Over intervals, periodic functions can be infinite sine and cosine series {trigonometric series}.
Trigonometric series {Fourier series} {Fourier integral} can represent function over interval: (1 / pi) * (integral from -a = infinity to a = +infinity of F(a) * da) * (integral from a = 0 to a = infinity of cos(q * (x - a)) * da). Complex waveforms over time or position can be finite or infinite series of harmonic sine and cosine waves {Fourier analysis}|: f(x) = (2 * pi)^-0.5 * (integral from -infinity to +infinity of g(p) * e^(i*x*p) * dp), where g(p) is density {Fourier transform, series}. Complex function can have g(p) = 0 for p >= 0 {positive frequency function}.
convergence
If Fourier series is single-valued, has a bound, is piecewise continuous, and has finite numbers of discontinuities, maxima, and minima {Dirichlet condition}, it converges.
domain
Domain can be circle whose circumference is period or wavelength.
range
Series can be on unit circle in complex plane.
theorem
Fourier-series coefficients can exist and have properties {Parseval's theorem}.
Sines and cosines of Fourier series can be complex exponentials {Laurent series}: F(z) = F(e^i * a * x) = sum from -infinity to + infinity of Ar * z^r, where z is complex number, Ar is general term, and r is convergence radius. Laurent series has convergence annulus on Riemann sphere. Convergence circle can be for positive-term sum {positive frequency part}. Convergence circle can be for negative-term sum {negative frequency part}.
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Date Modified: 2022.0225