# Fourier's theorem

## Fourier's theorem

[‚fu̇r·ē‚āz ‚thir·əm]
(mathematics)
If ƒ(x) satisfies the Dirichlet conditions on the interval -π <>x < π,="" then="" its="" fourier="" series="" converges="" to="" ƒ(x="" for="" all="" values="" of="">x in this interval at which ƒ(x) is continuous, and approaches 1/2[ƒ(x + 0) + ƒ(x- 0)at points at which ƒ(x) is discontinuous, where ƒ(x- 0) is the limit on the left of ƒ at x and ƒ(x + 0) is the limit on the right of ƒ at x.
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References in periodicals archive ?
Mathematics: ABEGG'S RULE, ABEL'S THEOREM, ARCHIMEDES' PROBLEM, BERNOULLI'S THEOREM, DE MOIVRE'S THEOREM, DE MORGAN'S THEOREM, DESARGUES' THEOREM, DESCARTES' RULE OF SIGNS, EUCLID'S ALGORITHM, EULER'S EQUATION/FORMULA, FERMAT'S PRINCIPLE, FOURIER'S THEOREM, GAUSS'S THEOREM, GOLDBACH'S CONJECTURE, HUDDE'S RULES, LAPLACE'S EQUATIONS, NEWTON'S METHOD/PARALLELOGRAM, PASCAL'S LAW/TRIANGLE, RIEMANN'S HYPOTHESIS
of London) provides an introductory textbook for physicists and engineers on AC and DC circuits, phasors, complex numbers, impedance, feedback and operational amplifiers, the physics of transistor operation, digital logic, resonance and transients, gates, Fourier's theorem, and transformers and three-phase supplies.
This article describes sound waves, their basis in the sine curve, Fourier's theorem of infinite series, the fractal equation and its application to the composition of music, together with algorithms (such as those employed by meteorologist Edward Lorenz in his discovery of chaos theory) that are now being used to compose fractal music on computers (or synthesisers) today.
Fourier's theorem gives a mathematical description of any phenomenon, such as a sound wave, that can be considered as a "periodic" function of time, that is, a function that keeps on repeating a cycle of values (ibid.).
Fourier's theorem states (Devlin, 1994) that if y is a periodic function of time (that keeps on repeating some cycles of values) and if the frequency of its period is, say, 100 times per second, then y can be expressed as: y = 4 sin200[pi]t + 0.1 sin400[pi]t + 0.3 sin600 [pi]t + ...

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