pade returns a rational 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 :
^6,3)Output :
^2+24*x+60)/(-x^3+9*x^2-36*x+60)To verify input :
^2+24*x+60)/(-x^3+9*x^2-36*x+60))Output :
^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 :
^15+x+1)/(x^12+1),x,12,3)or :
^15+x+1)/(x^12+1),x,x^13,3)Output :
Input :
^15+x+1)/(x^12+1),x,14,4)or :
^15+x+1)/(x^12+1),x,x^15,4)Output :
^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 :
Output :
^12-x^13+2x^15+x^16*order_size(x)then input :
^15+x+1)/(x^12+1),x=0,15)Output :
^12-x^13+x^15+x^16*order_size(x)These two expressions have the same 14th-order series expansion at x=0.