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transc1
http://www.openmath.org/cd
http://www.openmath.org/cd/transc1.ocd
2006-03-30
2004-03-30
3
1
Author: OpenMath Consortium
SourceURL: https://github.com/OpenMath/CDs
official
This CD holds the definitions of many transcendental
functions. They are defined as in Abromowitz and Stegun (ninth
printing on), with precise reductions to logs in the case of
inverse functions.
Note that, if signed zeros are supported, some strict
inequalities have to become weak
.
It is intended to be `compatible' with the MathML elements
denoting trancendental functions.
Some additional functions are in the CD transc2.
log
application
This symbol represents a binary log function; the first argument is the base,
to which the second argument is log'ed.
It is defined in Abramowitz and Stegun, Handbook of Mathematical
Functions, section 4.1
a^b = c implies log_a c = b
log 100 to base 10 (which is 2).
ln
application
This symbol represents the ln function (natural logarithm) as
described in Abramowitz and Stegun, section 4.1. It takes one
argument. Note the description in the CMP/FMP of the branch cut. If
signed zeros are in use, the inequality needs to be non-strict.
-pi < Im ln x <= pi
ln 1 (which is 0).
exp
application
This symbol represents the exponentiation function as described in
Abramowitz and Stegun, section 4.2. It takes one argument.
for all k if k is an integer then e^(z+2*pi*k*i)=e^z
2
sin
application
This symbol represents the sin function as described in Abramowitz and
Stegun, section 4.3. It takes one argument.
sin(x) = (exp(ix)-exp(-ix))/2i
2
sin(A + B) = sin A cos B + cos A sin B
sin A = - sin(-A)
cos
application
This symbol represents the cos function as described in Abramowitz and
Stegun, section 4.3. It takes one argument.
cos(x) = (exp(ix)+exp(-ix))/2
2
cos 2A = cos^2 A - sin^2 A
2
2
2
cos A = cos(-A)
tan
application
This symbol represents the tan function as described in Abramowitz and
Stegun, section 4.3. It takes one argument.
tan A = sin A / cos A
sec
application
This symbol represents the sec function as described in Abramowitz and
Stegun, section 4.3. It takes one argument.
sec A = 1/cos A
csc
application
This symbol represents the csc function as described in Abramowitz and
Stegun, section 4.3. It takes one argument.
csc A = 1/sin A
cot
application
This symbol represents the cot function as described in Abramowitz and
Stegun, section 4.3. It takes one argument.
cot A = 1/tan A
sinh
application
This symbol represents the sinh function as described in Abramowitz
and Stegun, section 4.5. It takes one argument.
sinh A = 1/2 * (e^A - e^(-A))
2
cosh
application
This symbol represents the cosh function as described in Abramowitz
and Stegun, section 4.5. It takes one argument.
cosh A = 1/2 * (e^A + e^(-A))
2
tanh
application
This symbol represents the tanh function as described in Abramowitz
and Stegun, section 4.5. It takes one argument.
tanh A = sinh A / cosh A
sech
application
This symbol represents the sech function as described in Abramowitz
and Stegun, section 4.5. It takes one argument.
sech A = 1/cosh A
csch
application
This symbol represents the csch function as described in Abramowitz
and Stegun, section 4.5. It takes one argument.
csch A = 1/sinh A
coth
application
This symbol represents the coth function as described in Abramowitz
and Stegun, section 4.5. It takes one argument.
coth A = 1/tanh A
arcsin
application
This symbol represents the arcsin function. This is the inverse of the
sin function as described in Abramowitz and Stegun, section 4.4. It
takes one argument.
arcsin(z) = -i ln (sqrt(1-z^2)+iz)
2
2
x in [-(pi/2),(pi/2)] implies arcsin(sin x) = x
2
2
arccos
application
This symbol represents the arccos function. This is the inverse of the
cos function as described in Abramowitz and Stegun, section 4.4. It
takes one argument.
arccos(z) = -i ln(z+i \sqrt(1-z^2))
2
2
x in [0,pi] implies arccos(cos x) = x
arctan
application
This symbol represents the arctan function. This is the inverse of the
tan function as described in Abramowitz and Stegun, section 4.4. It
takes one argument.
arctan(z) = (i/2)ln((1-iz)/(1+iz))
2
x in (-(pi/2),(pi/2)) implies arctan(tan x) = x
2
2
arcsec
application
This symbol represents the arcsec function as described in Abramowitz
and Stegun, section 4.4.
arcsec(z) = -i ln(1/z + i \sqrt(1-1/z^2))
2
2
for all z | arcsec z = i * arcsech z
arccsc
application
This symbol represents the arccsc function as described in Abramowitz
and Stegun, section 4.4.
arccsc(z) = -i ln(i/z + \sqrt(1 - 1/z^2))
2
2
arccsc(z) = i * arccsch(i * z)
arccsc(-z) = - arccsc(z)
arccot
application
This symbol represents the arccot function as described in Abramowitz
and Stegun, section 4.4.
arccot(-z) = - arccot(z)
arccot(x) = (i/2) * ln ((x - i)/(x + i))
2
arcsinh
application
This symbol represents the arcsinh function as described in Abramowitz
and Stegun, section 4.6.
arcsinh z = ln(z + \sqrt(1+z^2))
2
2
arcsinh(z) = - i * arcsin(i * z)
arccosh
application
This symbol represents the arccosh function as described in Abramowitz
and Stegun, section 4.6.
arccosh(z) = 2*ln(\sqrt((z+1)/2) + \sqrt((z-1)/2))
2
2
2
2
2
arccosh z = i * (pi - arccos z)
arctanh
application
This symbol represents the arctanh function as described in Abramowitz
and Stegun, section 4.6.
arctanh(z) = - i * arctan(i * z)
for all x where 0 <= x^2 < 1 |
arctanh(x) = 1/2 * ln((1 + x)/(1 - x))
2
2
1
2
arcsech
application
This symbol represents the arcsech function as described in Abramowitz
and Stegun, section 4.6.
arcsech(z) = 2 ln(\sqrt((1+z)/(2z)) + \sqrt((1-z)/(2z)))
2
2
2
2
2
for all x in (0..1] | arcsech x = ln(1/x + (1/(x^2) - 1)^(1/2))
0 1
2
1 2
arccsch
application
This symbol represents the arccsch function as described in Abramowitz
and Stegun, section 4.6.
arccsch(z) = ln(1/z + \sqrt(1+(1/z)^2))
2
2
arccsch(z) = i * arccsc(i * z)
arccoth
application
This symbol represents the arccoth function as described in Abramowitz
and Stegun, section 4.6.
arccoth(z) = (ln(-1-z)-ln(1-z))/2
2
for all z | if z is not zero then arccoth(z) = i * arccot(i * z)