Physlib

Physlib.SpaceAndTime.SpaceTime.SpaceTimeAlgebra.TaylorSeries

Taylor series of functions on spacetime

i. Overview

The Taylor series of a complex-valued function `f` on spacetime at a point `x₀` is the element of the spacetime algebra whose base-point derivative values are the iterated derivatives of `f` at `x₀`. For smooth `f` it is additive and `ℂ`-homogeneous in `f`, and it takes the derivative of `f` along a direction to the formal partial derivative of the series along the same direction.

ii. Key results

  • `SpaceTimeAlgebra.taylorSeries` : the Taylor series of a function at a point.
  • `SpaceTimeAlgebra.constantCoeff_iteratedPDeriv_taylorSeries` : derivative values of `f` at `x₀`.
  • `SpaceTimeAlgebra.taylorSeries_deriv` : derivatives become formal partial derivatives.

iii. Table of contents

  • A. The Taylor series of a function
  • B. Taylor series of derivatives

iv. References

* None.

A. The Taylor series of a function

The iterated derivatives `iteratedDeriv s f` of a function are indexed by multisets of directions, like the iterated formal partial derivatives of a series. The Taylor series of `f` at `x₀` is the series `ofDerivValues` built from their values at `x₀`, so that its coefficient of `x^s` is `(∂^s f)(x₀)` divided by `s!` and the base point of the spacetime algebra is identified with `x₀`. For smooth functions, iterated derivatives distribute over sums and scalar multiples, and so does the Taylor series.

B. Taylor series of derivatives

Applying `∂^s` after `∂_μ` is the same as applying `∂^(μ ::ₘ s)`, both for smooth functions and for series. Hence the Taylor series of `∂_μ f` and the formal partial derivative `pderiv μ` of the Taylor series of `f` have the same base-point derivative values, namely `(∂^(μ ::ₘ s) f)(x₀)` at each `s`. A series is determined by its base-point derivative values, so the two are equal.

6 declarations

definition

Taylor series of ff at x0x_0

For a complex-valued function f:SpaceTime→Cf: \text{SpaceTime} \to \mathbb{C} and a base point x0∈SpaceTimex_0 \in \text{SpaceTime}, the Taylor series of ff at x0x_0 is the element of the spacetime algebra C[[xμ]]\mathbb{C}[[x^\mu]] whose coefficients are determined by the values of the iterated derivatives of ff at x0x_0. Specifically, it is the formal power series whose coefficient for the monomial xmx^m is 1m!(∂mf)(x0)\frac{1}{m!} (\partial^m f)(x_0), where ∂m\partial^m denotes the iterated partial derivative corresponding to the multi-index mm.

theorem

The coefficient of xmx^m in the Taylor series of ff at x0x_0 is 1m!∂mf(x0)\frac{1}{m!} \partial^m f(x_0)

Let x0∈SpaceTimex_0 \in \text{SpaceTime} be a point and f:SpaceTime→Cf: \text{SpaceTime} \to \mathbb{C} be a complex-valued function. For any multi-index mm (mapping spacetime indices to natural numbers), the coefficient of the monomial xmx^m in the Taylor series of ff centered at x0x_0 is given by coeffm(taylorSeries(x0,f))=1m!(∂mf)(x0) \text{coeff}_m(\text{taylorSeries}(x_0, f)) = \frac{1}{m!} (\partial^m f)(x_0) where m!=∏νmν!m! = \prod_{\nu} m_\nu! and ∂mf(x0)\partial^m f(x_0) is the iterated partial derivative of ff evaluated at x0x_0 corresponding to the multi-index mm.

theorem

The constant coefficients of the formal derivatives of the Taylor series of ff at x0x_0 are the derivative values of ff at x0x_0

For a point x0x_0 in spacetime, a complex-valued function f:SpaceTime→Cf: \text{SpaceTime} \to \mathbb{C}, and a multiset ss of spacetime indices representing an iterated derivative, the constant coefficient of the formal iterated partial derivative ∂s\partial^s applied to the Taylor series of ff at x0x_0 is equal to the value of the iterated derivative ∂sf\partial^s f evaluated at x0x_0. In formulas: constantCoeff(∂s(taylorSeries x0f))=(∂sf)(x0)\text{constantCoeff}(\partial^s (\text{taylorSeries } x_0 f)) = (\partial^s f)(x_0)

theorem

Additivity of Taylor Series: taylorSeries(f+g)=taylorSeriesf+taylorSeriesg\text{taylorSeries}(f + g) = \text{taylorSeries} f + \text{taylorSeries} g

Let x0x_0 be a point in spacetime, and let f,g:SpaceTime→Cf, g: \text{SpaceTime} \to \mathbb{C} be infinitely differentiable (C∞C^\infty) functions. The Taylor series of the sum of these functions at x0x_0 is equal to the sum of their individual Taylor series at x0x_0, i.e., taylorSeries x0(f+g)=taylorSeries x0f+taylorSeries x0g.\text{taylorSeries } x_0 (f + g) = \text{taylorSeries } x_0 f + \text{taylorSeries } x_0 g.

theorem

The Taylor series of c⋅fc \cdot f equals cc times the Taylor series of ff

For any point x0∈SpaceTimex_0 \in \text{SpaceTime}, any complex scalar c∈Cc \in \mathbb{C}, and any smooth function f:SpaceTime→Cf: \text{SpaceTime} \to \mathbb{C}, the Taylor series of the function c⋅fc \cdot f at x0x_0 is equal to cc times the Taylor series of ff at x0x_0. The Taylor series is an element of the spacetime algebra C[[xμ]]\mathbb{C}[[x^\mu]].

theorem

Tx0(∂μf)=∂∂xμTx0(f)\mathcal{T}_{x_0}(\partial_\mu f) = \frac{\partial}{\partial x^\mu} \mathcal{T}_{x_0}(f)

Let f:SpaceTime→Cf : \text{SpaceTime} \to \mathbb{C} be a smooth (C∞C^\infty) function. For any point x0∈SpaceTimex_0 \in \text{SpaceTime} and any coordinate index μ∈{0,1,2,3}\mu \in \{0, 1, 2, 3\}, the Taylor series of the partial derivative ∂μf\partial_\mu f at x0x_0 is equal to the formal partial derivative of the Taylor series of ff at x0x_0 with respect to the μ\mu-th variable. That is, Tx0(∂μf)=∂∂xμTx0(f) \mathcal{T}_{x_0}(\partial_\mu f) = \frac{\partial}{\partial x^\mu} \mathcal{T}_{x_0}(f) where Tx0(f)\mathcal{T}_{x_0}(f) denotes the Taylor series of ff at x0x_0 as an element of the spacetime algebra C[[xμ]]\mathbb{C}[[x^\mu]], and the operator on the right denotes the formal derivative of the power series.