.\" Copyright 1993 David Metcalfe (david@prism.demon.co.uk)
.\" and Copyright 2008, Linux Foundation, written by Michael Kerrisk
.\"     <mtk.manpages@gmail.com>
.\"
.\" %%%LICENSE_START(VERBATIM)
.\" Permission is granted to make and distribute verbatim copies of this
.\" manual provided the copyright notice and this permission notice are
.\" preserved on all copies.
.\"
.\" Permission is granted to copy and distribute modified versions of this
.\" manual under the conditions for verbatim copying, provided that the
.\" entire resulting derived work is distributed under the terms of a
.\" permission notice identical to this one.
.\"
.\" Since the Linux kernel and libraries are constantly changing, this
.\" manual page may be incorrect or out-of-date.  The author(s) assume no
.\" responsibility for errors or omissions, or for damages resulting from
.\" the use of the information contained herein.  The author(s) may not
.\" have taken the same level of care in the production of this manual,
.\" which is licensed free of charge, as they might when working
.\" professionally.
.\"
.\" Formatted or processed versions of this manual, if unaccompanied by
.\" the source, must acknowledge the copyright and authors of this work.
.\" %%%LICENSE_END
.\"
.\" 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  2021-03-22 "" "Linux Programmer's Manual"
.SH NAME
y0, y0f, y0l, y1, y1f, y1l, yn, ynf, ynl \-
Bessel functions of the second kind
.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
Link with \fI\-lm\fP.
.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
.BR 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 http://sources.redhat.com/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
.BR 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).
.ad l
.nh
.TS
allbox;
lbx lb lb
l l l.
Interface	Attribute	Value
T{
.BR y0 (),
.BR y0f (),
.BR y0l ()
T}	Thread safety	MT-Safe
T{
.BR y1 (),
.BR y1f (),
.BR y1l ()
T}	Thread safety	MT-Safe
T{
.BR yn (),
.BR ynf (),
.BR ynl ()
T}	Thread safety	MT-Safe
.TE
.hy
.ad
.sp 1
.SH CONFORMING TO
The functions returning
.I double
conform to SVr4, 4.3BSD,
POSIX.1-2001, POSIX.1-2008.
The others are nonstandard functions that also exist on the BSDs.
.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
.BR ERANGE
and no
.B FE_DIVBYZERO
exception was raised.
.PP
Before glibc 2.17,
.\" http://sources.redhat.com/bugzilla/show_bug.cgi?id=6808
did not set
.I errno
for "range error: result underflow".
.PP
In glibc version 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 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)