transc1 http://www.openmath.org/cd/transc1.ocd 2003-04-01 2001-03-12 2 0 official alg1 arith1 interval1 logic1 nums1 quant1 relation1 set1 setname1 complex1 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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) = (ln(1+iz)-ln(1-iz))/2i 2 x in (-(pi/2),(pi/2)) implies arctan(tan x) = x 2 2 arcsec 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 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 This symbol represents the arccot function as described in Abramowitz and Stegun, section 4.4. arccot(-z) = - arccot(z) for all real x | arccot(x) = 1/(2*i) * ln ((x + i)/(x - i)) 2 arcsinh 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 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 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 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 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 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)