQ
.BR yn ()
POSIX.1-2008.
.TP
Others:
BSD.
.SH HISTORY
.TP
.BR y0 ()
.TQ
.BR y1 ()
.TQ
.BR yn ()
SVr4, 4.3BSD,
POSIX.1-2001.
.TP
Others:
BSD.
.SH BUGS
Before glibc 2.19,
.\" http://sourceware.org/bugzilla/show_bug.cgi?id=6807
these functions misdiagnosed pole errors:
.I errno
was set to
.BR EDOM ,
instead of
.B ERANGE
and no
.B FE_DIVBYZERO
exception was raised.
.PP
Before glibc 2.17,
.\" https://www.sourceware.org/bugzilla/show_bug.cgi?id=6808
did not set
.I errno
for "range error: result underflow".
.PP
In glibc 2.3.2 and earlier,
.\" Actually, 2.3.2 is the earliest test result I have; so yet
.\" to confirm if this error occurs only in glibc 2.3.2.
these functions do not raise an invalid floating-point exception
.RB ( FE_INVALID )
when a domain error occurs.
.SH SEE ALSO
.BR j0 (3)