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.\" Copyright 2002 Walter Harms (walter.harms@informatik.uni-oldenburg.de)
.\" Distributed under GPL
.\" Based on glibc infopages
.\" and Copyright 2008, Linux Foundation, written by Michael Kerrisk
.\"     <mtk.manpages@gmail.com>
.\" Modified 2004-11-15, fixed error noted by Fabian Kreutz
.\"      <kreutz@dbs.uni-hannover.de>
.\"
.\" Japanese Version Copyright (c) 2002 Akihiro MOTOKI
.\"         all rights reserved.
.\" Translated Sun Sep 22 09:31:52 2002
.\"         by Akihiro MOTOKI <amotoki@dd.iij4u.or.jp>
.\" Updated & Modified Sat Feb 12 12:00:00 2005
.\"         by SAITOH Akira <s-akira@users.sourceforge.net>
.\" Updated 2008-09-16, Akihiro MOTOKI <amotoki@dd.iij4u.or.jp>
.\"
.TH TGAMMA 3 2008-08-11 "GNU" "Linux Programmer's Manual"
.\"O .SH NAME
.SH ̾Á°
.\"O tgamma, tgammaf, tgammal \- true gamma function
tgamma, tgammaf, tgammal \- ËÜÅö¤Î¥¬¥ó¥Þ´Ø¿ô
.\"O .SH SYNOPSIS
.SH ½ñ¼°
.B #include <math.h>
.sp
.BI "double tgamma(double " x );
.br
.BI "float tgammaf(float " x );
.br
.BI "long double tgammal(long double " x );
.sp
.\"O Link with \fI\-lm\fP.
\fI\-lm\fP ¤Ç¥ê¥ó¥¯¤¹¤ë¡£
.sp
.in -4n
Feature Test Macro Requirements for glibc (see
.BR feature_test_macros (7)):
.in
.sp
.ad l
.BR tgamma (),
.BR tgammaf (),
.BR tgammal ():
_XOPEN_SOURCE\ >=\ 600 || _ISOC99_SOURCE; or
.I cc\ -std=c99
.ad b
.\"O .SH DESCRIPTION
.SH ÀâÌÀ
.\"O The Gamma function is defined by
.\"O .sp
.\"O     Gamma(x) = integral from 0 to infinity of t^(x\-1) e^\-t dt
.\"O .sp
.\"O It is defined for every real number except for nonpositive integers.
¥¬¥ó¥Þ´Ø¿ô¤Ï°Ê²¼¤Î¤è¤¦¤ËÄêµÁ¤µ¤ì¤ë:
.sp
     Gamma(x) = t^(x\-1) e^\-t dt ¤Î 0 ¤«¤é̵¸ÂÂç¤Þ¤Ç¤ÎÀÑʬ
.sp
¤³¤Î´Ø¿ô¤ÏÀµ¤Ç¤Ê¤¤À°¿ô¤ò½ü¤¯¤¹¤Ù¤Æ¤Î¼Â¿ô¤ËÂФ·¤ÆÄêµÁ¤µ¤ì¤Æ¤¤¤ë¡£
.\"O For nonnegative integral \fIm\fP one has
.\"O .sp
.\"O     Gamma(m+1) = m!
.\"O .sp
.\"O and, more generally, for all \fIx\fP:
.\"O .sp
.\"O     Gamma(x+1) = x * Gamma(x)
.\"O .sp
.\"O Furthermore, the following is valid for all values of \fIx\fP
.\"O outside the poles:
.\"O .sp
.\"O     Gamma(x) * Gamma(1 \- x) = PI / sin(PI * x)
.\"O .PP
ÈóÉé¤ÎÀ°¿ô \fIm\fP ¤Ë´Ø¤·¤Æ¡¢°Ê²¼¤¬À®Î©¤¹¤ë:
.sp
    Gamma(m+1) = m!
.sp
¤è¤ê°ìÈÌŪ¤Ë¤Ï¡¢¤¹¤Ù¤Æ¤Î \fIx\fP ¤Ë´Ø¤·¤Æ°Ê²¼¤¬À®Î©¤¹¤ë:
.sp
    Gamma(x+1) = x * Gamma(x)
.sp
¤µ¤é¤Ë¡¢¶Ë¤ò½ü¤¯¤¹¤Ù¤Æ¤Î \fIx\fP ¤Ç¼¡¼°¤âÀ®Î©¤¹¤ë:
.sp
    Gamma(x) * Gamma(1 \- x) = PI / sin(PI * x)
.PP
.\"O .SH RETURN VALUE
.SH ÊÖ¤êÃÍ
.\"O On success, these functions return Gamma(x).
À®¸ù¤¹¤ë¤È¡¢¤³¤ì¤é¤Î´Ø¿ô¤Ï Gamma(x) ¤òÊÖ¤¹¡£

.\"O If
.\"O .I x
.\"O is a NaN, a NaN is returned.
.I x
¤¬ NaN ¤Î¾ì¹ç¡¢NaN ¤¬ÊÖ¤µ¤ì¤ë¡£

.\"O If
.\"O .I x
.\"O is positive infinity, positive infinity is returned.
.I x
¤¬Àµ¤Î̵¸ÂÂç¤Î¾ì¹ç¡¢Àµ¤Î̵¸ÂÂ礬ÊÖ¤µ¤ì¤ë¡£

.\"O If
.\"O .I x
.\"O is a negative integer, or is negative infinity,
.\"O a domain error occurs,
.\"O and a NaN is returned.
.I x
¤¬Éé¤ÎÀ°¿ô¤«Éé¤Î̵¸ÂÂç¤Î¾ì¹ç¡¢Îΰ襨¥é¡¼ (domain error) ¤¬È¯À¸¤·¡¢
NaN ¤¬ÊÖ¤µ¤ì¤ë¡£

.\"O If the result overflows,
.\"O a range error occurs,
.\"O and the functions return
.\"O .BR HUGE_VAL ,
.\"O .BR HUGE_VALF ,
.\"O or
.\"O .BR HUGE_VALL ,
.\"O respectively, with the correct mathematical sign.
·ë²Ì¤¬¥ª¡¼¥Ð¡¼¥Õ¥í¡¼¤¹¤ë¾ì¹ç¡¢ÈÏ°Ï¥¨¥é¡¼ (range error) ¤¬È¯À¸¤·¡¢
³Æ´Ø¿ô¤Ï¤½¤ì¤¾¤ì
.BR HUGE_VAL ,
.BR HUGE_VALF ,
.BR HUGE_VALL ,
¤òÊÖ¤¹¡£¤³¤ÎºÝ¡¢¿ô³ØŪ¤ËÀµ¤·¤¤Éä¹æ¤¬ÉÕÍ¿¤µ¤ì¤ë¡£

.\"O If the result underflows,
.\"O a range error occurs,
.\"O and the functions return 0, with the correct mathematical sign.
·ë²Ì¤¬¥¢¥ó¥À¡¼¥Õ¥í¡¼¤¹¤ë¾ì¹ç¡¢ÈÏ°Ï¥¨¥é¡¼ (range error) ¤¬È¯À¸¤·¡¢
´Ø¿ô¤Ï 0 ¤ò¤òÊÖ¤¹¡£¤³¤ÎºÝ¡¢¿ô³ØŪ¤ËÀµ¤·¤¤Éä¹æ¤¬ÉÕÍ¿¤µ¤ì¤ë¡£

.\"O If
.\"O .I x
.\"O is -0 or +0,
.\"O a pole error occurs,
.\"O and the functions return
.\"O .BR HUGE_VAL ,
.\"O .BR HUGE_VALF ,
.\"O or
.\"O .BR HUGE_VALL ,
.\"O respectively, with the same sign as the 0.
.I x
¤¬ \-0 ¤« +0 ¤Î¾ì¹ç¡¢¶Ë¥¨¥é¡¼ (pole error) ¤¬È¯À¸¤·¡¢
³Æ´Ø¿ô¤Ï¤½¤ì¤¾¤ì
.BR HUGE_VAL ,
.BR HUGE_VALF ,
.BR HUGE_VALL ,
¤òÊÖ¤¹¡£
0 ¤ÈƱ¤¸Éä¹æ¤¬ÉÕÍ¿¤µ¤ì¤ë¡£
.\"O .SH ERRORS
.SH ¥¨¥é¡¼
.\"O See
.\"O .BR math_error (7)
.\"O for information on how to determine whether an error has occurred
.\"O when calling these functions.
¤³¤ì¤é¤Î´Ø¿ô¤ò¸Æ¤Ó½Ð¤·¤¿ºÝ¤Ë¥¨¥é¡¼¤¬È¯À¸¤·¤¿¤«¤ÎȽÄêÊýË¡¤Ë¤Ä¤¤¤Æ¤Î¾ðÊó¤Ï
.BR math_error (7)
¤ò»²¾È¤Î¤³¤È¡£
.PP
.\"O The following errors can occur:
°Ê²¼¤Î¥¨¥é¡¼¤¬È¯À¸¤¹¤ë²ÄǽÀ­¤¬¤¢¤ë¡£
.TP
.\"O Domain error: \fIx\fP is a negative integer, or negative infinity
Îΰ襨¥é¡¼: \fIx\fP ¤¬Éé¤ÎÀ°¿ô¤«Éé¤Î̵¸ÂÂç
.\" FIXME . errno is not set to EDOM for x == -inf
.\" Bug raised: http://sources.redhat.com/bugzilla/show_bug.cgi?id=6809
.\"O .I errno
.\"O is set to
.\"O .BR EDOM .
.\"O An invalid floating-point exception
.\"O .RB ( FE_INVALID )
.\"O is raised (but see BUGS).
.I errno
¤Ë
.B EDOM
¤¬ÀßÄꤵ¤ì¤ë¡£
ÉÔÀµ (invalid) ÉâÆ°¾®¿ôÅÀÎã³°
.RB ( FE_INVALID )
¤¬¾å¤¬¤ë (¡Ö¥Ð¥°¡×¤ÎÀá¤ò»²¾È)¡£
.TP
.\"O Pole error: \fIx\fP is +0 or \-0
¶Ë¥¨¥é¡¼ (pole error): \fIx\fP ¤¬ +0 ¤« \-0
.\"O .I errno
.\"O is set to
.\"O .BR ERANGE .
.\"O A divide-by-zero floating-point exception
.\"O .RB ( FE_DIVBYZERO )
.\"O is raised.
.I errno
¤Ë
.B ERANGE
¤¬ÀßÄꤵ¤ì¤ë¡£
0 ¤Ë¤è¤ë½ü»» (divide-by-zero) ÉâÆ°¾®¿ôÅÀÎã³°
.RB ( FE_DIVBYZERO )
¤¬¾å¤¬¤ë¡£
.TP
.\"O Range error: result overflow
ÈÏ°Ï¥¨¥é¡¼ (range error): ·ë²Ì¤Î¥ª¡¼¥Ð¡¼¥Õ¥í¡¼
.\"O .I errno
.\"O is set to
.\"O .BR ERANGE .
.\"O An overflow floating-point exception
.\"O .RB ( FE_OVERFLOW )
.\"O is raised.
.I errno
¤Ë
.B ERANGE
¤¬ÀßÄꤵ¤ì¤ë¡£
¥ª¡¼¥Ð¡¼¥Õ¥í¡¼ÉâÆ°¾®¿ôÅÀÎã³°
.RB ( FE_OVERFLOW )
¤¬¾å¤¬¤ë¡£
.PP
.\"O glibc also gives the following error which is not specified
.\"O in C99 or POSIX.1-2001.
glibc ¤Ç¤Ï¡¢C99 ¤ä POSIX.1-2001 ¤Çµ¬Äꤵ¤ì¤Æ¤¤¤Ê¤¤°Ê²¼¤Î¥¨¥é¡¼¤â
µ¯¤³¤êÆÀ¤ë¡£
.TP
.\"O Range error: result underflow
ÈÏ°Ï¥¨¥é¡¼ (range error): ·ë²Ì¤Î¥¢¥ó¥À¡¼¥Õ¥í¡¼
.\" e.g., tgamma(-172.5) on glibc 2.8/x86-32
.\" .I errno
.\" is set to
.\" .BR ERANGE .
.\"O An underflow floating-point exception
.\"O .RB ( FE_UNDERFLOW )
.\"O is raised.
.\"O .IP
.\"O .I errno
.\"O is not set for this case.
¥¢¥ó¥À¡¼¥Õ¥í¡¼ÉâÆ°¾®¿ôÅÀÎã³°
.RB ( FE_UNDERFLOW )
¤¬¾å¤¬¤ë¡£
¤³¤Î¾ì¹ç¤Ï
.I errno
¤ÏÀßÄꤵ¤ì¤Ê¤¤¡£
.\" FIXME . Is it intentional that errno is not set:
.\" Bug raised: http://sources.redhat.com/bugzilla/show_bug.cgi?id=6810
.\"
.\" glibc (as at 2.8) also supports and an inexact
.\" exception for various cases.
.\"O .SH VERSIONS
.SH ¥Ð¡¼¥¸¥ç¥ó
.\"O These functions first appeared in glibc in version 2.1.
¤³¤ì¤é¤Î´Ø¿ô¤Ï glibc ¥Ð¡¼¥¸¥ç¥ó 2.1 ¤Ç½é¤á¤ÆÅо줷¤¿¡£
.\"O .SH "CONFORMING TO"
.SH ½àµò
C99, POSIX.1-2001.
.\"O .SH NOTES
.SH Ãí°Õ
.\"O This function had to be called "true gamma function"
.\"O since there is already a function
.\"O .BR gamma (3)
.\"O that returns something else (see
.\"O .BR gamma (3)
.\"O for details).
¤³¤Î´Ø¿ô¤ò¡ÖËÜÅö¤Î (true) ¥¬¥ó¥Þ´Ø¿ô¡×¤È¸Æ¤Ð¤Ê¤±¤ì¤Ð¤Ê¤é¤Ê¤«¤Ã¤¿¡£
¤Ê¤¼¤Ê¤é¡¢Â¾¤ÎÃͤòÊÖ¤¹
.BR gamma (3)
¤È¤¤¤¦´Ø¿ô¤¬¤¹¤Ç¤Ë¸ºß¤¹¤ë¤«¤é¤Ç¤¢¤ë (¾ÜºÙ¤Ë¤Ä¤¤¤Æ¤Ï
.BR gamma (3)
¤ò»²¾È)¡£
.\"O .SH BUGS
.SH ¥Ð¥°
.\"O If
.\"O .I x
.\"O is negative infinity,
.\"O .I errno
.\"O is not set (it should be set to
.\"O .BR EDOM ).
.I x
¤¬Éé¤Î̵¸ÂÂç¤Î¾ì¹ç¡¢
.I errno
¤ÏÀßÄꤵ¤ì¤Ê¤¤
.RB ( EDOM
¤¬ÀßÄꤵ¤ì¤ë¤Ù¤­¤Ç¤¢¤ë)¡£
.\" Bug raised: http://sources.redhat.com/bugzilla/show_bug.cgi?id=6809

.\"O In glibc versions 2.3.3 and earlier,
.\"O an argument of +0 or -0 incorrectly produced a domain error
.\"O .RI ( errno
.\"O set to
.\"O .B EDOM
.\"O and an
.\"O .B FE_INVALID
.\"O exception raised), rather than a pole error.
glibc ¥Ð¡¼¥¸¥ç¥ó 2.3.3 °ÊÁ°¤Ç¤Ï¡¢
°ú¤­¿ô¤Ë +0 ¤ä \-0 ¤òÅϤ¹¤È¡¢¶Ë¥¨¥é¡¼¤Ç¤Ï¤Ê¤¯¡¢
Îΰ襨¥é¡¼¤ò´Ö°ã¤Ã¤ÆȯÀ¸¤·¤Æ¤¤¤¿
(Îΰ襨¥é¡¼¤Î¾ì¹ç¡¢
.I errno
¤Ë
.B EDOM
¤òÀßÄꤵ¤ì¡¢
.B FE_INVALID
Îã³°¤¬È¯À¸¤¹¤ë)¡£
.\"O .SH "SEE ALSO"
.SH ´ØÏ¢¹àÌÜ
.BR gamma (3),
.BR lgamma (3)