A Padé rational approximation to f(x) on [a,b] is the quotient of two polynomials and of degrees n and m, respectively. We use the notation to denote this quotient:

        .  
        
We attribute much of the founding theory to Henri Eugène Padé (1863-1953).  

Theorem ( Padé Approximation ).  Assume that  , and that Maclaurin polynomial expansion of degree at least .  Then

    ,  

where and are polynomials of degree n and m, respectively.