suivant: Intervals monter: Taylor and asymptotic expansions précédent: Résidu d'une expression en   Table des matières   Index

• an expression,
• the variable name the expression depends on,
• an integer n or a polynomial N,
• an integer p.
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 :
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.

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