If the Fourier coefficient bn of a periodic function is zero, then it must possess even symmetry.

Here's why:

1. The Fourier series of a function is a way to represent the function as an infinite sum of sine and cosine waves. The coefficients of these waves, an and bn, are called the Fourier coefficients.

2. The bn coefficients are associated with the sine terms in the Fourier series. Sine is an odd function, meaning that it is symmetric about the origin.

3. If all the bn coefficients are zero, this means that there are no sine terms in the Fourier series.

4. Therefore, the function can only be composed of cosine terms, which are even functions.

5. Hence, if all the bn coefficients are zero, the function must possess even symmetry.

Question

If the Fourier coefficient bn of a periodic function is zero, then it must possess even symmetry.

Here's why:

1. The Fourier series of a function is a way to represent the function as an infinite sum of sine and cosine waves. The coefficients of these waves, an and bn, are called the Fourier coefficients.

2. The bn coefficients are associated with the sine terms in the Fourier series. Sine is an odd function, meaning that it is symmetric about the origin.

3. If all the bn coefficients are zero, this means that there are no sine terms in the Fourier series.

4. Therefore, the function can only be composed of cosine terms, which are even functions.

5. Hence, if all the bn coefficients are zero, the function must possess even symmetry.

Knowee AI · Accepted Answer

If the Fourier coefficient bn of a periodic function is zero, then it must possess even symmetry.

Here's why:

1. The Fourier series of a function is a way to represent the function as an infinite sum of sine and cosine waves. The coefficients of these waves, an and bn, are called the Fourier coefficients.

2. The bn coefficients are associated with the sine terms in the Fourier series. Sine is an odd function, meaning that it is symmetric about the origin.

3. If all the bn coefficients are zero, this means that there are no sine terms in the Fourier series.

4. Therefore, the function can only be composed of cosine terms, which are even functions.

5. Hence, if all the bn coefficients are zero, the function must possess even symmetry.

If the fourier coefficient bn of a periodic function is zero, then it must possess ----------- symmetry.

Question

If the fourier coefficient `b_n` of a periodic function is zero, then it must possess ----------- symmetry.

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