This document is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. The copyright holder grants you permission to redistribute this document freely as a verbatim copy. Furthermore, the copyright holder permits you to develop any derived work from this document provided that the following conditions are met. a) The derived work acknowledges the fact that it is derived from this document, and maintains a prominent reference in the work to the original source. b) The fact that the derived work is not the original OpenMath document is stated prominently in the derived work. Moreover if both this document and the derived work are Content Dictionaries then the derived work must include a different CDName element, chosen so that it cannot be confused with any works adopted by the OpenMath Society. In particular, if there is a Content Dictionary Group whose name is, for example, `math' containing Content Dictionaries named `math1', `math2' etc., then you should not name a derived Content Dictionary `mathN' where N is an integer. However you are free to name it `private_mathN' or some such. This is because the names `mathN' may be used by the OpenMath Society for future extensions. c) The derived work is distributed under terms that allow the compilation of derived works, but keep paragraphs a) and b) intact. The simplest way to do this is to distribute the derived work under the OpenMath license, but this is not a requirement. If you have questions about this license please contact the OpenMath society at http://www.openmath.org. transc3 http://www.openmath.org/cd http://www.openmath.org/cd/transc3.ocd 2006-03-30 2004-03-30 2 1 experimental This CD holds the definitions of many transcendental and related functions. They are defined as multi-valued functions with precise reductions to logs in the case of inverse functions. Note that we use the same names as in the single-valued case, even though it would be traditional to render them with capital letters. In sum <OMS cd="transc3" name="ln"/> is multi-valued, while <OMS cd="transc1" name="ln"/> is single-valued. Note that in many cases A+S only states the log restrictions under some circumstances: JHD has proved (22.8.2002) all the inverse trig. ones 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 is equivalent to b in Log_a c log 100 to base 10 (which is {2+2n\pi i}). ln application This symbol represents the ln function (natural logarithm) as a multivalued function. y in Ln(x) <=> exp(y)=x 2 Ln 1 (which is 0+2n\pi i). arcsin application This symbol represents the arcsin function. This is the multi-valued inverse of the sin function as described in Abramowitz and Stegun, section 4.4. It takes one argument. y in Arcsin(x) <=> sin(y)=x arcsin(z) = -i ln (sqrt(1-z^2)+iz), but the multivalued equivalent is Arcsin(z) = -i Ln (Sqrt(1-z^2)+iz), which we translate into OpenMath as Arcsin(z) = -i [ Ln (sqrt(1-z^2)+iz) union Ln (-sqrt(1-z^2)+iz)], Only stated in A+S for \z^2|\le 1, but proved for all z in JHD's OpenMath deliverable. 2 2 2 2 arccos application This symbol represents the arccos function. This is the multivalued inverse of the cos function. y in Arccos(x) <=> cos(y)=x arccos(z) = -i ln(z+i \sqrt(1-z^2)), so the multi-valued equivalent is Arccos(z) = -i Ln(z+i \Sqrt(1-z^2)), encoded as Arccos(z) = -i(ln(z+i \sqrt(1-z^2)) union ln(z-i \sqrt(1-z^2))) Only stated in A+S for \z^2|\le 1, but proved for all z in JHD's OpenMath deliverable. 2 2 2 2 arctan application This symbol represents the arctan function. This is the multi-valued inverse of the tan function. y in Arctan(x) <=> tan(y)=x arctan(z) = (i/2)ln((1-iz)/(1+iz)), so the multi-valued equivalent is Arctan(z) = (i/2)Ln((1-iz)/(1+iz)), 2 arcsec application This symbol represents the multivalued arcsec function as the inverse of sec. y in Arcsec(x) <=> sec(y)=x arcsec(z) = -i ln(1/z+i \sqrt(1-1/z^2)), so the multi-valued equivalent is Arcsec(z) = -i Ln(1/z+i \Sqrt(1-1/z^2)), encoded as Arcsec(z) = -i(ln(1/z+i \sqrt(1-1/z^2)) union ln(1/z-i \sqrt(1-1/z^2))) 2 2 2 2 arccsc application This symbol represents the multivalued arccsc function as the inverse of csc. y in Arccsc(x) <=> csc(y)=x arccsc(z) = -i ln (sqrt(1-1/z^2)+i/z), but the multivalued equivalent is Arccsc(z) = -i Ln (Sqrt(1-1/z^2)+i/z), which we translate into OpenMath as Arccsc(z) = -i [ Ln (sqrt(1-1/z^2)+i/z) union Ln (-sqrt(1-1/z^2)+i/z), 2 2 2 2 arccot application This symbol represents the multi-valued arccot function as the inverse of cot y in Arccot(x) <=> cot(y)=x arccot(-z) = - arccot(z) arccot(z) = (i/2)ln((1+iz)/(1-iz)), so the multi-valued equivalent is Arccot(z) = (i/2)Ln((1+iz)/(1-iz)), 2 arcsinh application This symbol represents the Arcsinh function as described in Abramowitz and Stegun, section 4.6. y in Arcsinh(x) <=> sinh(y)=x Arcsinh z = ln(z +-\sqrt(1+z^2)) 2 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. y in Arccosh(x) <=> cosh(y)=x Arccosh z = ln(z +-\sqrt(z^2-1)) 2 2 2 2 Arccosh(z) = i * Arccos(i * z) A+S says +/- i ..., but this is irrelevant since Arccos(iz)=-Arccos(iz) arctanh application This symbol represents the Arctanh function as described in Abramowitz and Stegun, section 4.6. y in Arctanh(x) <=> tanh(y)=x Arctanh(z) = - i * Arctan(i * z) for all x arctanh(x) = 1/2 * ln((1 + x)/(1 - x)) The condition 0\le x^2 < 1 in A+S is not necessary The proof for Arctan is in JHD's OpenMath deliverable 1 2 arcsech application This symbol represents the Arcsech function as described in Abramowitz and Stegun, section 4.6. y in Arcsech(x) <=> sech(y)=x Arcsech z = ln(1/z +-\sqrt(1/z^2-1)) 2 2 2 2 Arcsech(z) = i * Arcsec(i * z) A+S says +/- i ..., but this is irrelevant since Arcsec(iz)=-Arcsec(iz) arccsch application This symbol represents the Arccsch function as described in Abramowitz and Stegun, section 4.6. y in Arccsch(x) <=> csch(y)=x Arccsch z = ln(1/z +-\sqrt(1+1/z^2)) 2 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. y in Arccoth(x) <=> coth(y)=x Arccoth(z) = i * Arccot(i * z) for all x arccoth(x) = 1/2 * ln((x + 1)/(x - 1)) The condition 0\le x^2 < 1 in A+S is not necessary The proof for Arctan is in JHD's OpenMath deliverable 1 2