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Padé expansion: pade

pade takes 4 arguments pade returns a rationnal fraction P/Q such that degree(P)< p and P/Q = f(mod xn+1) or P/Q = f(mod N). In the first case, it means that P/Q and f have the same Taylor expansion at 0 up to order n.
Input :
pade(exp(x),x,5,3)
Or :
pade(exp(x),x,x^6,3)
Output :
(3*x^2+24*x+60)/(-x^3+9*x^2-36*x+60)
To verify input :
taylor((3*x^2+24*x+60)/(-x^3+9*x^2-36*x+60))
Output :
1+x+1/2*x^2+1/6*x^3+1/24*x^4+1/120*x^5+x^6*order_size(x)
which is the 5th-order series expansion of exp(x) at x = 0.
Input :
pade((x^15+x+1)/(x^12+1),x,12,3)
Or :
pade((x^15+x+1)/(x^12+1),x,x^13,3)
Output :
x+1
Input :
pade((x^15+x+1)/(x^12+1),x,14,4)
Or :
pade((x^15+x+1)/(x^12+1),x,x^15,4)
Output :
(-2*x^3-1)/(-x^11+x^10-x^9+x^8-x^7+x^6-x^5+x^4- x^3-x^2+x-1)
To verify, input :
series(ans(),x=0,15)
Output :
1+x-x^12-x^13+2x^15+x^16*order_size(x)
then input :
series((x^15+x+1)/(x^12+1),x=0,15)
Output :
1+x-x^12-x^13+x^15+x^16*order_size(x)
These two expressions have the same 14th-order series expansion at x = 0.


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suivant: Intervals monter: Taylor and asymptotic expansions précédent: Résidu d'une expression en   Table des matières   Index
giac documentation written by Renée De Graeve and Bernard Parisse