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  "body": "\n## Summary\n\nPeriodic boundary conditions ask a solution on an interval $[a, b]$ to take the same value and the same derivatives at both ends.\nThey turn a [[boundary value problem|TM8VQJWGHZ]] on an interval into a problem on a circle.\nThey appear in the study of equations with periodic coefficients and in Fourier series.\nThe eigenfunctions of $-y''$ with these conditions are the sines and cosines.\n\n## Explanations\n\n### L1 human\n\nImagine bending the interval $[a, b]$ into a ring so that its two ends meet.\nA function that fits smoothly around the ring must have the same value and the same slope at both ends.\nThat is what periodic boundary conditions demand.\nFor example, on $[0, 2\\pi]$ the function $\\cos x$ satisfies them, since $\\cos 0 = \\cos 2\\pi$ and $-\\sin 0 = -\\sin 2\\pi$.\nThe function $x$ does not, since $0 \\ne 2\\pi$.\n",
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