At a jump discontinuity in the square wave's Fourier series, what value does the partial sum approach?

Conditions

• The function satisfies Dirichlet conditions.
• There is a finite jump discontinuity at the point of evaluation.

Reasoning, step by step

1. Identify the left-hand limit $f(x^-)$ and right-hand limit $f(x^+)$ at the jump.
2. Calculate the average: $\frac{1}{2}(f(x^-) + f(x^+))$.
3. Compare this with the actual assigned value of the function at that point (which may be different).
4. Conclude that the series converges to this average, not necessarily the function's defined value.

Example

The script states: 'At a jump they approach the midpoint, zero in this example.' The card adds: 'Values assigned at isolated jump points do not affect the Fourier coefficients.'

Common misconceptions

• Expecting the Fourier series to converge to the exact value of the function at the discontinuity if it was defined there.
• Thinking the series diverges at jumps.

Watch the explanation

Explore next

• Does changing the value of the function at the single jump point alter the Fourier coefficients?
• How does this behavior differ from continuous points?