In the Fourier series expansion of the even function f(x), the cosine coefficients are equal to zero.

Question

In the Fourier series expansion of the even function f(x), the cosine coefficients  are equal to zero.

BlackTom AI · Accepted Answer

The statement concerns the Fourier series expansion of an even function f(x).

First, recall the standard Fourier series on a symmetric interval: for an even function, the series contains only cosine terms, and the sine terms vanish. This means the sine coefficients are zero for an even function, not the cosine coefficients.

Option analysis:
- The claim that the cosine coefficients are zero: This is generally false. While it is true that sine coefficients vanish for even functions, the cosine coefficients a_n generally are not zero. They encode the even part of f(x) through the projection onto cos(n x) basis functions. 
- Why sine terms vanish for even functions: sin(n x) are odd functions, and the product of an even function with an odd function is odd. When integrated over a symmetric interval, the integral of an odd function is zero, hence the sine coefficients b_n = 0 for even f. 
- Why cosine terms can be nonzero: cos(n x) are even, and their projection onto an even function can yield nonzero results, representing the even shape of f(x). The a_0 (the average value) and a_n terms reflect the amplitude of the even components.

Common misconceptions to avoid:
- Believing that evenness forces all coefficients to be zero; instead, only the sine coefficients vanish due to parity considerations.
- Thinking that only sine terms appear for even functions; the correct pattern is the opposite: cosine terms survive, sine terms drop out.

In summary, for an even function f(x), the Fourier series consists of cosine terms with nonzero coefficients (in general), while the sine coefficients are zero. Therefore, the statement that cosine coefficients are equal to zero is incorrect.

In the Fourier series expansion of the even function f(x), the cosine coefficients are equal to zero.判断题

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