- #1
Tosh5457
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Hi, I don't understand why does n goes from -∞ to +∞ in the complex Fourier series, but it goes from n=1 to n=+∞ in the real Fourier series?
HallsofIvy said:"Real Fourier Series" are in the form [itex]\sum a_ncos(nx)+ b_nsin(nx)[tex]
cosine is an even function and sine is an odd function so that if we did use negative values for n, it wouldn't give us anything new: [itex]a_{-n}cos(-nx)+ b_n sin(-nx)= a_{-n}cos(nx)- b_{-n} sin(nx)[/itex] and would can then combine that with the corresponding "n" term: [itex]( a_n+ a_{-n})cos(nx)+ (b_n- b_{-n})sin(nx)[/itex]
Another, but equivalent, way of looking at it is that [itex]cos(nx)= (e^{inx}+ e^{-inx})/2[/itex] and [itex]sin(nx)= (e^{inx}+ e^{-inx})/2i[/itex] so that sin(nx) and cosine(nx) with only positive n includes exponentials with both positive and negative n.
A complex Fourier series is a mathematical representation of a periodic function as an infinite sum of complex exponential functions. It is used to analyze and represent the frequency components of a signal or function.
The formula for a complex Fourier series is given by:
f(x) = ∑[n = -∞ to +∞] (cn * ei*n*x)
where cn represents the complex coefficients and e represents the complex exponential function.
Complex Fourier series have numerous applications in mathematics, physics, engineering, and signal processing. They are used to analyze periodic signals, solve differential equations, and model physical systems.
A complex Fourier series includes both real and imaginary components, whereas a real Fourier series only includes real components. This means that a complex Fourier series can represent more complex functions and signals.
The convergence of a complex Fourier series depends on the properties of the function being represented. If the function is continuous and piecewise smooth, the complex Fourier series will converge to the function. However, for discontinuous functions, the complex Fourier series may only converge in a certain sense.