real valued function ''s'' converse true transform odd symmetric function ''i'' ' 's'' imaginary valued function ''i'' ''s'' converse true history early form harmonic series dates back ancient babylonian mathematics used compute ephemerides tables astronomical positions ref name prestini ref name rota ref name neugebauer ref name brack classical greek concepts deferent epicycle ptolemaic system astronomy related fourier series see modern times variants discrete fourier transform used alexis clairaut
Formal science Mathematics Pure mathematics Analysis Fourier analysis