. Domain of Representation

  • Taylor Series: Works best for local approximations around a single point (Maclaurin series if centered at zero).
  • Fourier Series: Represents a function over an entire interval (typically [−L,L][-L, L] or [0,2π][0, 2\pi]).

Key Differences Between Taylor Series and Fourier Series

Aspect Taylor Series Fourier Series
Nature of Expansion Uses polynomials from derivatives at a single point. Uses sines and cosines (or complex exponentials) over an interval.
Convergence Conditions Requires infinite differentiability and a valid radius of convergence. Requires periodicity and Dirichlet conditions for convergence.
Domain of Representation Local approximation around a single point. Represents a function over an entire interval.
Basis Functions Powers of (x – a). Sinusoids (sines and cosines) or complex exponentials.
Handling of Discontinuities Poor handling; requires smoothness. Can approximate discontinuous functions (with Gibbs phenomenon).
Applications Local function approximation, differential equations, numerical analysis. Signal processing, wave analysis, heat conduction, quantum mechanics.