/usr/share/axiom-20170501/src/algebra/TRANFUN.spad is in axiom-source 20170501-3.
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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 | )abbrev category TRANFUN TranscendentalFunctionCategory
++ Author: Manuel Bronstein
++ Date Last Updated: 14 May 1991
++ Description:
++ Category for the transcendental elementary functions;
TranscendentalFunctionCategory() : Category == SIG where
TFC ==> TrigonometricFunctionCategory
ATFC ==> ArcTrigonometricFunctionCategory
HFC ==> HyperbolicFunctionCategory
AHFC ==> ArcHyperbolicFunctionCategory
EFC ==> ElementaryFunctionCategory
SIG ==> Join(TFC,ATFC,HFC,AHFC,EFC) with
pi : () -> $
++ pi() returns the constant pi.
add
if $ has Ring then
pi() == 2*asin(1)
acsch x ==
(a := recip x) case "failed" => error "acsch: no reciprocal"
asinh(a::$)
asech x ==
(a := recip x) case "failed" => error "asech: no reciprocal"
acosh(a::$)
acoth x ==
(a := recip x) case "failed" => error "acoth: no reciprocal"
atanh(a::$)
if $ has Field and $ has sqrt: $ -> $ then
asin x == atan(x/sqrt(1-x**2))
acos x == pi()/2::$ - asin x
acot x == pi()/2::$ - atan x
asinh x == log(x + sqrt(x**2 + 1))
acosh x == 2*log(sqrt((x+1)/2::$) + sqrt((x-1)/2::$))
atanh x == (log(1+x)-log(1-x))/2::$
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