Taylor series

The Taylor series is a series that expands a real or complex infinitely differentiable function f(x)f(x) around a point x0x_{0} using its derivatives

f(x)=∑n=0∞f(n)(x0)n!(x−x0)n=f(x0)+f(1)(x0)(x−x0)+12f(2)(x0)(x−x0)2+…\begin{align} f(x)&=\sum_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!}(x-x_{0})^{n} \\ &=f(x_{0})+ f^{(1)}(x_{0})(x-x_{0})+ \frac{1}{2}f^{(2)}(x_{0})(x-x_{0})^{2}+\ldots \end{align}

where f(n)f^{(n)} represents the nn-th derivative of ff. If x0=0x_{0}=0 the series is also known as a Maclaurin series:

f(x)=∑n=0∞f(n)(0)n!xn=f(0)+f(1)(0)x+12f(2)(0)x2+…\begin{align} f(x)&=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^{n} \\ &=f(0)+f^{(1)}(0)x+ \frac{1}{2}f^{(2)}(0)x^{2}+\ldots \end{align}

N dimensions

For a function f(x)f(\mathbf{x}) in NN dimensions, the series centered at a point x0\mathbf{x}_{0} becomes

f(x)=∑d=0∞1d!Qd(x−x0)=∑d=0∞1d![∑∣α∣=d∣α∣!α!Dαf(x0)(x−x0)α]f(\mathbf{x})=\sum\limits_{d=0}^{\infty} \frac{1}{d!}Q_{d}(\mathbf{x}-\mathbf{x}_{0})=\sum\limits_{d=0}^{\infty} \frac{1}{d!}\left[\sum\limits_{|\alpha|=d} \frac{|\alpha|!}{\alpha!}D^{\alpha}f(\mathbf{x}_{0})(\mathbf{x}-\mathbf{x}_{0})^\alpha\right]

α\alpha is the multi-index, defined as a vector (α1,…,αN)(\alpha_{1},\ldots,\alpha_{N}) in NN\mathbb{N}^{N} where each component is used to represent the number of differentiations with respect to that index. We define two operations on it:

  1. Its norm is ∣α∣≡α1+α2+…+αN\lvert \alpha \rvert\equiv\alpha_{1}+\alpha_{2}+\ldots+\alpha_{N} and is called the length of the multi-index.
  2. Its factorial is α!=α1!α2!…αN!\alpha!=\alpha_{1}!\alpha_{2}!\ldots\alpha_{N}!.

The notation xα\mathbf{x}^{\alpha} represents the monomial x1α1x2α2…xNαNx_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\ldots x_{N}^{\alpha_{N}}. DαD^{\alpha} is the differentiation operator corresponding to that multi-index such that

Dα=∂∣α∣∂α1x1…∂αNxND^{\alpha}=\frac{ \partial ^{\lvert \alpha \rvert } }{ \partial^{\alpha_{1}}x_{1}\ldots\partial^{\alpha_{N}}x_{N} }

For example, in N3\mathbb{N}^{3}, the multi-index (1,0,2)(1,0,2) represents a differentiation with respect to the first variable, none with respect to the second, and two with respect to the third, so it corresponds to the operator

D(1,0,2)=∂3∂x1∂2x3D^{(1,0,2)}=\frac{ \partial ^{3} }{ \partial x_{1}\partial ^{2}x_{3} }

and the associated monomial would be x(1,0,2)=x1x32\mathbf{x}^{(1,0,2)}=x_{1}x_{3}^{2}. The polynomial QdQ_{d} is

Qd=∑∣α∣=d∣α∣!α!Dαf(x0)(x−x0)αQ_{d}=\sum_{\lvert \alpha \rvert =d} \frac{\lvert \alpha \rvert !}{\alpha!} D^{\alpha}f(\mathbf{x}_{0})(\mathbf{x}-\mathbf{x}_{0})^{\alpha}

which includes all possible dd-th degree derivatives.

In the case x0=0\mathbf{x}_{0}=\mathbf{0}, we have

f(x)=∑d=0∞1d!Qd(x)=∑d=0∞1d![∑∣α∣=d∣α∣!α!Dαf(0)xα]f(\mathbf{x})=\sum_{d=0}^{\infty} \frac{1}{d!}Q_{d}(\mathbf{x})=\sum_{d=0}^{\infty} \frac{1}{d!}\left[ \sum_{\lvert \alpha \rvert =d} \frac{\lvert \alpha \rvert !}{\alpha!} D^{\alpha}f(\mathbf{0})\mathbf{x}^{\alpha} \right]