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Mathematics for Physics

315 Derivation of Taylor Expansion

A derivation that is used to obtain a polynomial approximation and exact remainder, where a remainder is the difference between the function and polynomial.

Taylor’s theorem. If a function is differentiable \(n+1\) times near \(a\), then it equals its Taylor polynomial plus a remainder.

The splitting into polynomial and remainder is

\[ f(x) = P_{n}(x) + R_{n+1}(x) \]

where

  • \(f\) is the function being expanded.
  • \(a\) is the expansion point.
  • \(x\) is a nearby evaluation point.
  • \(P_{n}(x)\) is the Taylor polynomial of degree \(n\).
  • \(R_{n+1}(x)\) is the remainder.

Taylor polynomial. A polynomial whose value and first \(n\) derivatives at \(a\) match those of \(f\).

The Taylor polynomial is

\[ P_{n}(x) = f(a) + f'(a)(x-a) + \dfrac{f''(a)}{2!}(x-a)^{2} + \cdots + \dfrac{f^{(n)}(a)}{n!}(x-a)^{n} \]

where

  • \(f^{(k)}(a)\) is the \(k\)th derivative of \(f\) at \(a\).
  • \(k!\) is the factorial of \(k\).

Remainder. The difference \(f(x)-P_{n}(x)\).

Auxiliary functions. Helper functions of a running point \(u\) that recast the remainder as a ratio of changes.

The auxiliary functions are

\[ \phi(u) = f(u) + f'(u)(x-u) + \dfrac{f''(u)}{2!}(x-u)^{2} + \cdots + \dfrac{f^{(n)}(u)}{n!}(x-u)^{n} \]

\[ \psi(u) = (x-u)^{n+1} \]

where

  • \(u\) is a running point between \(a\) and \(x\).
  • \(\phi(u)\) interpolates \(f\) from \(u\) toward \(x\).
  • \(\psi(u)\) is the \((n+1)\)st power of the remaining distance.

At the endpoints these helpers satisfy \(\phi(x)=f(x)\), \(\psi(x)=0\), \(\phi(a)=P_{n}(x)\), and \(\psi(a)=(x-a)^{n+1}\).

Cauchy’s mean-value theorem. The ratio of changes equals the ratio of derivatives at an interior point.

The Cauchy mean-value identity is

\[ \dfrac{\phi(x)-\phi(a)}{\psi(x)-\psi(a)} = \dfrac{\phi'(\xi)}{\psi'(\xi)} \]

where

  • \(\xi\) is a point strictly between \(a\) and \(x\).
  • \(\phi'\) and \(\psi'\) are the derivatives of the auxiliary functions.

Telescoping derivative. A derivative calculation that cancels all but the highest derivative term.

The derivatives of the auxiliary functions are

\[ \phi'(u) = \dfrac{f^{(n+1)}(u)}{n!}(x-u)^{n} \qquad \psi'(u) = -(n+1)(x-u)^{n} \]

where

  • \(f^{(n+1)}(u)\) is the \((n+1)\)st derivative of \(f\) at \(u\).

Lagrange remainder. The error expressed using the next derivative at an unknown interior point.

The Lagrange remainder is

\[ R_{n+1}(x) = \dfrac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1} \]

where

  • \(\xi\) is a point strictly between \(a\) and \(x\).
  • \(R_{n+1}(x)\) is the remainder after \(n\) Taylor terms.

Note: Also called Taylor’s formula. Also called Taylor’s theorem with remainder.

315.1 References

  1. Aleksandrov, A. D., Kolmogorov, A. N., & Lavrent’ev, M. A. Mathematics: Its Content, Methods and Meaning. Vol. 1. Dover, 1999. Chapter II, Section 9 — Taylor’s formula via Cauchy’s generalized mean-value theorem; Lagrange remainder.
  2. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. 5th ed. Matrix Editions, 2015. Appendix A12 — Taylor’s theorem with remainder.
  3. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists. 7th ed. Academic Press, 2013. Section 1.2 — Taylor’s expansion.