/* ----------[ M a c s y m a ]---------- */ /* ---------- Initialization ---------- */ showtime: all$ prederror: false$ /* ---------- Determining Zero Equivalence ---------- */ /* The following expressions are all equal to zero */ sqrt(997) - (997^3)^(1/6); sqrt(999983) - (999983^3)^(1/6); radcan(%); (2^(1/3) + 4^(1/3))^3 - 6*(2^(1/3) + 4^(1/3)) - 6; radcan(%); cos(x)^3 + cos(x)*sin(x)^2 - cos(x); trigsimp(%); /* See Joel Moses, ``Algebraic Simplification: A Guide for the Perplexed'', _Communications of the Association of Computing Machinery_, Volume 14, Number 8, August 1971, 527--537. This expression is zero if Re(x) is contained in the interval ((4 n - 1)/2 pi, (4 n + 1)/2 pi) for n an integer: ..., (-5/2 pi, -3/2 pi), (-pi/2, pi/2), (3/2 pi, 5/2 pi), ... */ expr: log(tan(1/2*x + %pi/4)) - asinh(tan(x)); radcan(exponentialize(logcontract(logarc(expr)))); q: logcontract(trigsimp(halfangles(trigexpand(logarc(expr))))); assume(-%pi/2 < x, x < %pi/2)$ trigsimp(q); forget(-%pi/2 < x, x < %pi/2)$ /* Use a roundabout method---show that expr is a constant equal to zero */ dexpr: diff(expr, x); radcan(exponentialize(dexpr)); q: factor(trigreduce(dexpr)); assume(-%pi/2 < x, x < %pi/2)$ ratsimp(q); forget(-%pi/2 < x, x < %pi/2)$ ev(expr, x = 0); remvalue(expr, dexpr, q)$ log((2*sqrt(r) + 1)/sqrt(4*r + 4*sqrt(r) + 1)); logcontract(radcan(%)); (4*r + 4*sqrt(r) + 1)^(sqrt(r)/(2*sqrt(r) + 1)) * (2*sqrt(r) + 1)^(1/(2*sqrt(r) + 1)) - 2*sqrt(r) - 1; radcan(%); /* [Gradshteyn and Ryzhik 9.535(3)] */ 2^(1 - z)*gamma(z)*zeta(z)*cos(z*%pi/2) - %pi^z*zeta(1 - z); ratsimp(%); /* ---------- Quit ---------- */ quit();