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  "body": "\n## Definition\n\nThe *Fresnel integrals* are the functions $S, C \\colon \\R \\to \\R$ given by the [[definite integrals|04TMP0XGMJ]]\n$$S(x) = \\int_0^x \\sin(t^2)\\, dt, \\qquad C(x) = \\int_0^x \\cos(t^2)\\, dt,$$\nwhere $\\sin$ and $\\cos$ are the [[sine and cosine|08WFZP6P89]].\n\n## Remarks\n\nExample: $S$ and $C$ are odd, since the integrands are even, and $S(0) = C(0) = 0$.\nThe curve $t \\mapsto (C(t), S(t))$ has velocity $(\\cos t^2, \\sin t^2)$ of length $1$, so $t$ is its arc length, and its direction angle $t^2$ gives curvature $2t$.\nThis curve is the Euler spiral.\n\nNon-example: $x \\mapsto \\int_0^x \\sin(t)^2\\, dt = \\frac{x}{2} - \\frac{\\sin 2x}{4}$ is not a Fresnel integral; the square must be inside the sine.\n\nAs $x \\to \\infty$, $S(x) \\to \\sqrt{\\pi/8}$ and $C(x) \\to \\sqrt{\\pi/8}$, as one sees by rotating the contour of the [[Gaussian integral|0DFT1GP1JP]].\nMany references use the normalisation $\\int_0^x \\sin\\left(\\frac{\\pi t^2}{2}\\right) dt$ and $\\int_0^x \\cos\\left(\\frac{\\pi t^2}{2}\\right) dt$ instead; then both limits are $\\frac12$.\n",
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