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* doc/misc/calc.texi (Advanced Math Functions): Correct calc

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algebraic syntax.
This commit is contained in:
Andreas Schwab 2023-06-15 16:18:58 +02:00
parent 92faf1f564
commit 0a1459f1d9
1 changed files with 5 additions and 5 deletions
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doc/misc/calc.texi
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@ -18613,7 +18613,7 @@ 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 @samp{gamma(a) = integ(t@sup{a-1} exp@sup{-t}, t, 0, inf)}.
@infoline @expr{gamma(a) = integ(t^(a-1) exp(-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 @samp{gammaP(a,x) = integ(t@sup{a-1} e@sup{t}, t, 0, x) / gamma(a)}.
@infoline @expr{gammaP(a,x) = integ(t^(a-1) exp(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 @samp{beta(a,b) = integ(t@sup{a-1} (1-t)@sup{b-1}, t, 0, 1)}.
@infoline @expr{beta(a,b) = integ(t^(a-1) (1-t)^(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 @samp{betaI(x,a,b) = integ(t@sup{a-1} (1-t)@sup{b-1}, t, 0, x) / beta(a,b)}.
@infoline @expr{betaI(x,a,b) = integ(t^(a-1) (1-t)^(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 @samp{erf(x) = 2 integ(exp(-(t@sup{2})), t, 0, x) / sqrt(pi)}.
@infoline @expr{erf(x) = 2 integ(exp(-(t^(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}.