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Fourier transform vs Fourier series

Short answer

Fourier transform: F(s)=∫_{-∞}^{∞} f(x) e^{-isx} dx, inverse (1/2π)∫ F(s) e^{isx} ds. Series represents periodic functions with discrete spectrum; transform handles non-periodic functions giving continuous spectrum. Example: transform of e^{-|ax|} is 2a/(a²+s²); a periodic function uses series.

Define Fourier transform of a function. How does it differ from Fourier series? Support your answer with suitable example.

Fourier transform

For a non-periodic, absolutely integrable function f(x), the Fourier transform is

F(s) = ∫_{−∞}^{∞} f(x) e^{−isx} dx

and the original function is recovered by the inverse transform

f(x) = (1/2π) ∫_{−∞}^{∞} F(s) e^{isx} ds.

(Some texts put 1/√(2π) in front of both formulas.) F(s) gives the amount of each frequency s in f(x), a continuous spectrum.

Fourier series vs Fourier transform

PointFourier seriesFourier transform
Applies toPeriodic functionsNon-periodic functions (or the limit of infinite period)
DomainInterval of one period [−L, L]Whole line (−∞, ∞)
SpectrumDiscrete (frequencies nπ/L)Continuous (all s)
RepresentationSum Σ cₙ e^{inπx/L}Integral ∫ F(s) e^{isx} ds
CoefficientsNumbers aₙ, bₙ or cₙA function F(s)
ConditionDirichlet conditionsf absolutely integrable

The transform is the limit of the series when the period 2L → ∞: the discrete frequencies come closer and the sum turns into an integral.

Examples

Fourier series. f(x) = x on (−π, π), period 2π:

x = 2 [ sin x − (sin 2x)/2 + (sin 3x)/3 − … ]

Fourier transform. f(x) = e^{−|x|} (not periodic):

F(s) = ∫ e^{−|x|} e^{−isx} dx = 2/(1 + s²)

The series gives a list of coefficients for a periodic function; the transform gives a continuous function of s for a single non-repeating pulse.

✓ Answer checked by Claude Opus 5.5 (high effort)

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