TranscendentalManipulations(R, F)¶

TranscendentalManipulations provides functions to simplify and expand expressions involving transcendental operators.

cos2sec: F -> F: cos2sec(f) converts every cos(u) appearing in f into 1/sec(u).

cosh2sech: F -> F: cosh2sech(f) converts every cosh(u) appearing in f into 1/sech(u).

cot2tan: F -> F: cot2tan(f) converts every cot(u) appearing in f into 1/tan(u).

cot2trig: F -> F: cot2trig(f) converts every cot(u) appearing in f into cos(u)/sin(u).

coth2tanh: F -> F: coth2tanh(f) converts every coth(u) appearing in f into 1/tanh(u).

coth2trigh: F -> F: coth2trigh(f) converts every coth(u) appearing in f into cosh(u)/sinh(u).

csc2sin: F -> F: csc2sin(f) converts every csc(u) appearing in f into 1/sin(u).

csch2sinh: F -> F: csch2sinh(f) converts every csch(u) appearing in f into 1/sinh(u).

expand: F -> F: expand(f) performs the following expansions on f: begin{items} item 1. logs of products are expanded into sums of logs. item 2. trigonometric and hyperbolic trigonometric functions of sums are expanded into sums of products of trigonometric and hyperbolic trigonometric functions. item 3. formal powers of the form (a/b)^c are expanded into a^c * b^(-c). end{items}

expandLog: F -> F: expandLog(f) converts every log(a/b) appearing in f into log(a) - log(b), and every log(a*b) into log(a) + log(b).

expandPower: F -> F: expandPower(f) converts every power (a/b)^c appearing in f into a^c * b^(-c).

expandTrigProducts: F -> F if F has PatternMatchable R and R has ConvertibleTo Pattern R and R has PatternMatchable R and F has ConvertibleTo Pattern R: expandTrigProducts(e) replaces sin(x)*sin(y) by (cos(x-y)-cos(x+y))/2, cos(x)*cos(y) by (cos(x-y)+cos(x+y))/2, and sin(x)*cos(y) by (sin(x-y)+sin(x+y))/2. Note: this operation uses pattern matcher, so it is relatively expensive. To avoid getting into an infinite loop the transformations are applied at most ten times.

htrigs: F -> F: htrigs(f) converts all the exponentials in f into hyperbolic sines and cosines.

removeCoshSq: F -> F: removeCoshSq(f) converts every cosh(u)^2 appearing in f into 1 - sinh(u)^2, and also reduces higher powers of cosh(u) with that formula.

removeCosSq: F -> F: removeCosSq(f) converts every cos(u)^2 appearing in f into 1 - sin(u)^2, and also reduces higher powers of cos(u) with that formula.

removeSinhSq: F -> F: removeSinhSq(f) converts every sinh(u)^2 appearing in f into 1 - cosh(u)^2, and also reduces higher powers of sinh(u) with that formula.

removeSinSq: F -> F: removeSinSq(f) converts every sin(u)^2 appearing in f into 1 - cos(u)^2, and also reduces higher powers of sin(u) with that formula.

sec2cos: F -> F: sec2cos(f) converts every sec(u) appearing in f into 1/cos(u).

sech2cosh: F -> F: sech2cosh(f) converts every sech(u) appearing in f into 1/cosh(u).

simplify: F -> F: simplify(f) performs the following simplifications on f: begin{items} item 1. rewrites trigs and hyperbolic trigs in terms of sin , cos, sinh, cosh. item 2. rewrites sin^2 and sinh^2 in terms of cos and cosh, item 3. rewrites exp(a)*exp(b) as exp(a+b). item 4. rewrites (a^(1/n))^m * (a^(1/s))^t as a single power of a single radical of a. end{items}

simplifyExp: F -> F: simplifyExp(f) performs the following simplifications on f: begin{items} item 1. rewrites exp(a)*exp(b) as exp(a+b). item 2. rewrites a^b*a^c as a^(b+c). item 3. rewrites exp(a)/exp(b) as exp(a-b). item 4. rewrites a^b/a^c as a^(b-c). end{items}

simplifyLog: F -> F: simplifyLog(f) converts every log(a) - log(b) appearing in f into log(a/b), every log(a) + log(b) into log(a*b) and every n*log(a) into log(a^n).

sin2csc: F -> F: sin2csc(f) converts every sin(u) appearing in f into 1/csc(u).

sinh2csch: F -> F: sinh2csch(f) converts every sinh(u) appearing in f into 1/csch(u).

tan2cot: F -> F: tan2cot(f) converts every tan(u) appearing in f into 1/cot(u).

tan2trig: F -> F: tan2trig(f) converts every tan(u) appearing in f into sin(u)/cos(u).

tanh2coth: F -> F: tanh2coth(f) converts every tanh(u) appearing in f into 1/coth(u).

tanh2trigh: F -> F: tanh2trigh(f) converts every tanh(u) appearing in f into sinh(u)/cosh(u).