Editing Integral equation (section)

=== Regularity ===
{{Em|Regular}}: An integral equation is called regular if the integrals used are all proper integrals.<ref name=":1" />

{{Em|Singular}} or {{Em|weakly singular}}: An integral equation is called singular or weakly singular if the integral is an improper integral.<ref name=":1" /> This could be either because at least one of the limits of integration is infinite or the kernel becomes unbounded, meaning infinite, on at least one point in the interval or domain over which is being integrated.<ref name=":0" />

Examples include:<ref name=":0" /><math display="block">F(\lambda) = \int_{-\infty}^\infty e^{-i\lambda x} u(x) \, dx</math><math display="block">L[u(x)] = \int_0^\infty e^{-\lambda x} u(x) \, dx</math>These two integral equations are the Fourier transform and the Laplace transform of ''u''(''x''), respectively, with both being Fredholm equations of the first kind with kernel <math>K(x,t)=e^{-i\lambda x}</math> and <math>K(x,t)=e^{-\lambda x}</math>, respectively.<ref name=":0" /> Another example of a singular integral equation in which the kernel becomes unbounded is:<ref name=":0" /> <math display="block">x^2= \int_0^x \frac{1}{\sqrt{x-t}} \, u(t) \, dt.</math>This equation is a special form of the more general weakly singular Volterra integral equation of the first kind, called Abel's integral equation:<ref name=":1" /> <math display="block">g(x)=\int_a^{x} \frac{f(y)}{\sqrt{x-y}} \, dy</math>{{Em|Strongly singular}}: An integral equation is called strongly singular if the integral is defined by a special regularisation, for example, by the Cauchy principal value.<ref name=":1" />