Laurent Series
Source inspiration: (Mathew 2000-2019).
Description
The Laurent series of a function around a center \(x_0\) is
\[ f(x)=\sum_{k=-\infty}^{\infty} a_k (x-x_0)^k, \]
which extends Taylor series by allowing negative powers. Those negative-power terms form the principal part and capture pole-like behavior. In many smooth cases centered away from singularities, the principal part is zero and the Laurent series reduces to an ordinary Taylor expansion.
Animations
The Laurent animation for this topic reuses the same 27 function/interval/center cases used on the Padé page, but draws Laurent-series truncations instead of rational Padé forms.
Each frame uses a two-sided truncation with both negative and positive powers at the same time,
\[ \sum_{k=-N}^{N} a_k (x-x_0)^k, \]
with \(N\) increasing frame by frame.
Each case below follows the same case ordering as the Padé suite (Case 01 through Case 27).
Case 01 — \(f(x)=\sqrt{x}\), \(x_0=1\), \(x\in[0,10.5]\)
Case 02 — \(f(x)=\sqrt{x}\), \(x_0=4\), \(x\in[0,10.5]\)
Case 03 — \(f(x)=\sqrt{x}\), \(x_0=5\), \(x\in[0,10.5]\)
Case 04 — \(f(x)=\dfrac{1}{\sqrt{1-x}}\), \(x_0=0\), \(x\in[-1.5,1.5]\)
Case 05 — \(f(x)=\log(x)\), \(x_0=1\), \(x\in[-0.5,4.1]\)
Case 06 — \(f(x)=\log(x)\), \(x_0=2\), \(x\in[-0.5,4.1]\)
Case 07 — \(f(x)=\log(1+x)\), \(x_0=0\), \(x\in[-1.5,1.5]\)
Case 08 — \(f(x)=\sin(x)\), \(x_0=0\), \(x\in[-3\pi,3\pi]\)
Case 09 — \(f(x)=\cos(x)\), \(x_0=0\), \(x\in[-3\pi,3\pi]\)
Case 10 — \(f(x)=\sin(x)\), \(x_0=\pi/4\), \(x\in[-7\pi/4,9\pi/4]\)
Case 11 — \(f(x)=\cos(x)\), \(x_0=\pi/3\), \(x\in[-5\pi/3,7\pi/3]\)
Case 12 — \(f(x)=\tan(x)\), \(x_0=0\), \(x\in[-\pi,\pi]\)
Case 13 — \(f(x)=e^x\), \(x_0=0\), \(x\in[-2,3]\)
Case 14 — \(f(x)=e^{-x}\cos(x)\), \(x_0=0\), \(x\in[-2,4]\)
Case 15 — \(f(x)=\cosh(x)\), \(x_0=0\), \(x\in[-4,4]\)
Case 16 — \(f(x)=\arctan(x)\), \(x_0=0\), \(x\in[-2,2]\)
Case 17 — \(f(x)=\arcsin(x)\), \(x_0=0\), \(x\in[-1.5,1.5]\)
Case 18 — \(f(x)=J_0(x)\), \(x_0=0\), \(x\in[-10.2,10.2]\)
Case 19 — \(f(x)=J_1(x)\), \(x_0=0\), \(x\in[-10.2,10.2]\)
Case 20 — \(f(x)=\dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}\), \(x_0=0\), \(x\in[-3,3]\)
Case 21 — \(f(x)=\dfrac{1}{2}+\dfrac{1}{2}\operatorname{erf}\!\left(\dfrac{x}{\sqrt{2}}\right)\), \(x_0=0\), \(x\in[-3,3]\)
Case 22 — \(f(x)=\Gamma(x)\), \(x_0=1\), \(x\in[-0.2,5.2]\)
Case 23 — \(f(x)=\Gamma(x)\), \(x_0=2\), \(x\in[-0.2,5.2]\)
Case 24 — \(f(x)=\Gamma(x)\), \(x_0=3\), \(x\in[-0.2,5.2]\)
Case 25 — \(f(x)=Y_0(x)\), \(x_0=10\), \(x\in[0,22]\)
Case 26 — \(f(x)=Y_0(x)\), \(x_0=5\), \(x\in[0,22]\)
Case 27 — \(f(x)=Y_0(x)\), \(x_0=2\), \(x\in[0,22]\)
