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The expand command distributes products over sums. This is done for all polynomials. For quotients of polynomials, only sums in the numerator are expanded; products and powers are left alone. See the normal command for dealing with quotients of polynomials.

The expand command also expands most mathematical functions, including AngerJ , BesselI , BesselJ , BesselK , BesselY , Beta , ChebyshevT , ChebyshevU , CylinderD , CylinderU , CylinderV , Dirac , Ei , GAMMA , GegenbauerC , HankelH1 , HankelH2 , HermiteH , JacobiAM , JacobiCD , JacobiCN , JacobiCS , JacobiDC , JacobiDN , JacobiDS , JacobiNC , JacobiND , JacobiNS , JacobiP , JacobiSC , JacobiSD , JacobiSN , JacobiTheta1 , JacobiTheta2 , JacobiTheta3 , JacobiTheta4 , JacobiZeta , KelvinBei , KelvinBer , KelvinHei , KelvinHer , KelvinKei , KelvinKer , LaguerreL , LegendreP , LegendreQ , LerchPhi , Psi , StruveH , StruveL , WeberE , WeierstrassP , WeierstrassPPrime , WeierstrassSigma , WeierstrassZeta , Zeta , abs , arccos , arccosh , arccot , arccoth , arccsc , arccsch , arcsec , arcsech , arcsin , arcsinh , arctan , arctanh , bernoulli , binomial , cos , cosh , cot , coth , csc , csch , csgn , dilog , erf , erfc , euler , exp , factorial , ln , pochhammer , polylog , sec , sech , signum , sin , sinh , tan , and tanh .

The optional arguments expr1, expr2, ..., exprn are used to prevent particular subexpressions in expr from being expanded. To prevent all expressions from being expanded, use the frontend command: frontend(expand, [expr]);

The expand command is extensible: if the procedure `expand/f` is defined, then the call expand(f(x)); calls `expand/f`(x);

$\mathrm{expand}\left(\left(x+1\right)\left(x+2\right)\right)$

$x^2+3x+2$

$\mathrm{expand}\left(\frac{x+1}{x+2}\right)$

$\frac{x}{x+2}+\frac{1}{x+2}$

$\mathrm{expand}\left(\frac{1}{\left(x+1\right)x}\right)$

$\frac{1}{\left(x+1\right)x}$

$\mathrm{expand}\left(\sin \left(x+y\right)\right)$

$\sin \left(x\right)\cos \left(y\right)+\cos \left(x\right)\sin \left(y\right)$

$\mathrm{expand}\left(\cos \left(2x\right)\right)$

$2{\cos \left(x\right)}^2-1$

$\mathrm{expand}\left(\exp \left(a+\ln \left(b\right)\right)\right)$

${\ⅇ}^{a}b$

$\mathrm{expand}\left(\left(x+1\right)\left(y+z\right)\right)$

$xy+xz+y+z$

$\mathrm{expand}\left(\left(x+1\right)\left(y+z\right)\,x+1\right)$

$\left(x+1\right)y+\left(x+1\right)z$

$\mathrm{expand}\left(\ln \left(\frac{x}{{\left(1-x\right)}^2}\right)\right)$

$\ln \left(\frac{x}{{\left(1-x\right)}^2}\right)$

$\mathrm{assume}\left(x\,\mathrm{real}\right)$

$\mathrm{expand}\left(\ln \left(\frac{x}{{\left(1-x\right)}^2}\right)\right)$

$-\ln \left({\left(1-\mathrm{x~}\right)}^2\right)+\ln \left(\mathrm{x~}\right)$

$\mathrm{expand}\left(\mathrm{BesselJ}\left(2\,t\right)\right)$

$\frac{2\mathrm{BesselJ}\left(1\,t\right)}{t}-\mathrm{BesselJ}\left(0\,t\right)$

$\mathrm{expand}\left(\mathrm{LegendreQ}\left(2\,t\right)\right)$

$-\frac{\ln \left(t+1\right)}{4}+\frac{\ln \left(t-1\right)}{4}+\frac{3t^2\ln \left(t+1\right)}{4}-\frac{3t^2\ln \left(t-1\right)}{4}-\frac{3t}{2}$

$\mathrm{expand}\left(\mathrm{JacobiP}\left(3\,1\,\frac{1}{4}\,t\right)\right)$

$-\frac{307}{1024}-\frac{2247}{1024}t+\frac{1575}{1024}{t}^2+\frac{5075}{1024}{t}^3$

$\mathrm{ee}≔\left(a+b\right)z+\sin \left(a+b\right)+\exp \left(a+b\right)$

$\mathrm{ee}≔\left(a+b\right)z+\sin \left(a+b\right)+{\ⅇ}^{a+b}$

$\mathrm{expand}\left(\mathrm{ee}\right)$

$za+zb+\sin \left(a\right)\cos \left(b\right)+\cos \left(a\right)\sin \left(b\right)+{\ⅇ}^a{\ⅇ}^b$

$\mathrm{expand}\left(\mathrm{ee}\,\sin \left(a+b\right)\right)$

$za+zb+\sin \left(a+b\right)+{\ⅇ}^a{\ⅇ}^b$

$\mathrm{expand}\left(\mathrm{ee}\,\mathrm{op}\left(\mathrm{indets}\left(\mathrm{ee}\,\mathrm{specfunc}\left(\mathrm{anything}\,\exp \right)\right)\right)\right)$

$za+zb+\sin \left(a\right)\cos \left(b\right)+\cos \left(a\right)\sin \left(b\right)+{\ⅇ}^{a+b}$

$\mathrm{expand}\left(\mathrm{ee}\,\mathrm{op}\left(\mathrm{indets}\left(\mathrm{ee}\,\mathrm{specfunc}\left(\mathrm{anything}\,\left[\exp \,\sin \right]\right)\right)\right)\right)$

$za+zb+{\ⅇ}^{a+b}+\sin \left(a+b\right)$

$\mathrm{expand}\left(\mathrm{ee}\,\mathrm{op}\left(\mathrm{indets}\left(\mathrm{ee}\,\mathrm{function}\right)\right)\right)$

$za+zb+{\ⅇ}^{a+b}+\sin \left(a+b\right)$