# Padé expansion: pade

### 5.36.5 Padé expansion: pade

The pade command finds a rational expression which agrees with a function up to a given order.

- pade takes 4 arguments
  - expr, an expression.
  - x, the variable name.
  - n, an integer or R, a polynomial.
  - p, an integer.
- pade(expr,x,n,p) or pade(expr,x,P,p) returns a rational function P/Q such that degree(P)<p and P/Q=expr (mod xn+1 ) (meaning that P/Q and f have the same Taylor expansion at 0 up to order n) or P/Q=exprf (mod R ), respecively.

#### Examples.

- Input:
  pade(exp(x),x,5,3)
  or:
  pade(exp(x),x,x^6,3)
  Output:

  −3 x2−24 x−60
  x3−9 x2+36 x−60

  To verify:
  Input:
  taylor((3*x^2+24*x+60)/(-x^3+9*x^2-36*x+60))
  Output:

  1+x+

  x2
  2

  +

  x3
  6

  +

  x4
  24

  +

  x5
  120

  +x6 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 x3+1
  x11−x10+x9−x8+x7−x6+x5−x4+x3+x2−x+1

  To verify:
  Input:
  series(ans(),x=0,15)
  Output:

  1+x−x12−x13+2 x15+x16 order_size | ⎛ ⎝ | x | ⎞ ⎠

  Then:
  Input:
  series((x^15+x+1)/(x^12+1),x=0,15)
  Output:

  1+x−x12−x13+x15+x16 order_size | ⎛ ⎝ | x | ⎞ ⎠

  These two expressions have the same 14th-order series expansion at x=0.