You can make the Fourier series solution for a

QUESTION:

You can make the Fourier series solution for a periodically driven oscillator a bit tidier if you don't mind using complex numbers. Obviously the periodic force of Equation (5.90) can be written as f = Re(g), where the complex function g is cx) g(t) = fneinwt. n=0 Show that the real solution for the oscillator's motion can likewise be written as x = Re(z), where z(t),Ecneinwt n=0 and cn = fn (02 n2 co2 + 2i finco 0 This solution avoids our having to worry about the real amplitude An and phase shift 5, separately. (Of course An and 8n are hidden inside the complex number Ca.)