Singular integral equation

Remarks

Example: on the unit circle, with $a = 0$, $b = 1$ and $k = 0$, the equation $\frac{1}{\pi i} \, \mathrm{p.v.} \int_\Gamma \frac{\varphi(\tau)}{\tau - t} \, d\tau = f(t)$ is of normal type, since $a^2 - b^2 = -1$. For $\varphi(\tau) = \tau^n$ with $n \ge 0$ the left side is $t^n$, and for $n < 0$ it is $-t^n$; so $f(t) = t^2$ is solved by $\varphi(\tau) = \tau^2$. Non-example: if the kernel is $k(t, \tau)$ alone, continuous, with $b = 0$ and $a = 1$, the equation is a Fredholm equation of the second kind and is not singular. The term $\frac{1}{\tau - t}$ makes the integral divergent as an ordinary integral at $\tau = t$ unless $\varphi(t) = 0$; the Hölder condition makes the principal value exist. More generally, the name covers equations with other kernels that are integrated as principal values, including kernels on open arcs and in several variables.