Why does odd symmetry remove the constant and cosine terms from the Fourier expansion of a square wave?

Conditions

• The function is $2\pi$-periodic.
• The function is odd, meaning $f(-x) = -f(x)$.

Reasoning, step by step

1. Identify the parity of the square wave function.
2. Recall that the constant term is the integral of the function over one period.
3. Note that the integral of an odd function over a symmetric interval is zero.
4. Recall that cosine functions are even.
5. Observe that the product of an odd function and an even function is odd.
6. Conclude that the integrals for cosine coefficients are zero.

Example

The script states: 'Odd symmetry removes the constant and cosine terms from its Fourier expansion.'

Common misconceptions

• Believing that odd symmetry affects sine coefficients.
• Thinking that the constant term is removed because the function is discontinuous.

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Connected concepts

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