Question

Find the Fourier Cosine Series for the given function.

(a) 1/2 a₀ + ∑(n=1 to [infinity]) aₙ cos(nx)
(b) a₀ + ∑(n=1 to [infinity]) aₙ cos(nx)
(c) 1/2 a₀ + ∑(n=1 to [infinity]) aₙ sin(nx)
(d) a₀ + ∑(n=1 to [infinity]) aₙ sin(nx)

Answer 1

The Fourier Cosine Series for a function defined on [0, L] is a_0/2 + sum(a_n cos(nπx/L)) for n=1 to infinity, with coefficients a_0 and a_n determined by specific integral formulas.

Step-by-step explanation:

The question seems to be asking to find the Fourier Cosine Series of a given function. The Fourier Cosine Series for a function defined on [0, L] is a_0/2 + sum(a_n cos(nπx/L)) for n=1 to infinity, with coefficients a_0 and a_n determined by specific integral formulas. For a periodic function f(x), defined in the interval [0,L], the Fourier Cosine Series is of the form a0/2 + ∑(an cos(nπx/L)) for n=∞.

The coefficients a0 and an are calculated using integral formulas where a0 = (2/L)∫0Lf(x)dx and an = (2/L)∫0Lf(x) cos(nπx/L)dx for n=1,2,3,... The cosine function and its phase shift implications pertain to how the function may be horizontally translated, which is relevant when considering the cos(nx) term in the series.