A numerical tour of wave propagation

Taylor and Páde expansion

The Taylor expansion of a function $f(x+h)$ at $x$ is written as

$$f(x+h)=f(x)+\frac{\partial f(x)}{\partial x}h+\frac{1}{2!}\frac{\... ...)}{\partial x^2}h^2+\frac{1}{3!}\frac{\partial^3 f(x)}{\partial x^3}h^3+\ldots.$$ (10)

A popular example is

$$(1+x)^{\alpha}=1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^2+\ldots+\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}x^n+\ldots.$$ (11)

Here we mainly consider the following expansion formula:

$$(1-x)^{\frac{1}{2}}=1-\frac{1}{2}x-\frac{1}{8}x^2-\frac{1}{16}x^3-\frac{5}{128}x^4-\ldots, \vert x\vert<1.$$ (12)

The Páde expansion of Eq. (12) follows from expansion in continuous fractions:

$$(1-x)^{\frac{1}{2}}=1-\frac{x/2}{1-\frac{x/4}{1-\frac{x/4}{\ldots}}}$$ (13)

I provide an informal derivation:

$$\begin{split} y&=(1-x)^{\frac{1}{2}}\Rightarrow x=1-y^2=(1-y... ...-\frac{x/2}{1-\frac{x/4}{1-\frac{x/2}{1+y}}}=\ldots \end{split}$$

The 1st-order Pade expansion is:

$$(1-x)^{\frac{1}{2}}=1-\frac{x}{2}$$ (14)

The 2nd-order Pade expansion is:

$$(1-x)^{\frac{1}{2}}=1-\frac{x/2}{1-\frac{x}{4}}.$$ (15)

And the 3rd-order one is:

$$(1-x)^{\frac{1}{2}}=1-\frac{x/2}{1-\frac{x/4}{1-\frac{x}{4}}}.$$ (16)

A numerical tour of wave propagation