Pendelreihen

Pendelreihen, also known as Fourier series, are mathematical representations of periodic functions as sums of oscillating terms. They were first introduced by Joseph Fourier in the early 19th century as part of his studies on heat conduction. A Fourier series decomposes a periodic function into a sum of sines and cosines, each with a specific frequency and amplitude.

The general form of a Fourier series for a function f(x) with period 2π is given by:

f(x) = a0/2 + ∑ [an * cos(nx) + bn * sin(nx)], n=1 to ∞

where a0, an, and bn are the Fourier coefficients, determined by the function f(x). The coefficients are calculated using integral formulas:

a0 = (1/π) ∫ from -π to π f(x) dx

an = (1/π) ∫ from -π to π f(x) * cos(nx) dx

bn = (1/π) ∫ from -π to π f(x) * sin(nx) dx

Pendelreihen have wide applications in various fields, including signal processing, image compression, and solving partial differential equations. They are particularly useful for analyzing and synthesizing periodic signals, as well as for approximating non-periodic functions using truncated Fourier series.

The convergence of a Fourier series to the original function depends on certain conditions, such as the function being piecewise continuous and having a finite number of discontinuities. If these conditions are met, the Fourier series converges pointwise to the function at all points where the function is continuous, and converges to the average of the left and right limits at points of discontinuity.