帕德近似1890

例如 1/(1-x) 的泰勒级数 1+x+x²+x³+⋯ 只有在 -1<x<1 时收敛，不如原函数广泛。

定义

给定自然数m和正整数n, 函数 f(x) 的 [m,n] 阶帕德近似为 （公式略）

并且 （公式略）

对于给定的 m,n 函数 f(x) 的 [m,n] 阶帕德近似是唯一的。

函数 f(x) 的帕德近似记为 [m/n]_(f)(x).

例子

正弦函数： （公式略）

[ [6/6]_(sin(x)) 的6+6=12阶泰勒级数展开为 x-(1/6)*x³+(1/120)*x⁵-(1/5040)*x⁷+(1/362880)*x⁹-(1/39916800)*x¹¹+O(x¹³)

和 sin(x) 的12阶泰勒级数全同： sin(x)≈ x-(1/6)*x³+(1/120)*x⁵-(1/5040)*x⁷+(1/362880)*x⁹-(1/39916800)*x¹¹+O(x¹³)

• 指数函数： （公式略）
• 其泰勒级数为 （公式略）
• 与exp(x)本身的泰勒级数展开的前10阶完全等同： （公式略）
• 又如 （公式略）
• [ （公式略）
• 雅可比椭圆函数： sn (x;3)
• − （公式略）
• 第一类 5 阶贝塞尔函数： J₅(x)
• − （公式略）
• 误差函数： （公式略）
• 菲涅耳积分： C(x)
• ( （公式略）

Maple计算

• Maple中
• pade(f(x),x,[m,n]);
• 其中 m,n 分别表示 分子、分母的级数；

In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series.

The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results, ad hoc methods—in some sense inspired by the Padé theory—typically replace them. Since a Padé approximant is a rational function, an artificial singular point may occur as an approximation, but this can be avoided by Borel–Padé analysis.

The reason the Padé approximant tends to be a better approximation than a truncating Taylor series is clear from the viewpoint of the multi-point summation method (see section 'Two-points Padé approximant', below). Since there are many cases in which the asymptotic expansion at infinity becomes 0 or a constant, it can be interpreted as the "incomplete two-point Padé approximation", in which the ordinary Padé approximation improves on the method of truncating a Taylor series.

Generalizations

A Padé approximant approximates a function in one variable. An approximant in two variables is called a Chisholm approximant (after J. S. R. Chisholm), in multiple variables a Canterbury approximant (after Graves-Morris at the University of Kent).

Examples

• sin(x) （公式略）
• exp(x) （公式略）
• ln(1+x) ln(1+x)≈ (x+1/2x²)/(1+x+1/6x²)
• Jacobi sn(z|3) （公式略）
• Bessel J5(x) （公式略）
• Error erf(x) （公式略）
• Fresnel C(x) （公式略）