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  "body": "\n## Example\n\nThe *airfoil equation* asks for a function $\\varphi$ on $(-1, 1)$ with\n$$\\frac{1}{\\pi}\\, \\mathrm{PV}\\!\\int_{-1}^{1} \\frac{\\varphi(t)}{t - x}\\, dt = f(x), \\quad -1 \u003c x \u003c 1,$$\nfor a given function $f$, where $\\mathrm{PV}$ is the [[Cauchy principal value|0SYR10HD4R]].\nSo it inverts the [[finite Hilbert transform|5JYBXH9BP6]], and it is a [[singular integral equation|E30EA6ETZ4]] of the first kind.\nIn thin airfoil theory the chord of the wing is scaled to $[-1, 1]$, and $\\varphi$ is the density of vorticity along the chord.\nThe right side $f$ is fixed by the free stream speed, the angle of attack and the slope of the camber line.\n\nFor $f$ smooth enough, the solutions are\n$$\\varphi(x) = -\\frac{1}{\\pi \\sqrt{1 - x^2}}\\, \\mathrm{PV}\\!\\int_{-1}^{1} \\frac{\\sqrt{1 - t^2}\\, f(t)}{t - x}\\, dt + \\frac{C}{\\sqrt{1 - x^2}},$$\nwith an arbitrary constant $C$.\n\nCheck on Chebyshev polynomials.\nLet $T_n$ and $U_{n-1}$ be the Chebyshev polynomials of the first and second kind, $n \\ge 1$.\nFor $-1 \u003c x \u003c 1$ one has\n$$\\frac{1}{\\pi}\\, \\mathrm{PV}\\!\\int_{-1}^{1} \\frac{T_n(t)}{\\sqrt{1 - t^2}\\,(t - x)}\\, dt = U_{n-1}(x), \\qquad \\frac{1}{\\pi}\\, \\mathrm{PV}\\!\\int_{-1}^{1} \\frac{\\sqrt{1 - t^2}\\, U_{n-1}(t)}{t - x}\\, dt = -T_n(x).$$\nFor $f = U_{n-1}$ the second identity turns the formula into $\\varphi = T_n / \\sqrt{1 - x^2}$ plus the free term, and the first identity confirms that this solves the equation.\nThe free term solves the homogeneous equation, since $\\frac{1}{\\pi}\\, \\mathrm{PV}\\!\\int_{-1}^{1} \\frac{dt}{\\sqrt{1 - t^2}\\,(t - x)} = 0$ for $-1 \u003c x \u003c 1$.\n\n## Remarks\n\nThe constant $C$ is not fixed by the equation.\nIn aerodynamics it is fixed by the Kutta condition, which asks that $\\varphi$ stay bounded at the trailing edge $x = 1$.\nThe lift of the airfoil is proportional to $\\int_{-1}^{1} \\varphi(t)\\, dt$.\n\n## Explanations\n\n### L1 human\n\nA thin wing in a steady flow is replaced by a line of small vortices along its chord.\nThe flow must slide along the wing, and this condition gives one equation for the strength of the vortices.\nThat equation is the airfoil equation, and it has an explicit solution.\nThe solution is unique only after one adds a physical condition at the rear edge of the wing.\n",
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