Airfoil equation

Example

The airfoil equation asks for a function $\varphi$ on $(-1, 1)$ with $$\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\varphi(t)}{t - x}\, dt = f(x), \quad -1 < x < 1,$$ for a given function $f$, where $\mathrm{PV}$ is the Cauchy principal value. So it inverts the finite Hilbert transform, and it is a singular integral equation of the first kind. In thin airfoil theory the chord of the wing is scaled to $[-1, 1]$, and $\varphi$ is the density of vorticity along the chord. The right side $f$ is fixed by the free stream speed, the angle of attack and the slope of the camber line.

For $f$ smooth enough, the solutions are $$\varphi(x) = -\frac{1}{\pi \sqrt{1 - x^2}}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\sqrt{1 - t^2}\, f(t)}{t - x}\, dt + \frac{C}{\sqrt{1 - x^2}},$$ with an arbitrary constant $C$.

Check on Chebyshev polynomials. Let $T_n$ and $U_{n-1}$ be the Chebyshev polynomials of the first and second kind, $n \ge 1$. For $-1 < x < 1$ one has $$\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{T_n(t)}{\sqrt{1 - t^2}\,(t - x)}\, dt = U_{n-1}(x), \qquad \frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{\sqrt{1 - t^2}\, U_{n-1}(t)}{t - x}\, dt = -T_n(x).$$ For $f = U_{n-1}$ 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. The 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 $-1 < x < 1$.