Fourier Coefficients: Complex with Sound -- Help Coefficients cn in the complex Fourier series f(t) = ... + c-2 e-4 i π ν t + c-1 e-2 i π ν t + c0 + c1 e2 i π ν t + c2 e4 i π ν t + .... are selected using the windows at bottom. The windows at lower right record the values of |cn| and Arg(cn) by yellow diamonds and struts. The diamond and strut for the chosen value of n are gray. A mouse click on either window at lower right sets the value of n , and also the value of either |cn| or Arg(cn) . Sliders at left also control these values. The window at lower left records the value of cn on the complex plane, and the value of cn can be chosen by a click and drag on that window. A slider at lower right selects a value of a parameter φ . The graphing window at the top displays the graph of Re(e-i φ f(t)) . Thus if φ = 0 , the real part of f(t) is displayed, while if φ = π / 2, the imaginary part of f(t) is displayed. Red vertical lines on this window also mark t = -L and t = +L , where L = 1 / (2ν) , so that 2L is the period. A cursor rollover generates crosshairs and a readout of values of t and x . At the top, a [f(t) real] key toggles the restriction on the values of the coefficients making f(t) real valued. When [f(t) real] is selected, c-n is reset to equal the complex conjugate of cn. A slider at the top sets the frequency ν , in kilohertz. At top right, a [Reset] key resets all the coefficients to zero. A [Play]/[Pause] key toggles a sound production, at the frequency ν kilohertz. Be sure the volume is low when [Play] is selected. Also be aware that pure sine waves carry more energy than they sound like they do, and prolonged exposure through headphones can damage the ear.   © 2008 H. Miller and J.-M. Claus