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emacs-29 1952dbd4c5f: Fix Gamma function definition in calc.texi
From: |
Eli Zaretskii |
Subject: |
emacs-29 1952dbd4c5f: Fix Gamma function definition in calc.texi |
Date: |
Thu, 15 Jun 2023 03:40:28 -0400 (EDT) |
branch: emacs-29
commit 1952dbd4c5fe7f82d6e0d10e1ba719b7e6422969
Author: Eli Zaretskii <eliz@gnu.org>
Commit: Eli Zaretskii <eliz@gnu.org>
Fix Gamma function definition in calc.texi
* doc/misc/calc.texi (Advanced Math Functions): Fix definition
of Gamma function. Use @sup in @infoline lines. (Bug#64077)
---
doc/misc/calc.texi | 12 ++++++------
1 file changed, 6 insertions(+), 6 deletions(-)
diff --git a/doc/misc/calc.texi b/doc/misc/calc.texi
index db6cad3450c..dae071bbd3b 100644
--- a/doc/misc/calc.texi
+++ b/doc/misc/calc.texi
@@ -18612,8 +18612,8 @@ gamma function. For positive integer arguments, this
is related to the
factorial function: @samp{gamma(n+1) = fact(n)}. For general complex
arguments the gamma function can be defined by the following definite
integral:
-@texline @math{\Gamma(a) = \int_0^\infty t^{a-1} e^t dt}.
-@infoline @expr{gamma(a) = integ(t^(a-1) exp(t), t, 0, inf)}.
+@texline @math{\Gamma(a) = \int_0^\infty t^{a-1} e^{-t} dt}.
+@infoline @samp{gamma(a) = integ(t@sup{a-1} exp@sup{-t}, t, 0, inf)}.
(The actual implementation uses far more efficient computational methods.)
@kindex f G
@@ -18647,7 +18647,7 @@ The @kbd{f G} (@code{calc-inc-gamma}) [@code{gammaP}]
command computes
the incomplete gamma function, denoted @samp{P(a,x)}. This is defined by
the integral,
@texline @math{P(a,x) = \left( \int_0^x t^{a-1} e^t dt \right) / \Gamma(a)}.
-@infoline @expr{gammaP(a,x) = integ(t^(a-1) exp(t), t, 0, x) / gamma(a)}.
+@infoline @samp{gammaP(a,x) = integ(t@sup{a-1} exp@sup{t}, t, 0, x) /
gamma(a)}.
This implies that @samp{gammaP(a,inf) = 1} for any @expr{a} (see the
definition of the normal gamma function).
@@ -18682,7 +18682,7 @@ Euler beta function, which is defined in terms of the
gamma function as
@infoline @expr{beta(a,b) = gamma(a) gamma(b) / gamma(a+b)},
or by
@texline @math{B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt}.
-@infoline @expr{beta(a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, 1)}.
+@infoline @samp{beta(a,b) = integ(t@sup{a-1} (1-t)@sup{b-1}, t, 0, 1)}.
@kindex f B
@kindex H f B
@@ -18692,7 +18692,7 @@ or by
The @kbd{f B} (@code{calc-inc-beta}) [@code{betaI}] command computes
the incomplete beta function @expr{I(x,a,b)}. It is defined by
@texline @math{I(x,a,b) = \left( \int_0^x t^{a-1} (1-t)^{b-1} dt \right) /
B(a,b)}.
-@infoline @expr{betaI(x,a,b) = integ(t^(a-1) (1-t)^(b-1), t, 0, x) /
beta(a,b)}.
+@infoline @samp{betaI(x,a,b) = integ(t@sup{a-1} (1-t)@sup{b-1}, t, 0, x) /
beta(a,b)}.
Once again, the @kbd{H} (hyperbolic) prefix gives the corresponding
un-normalized version [@code{betaB}].
@@ -18704,7 +18704,7 @@ un-normalized version [@code{betaB}].
The @kbd{f e} (@code{calc-erf}) [@code{erf}] command computes the
error function
@texline @math{\hbox{erf}(x) = {2 \over \sqrt{\pi}} \int_0^x e^{-t^2} dt}.
-@infoline @expr{erf(x) = 2 integ(exp(-(t^2)), t, 0, x) / sqrt(pi)}.
+@infoline @samp{erf(x) = 2 integ(exp(-(t@sup{2})), t, 0, x) / sqrt(pi)}.
The complementary error function @kbd{I f e} (@code{calc-erfc}) [@code{erfc}]
is the corresponding integral from @samp{x} to infinity; the sum
@texline @math{\hbox{erf}(x) + \hbox{erfc}(x) = 1}.
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