Abstract
Integral equations arise in many of mathematical physics. It is not our intention to give an exhaustive discussion of all available techniques. We would rather like to concentrate on those in which complex integration plays an essential role. Special attention is paid to Wiener–Hopf decomposition and its application to singular integral equations.
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Notes
- 1.
The path of integration from \(a\) to \(b\) on the real axis could be extended to an arc, or a collection of arcs, or a closed contour in the complex plane.
- 2.
The integral itself is convergent for \(-\pi /2 < \text {Im} \, \lambda < \pi /2\).
- 3.
Obviously, this integral can also be calculated directly by means of the contour integration technique.
- 4.
In a slightly different form, namely as a problem
$$\begin{aligned} f(x) = (1/2) \int _0^{\infty } dy \, \int _0^1 e^{- |x-y|/\mu }\; \frac{d\mu }{\mu } f(y) \end{aligned}$$(3.15)it is referred to as Milne problem [4].
- 5.
Please note that the zero at \(s=i\) is unimportant because we never have to divide anything by \(R_+(s)\). See also discussion above.
- 6.
The condition of vanishing at \(\infty \) can be relaxed, we would like to keep it for simplicity though.
- 7.
In fact, it is sufficient to require that \(R(s)\) be bounded on infinity.
- 8.
Closing the contour in the lower half-plane gives, of course, the same result.
- 9.
The computation of the first part is simple after making sure that
$$\begin{aligned} \int _0^\infty d z \, \frac{\ln z^2}{z^2 + \zeta ^2} = \frac{\pi }{\zeta } \ln \zeta , \, \, \, \, \,\int _0^\infty d z \, \frac{\ln (1 + z^2) }{z^2 + \zeta ^2} = \frac{\pi }{\zeta } \ln ( 1 + \zeta ). \end{aligned}$$ - 10.
For full details see for instance Sect. 12.32 of [3].
- 11.
This is on no account a generic feature and is absent for more complicated kernels.
- 12.
These boundary conditions are generated by the equation
$$\begin{aligned} f(x)=\lambda \int \limits _a^be^{-\eta |x-y|}f(y) dy . \end{aligned}$$For the ODE to look the same we need to modify the relation between \(\lambda \) and \(p\) as \(\eta ^2-2\eta \lambda =p^2\) and similarly for \(\kappa \). The factor \(\eta \), as long as it is positive, can be re-absorbed by the re-scaling \(x\rightarrow x/\eta \), \(p\rightarrow \eta p\), and \(\lambda \rightarrow \eta \lambda \).
- 13.
If two functions coincide on a line in an analyticity domain then they coincide everywhere.
- 14.
This type of equations arises, for instance, in the aerofoil theory, see [24].
- 15.
The integral
$$\begin{aligned} \int \limits _0^{\pi } \frac{d \theta }{x-\cos {\theta }} \end{aligned}$$can be very conveniently performed via residua after substitution \(z=e^{i\theta }\). The integration angle can be extended to \([0,2\pi ]\), that only doubles the integral. Then the \(z\)-integration is along a unit circle around the coordinate origin. For \(x>1\) the integrand has two poles \(z_{1,2}= x \pm \sqrt{x^2 - 1}\), of which only one lies within the circle. Finally we then obtain for the integral \(\pi /\sqrt{x^2 - 1}\).
- 16.
It can in general be used with all integrals of the type
$$\begin{aligned} \mathrm{P} \int \limits _{-1}^1 \frac{\sqrt{1-t^2}}{t-x} g(t) dt, \end{aligned}$$where \(g(t)\) is a rational function.
- 17.
Although this requirement could in principle be relaxed.
- 18.
Original equation
$$\begin{aligned} f(x) = \int \limits _{0}^x d t \, \frac{u(t)}{\sqrt{x-t}} . \end{aligned}$$was obtained by Abel in the study of the motion of a particle in the gravitational field sliding in the vertical plane along a given curve. \(f(x)\) then has a meaning of the time of descent from the highest point to the lowest point on the curve, \(ds = u(t)dt\) is the curve element.
- 19.
Setting \(y=(z-x)/(z-\xi )\) and using definitions (2.10) and (2.11) of the Beta function as well as the complement formula (2.6) for the Gamma function, we have:
$$\begin{aligned} \int \limits _{\xi }^z \frac{dx}{(z-x)^{1-\mu } (x - \xi )^{\mu }} = \int \limits _0^1 y^{\mu -1}(1-y)^{-\mu } dy =B(\mu , 1-\mu ) = \Gamma (\mu ) \Gamma (1-\mu ) = \frac{\pi }{\sin {(\mu \pi )}}. \end{aligned}$$ - 20.
This kind of equation emerges in the problem of transients, see for instance [25, 26].
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Gogolin, A.O., Tsitsishvili, E.G., Komnik, A. (2014). Integral Equations. In: Tsitsishvili, E., Komnik, A. (eds) Lectures on Complex Integration. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00212-5_3
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