1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
|
'\" 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 2023-10-31 "Linux man-pages 6.06"
.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>
.P
.BI "double y0(double " x );
.BI "double y1(double " x );
.BI "double yn(int " n ", double " x );
.P
.BI "float y0f(float " x );
.BI "float y1f(float " x );
.BI "float ynf(int " n ", float " x );
.P
.BI "long double y0l(long double " x );
.BI "long double y1l(long double " x );
.BI "long double ynl(int " n ", long double " x );
.fi
.P
.RS -4
Feature Test Macro Requirements for glibc (see
.BR feature_test_macros (7)):
.RE
.P
.BR y0 (),
.BR y1 (),
.BR yn ():
.nf
_XOPEN_SOURCE
|| /* Since glibc 2.19: */ _DEFAULT_SOURCE
|| /* glibc <= 2.19: */ _SVID_SOURCE || _BSD_SOURCE
.fi
.P
.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 .
.P
The value of
.I x
must be positive.
.P
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 .
.P
If
.I x
is a NaN, a NaN is returned.
.P
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.)
.P
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.
.P
If the result underflows,
a range error occurs,
and the functions return 0.0
.P
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.
.P
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
.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.
.P
Before glibc 2.17,
.\" https://www.sourceware.org/bugzilla/show_bug.cgi?id=6808
did not set
.I errno
for "range error: result underflow".
.P
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)
|
