ABSTRACT
It is obvious from the previous section that you may arbitrarily deform the innite integration contour Γ in (1.27) within the marked strip, if necessary.
Now we are ready to apply the Wiener–Hopf method to the equation∫ ∞
0 K(x – ξ)ϕ(ξ) dξ = f (x), 0 < x < ∞. (1.28)
Equation (1.28) is equivalent to∫ ∞
–∞ K(x – ξ)ϕ+(ξ) dξ = f+(x) + f-(x), |x| < ∞, (1.29)
where ϕ+(x) =
{ ϕ(x), x > 0, 0, x < 0, f+(x) =
{ f (x), x > 0, 0, x < 0, (1.30)
and f-(x) is a certain additional unknown function. Equation (1.29) contains two unknown functions, ϕ+(x) and f-(x); however both of them can be determined, as we will see soon, from a single equation. It should be noted that we assume the functions f+(x) and f-(x) to satisfy properties 1◦ and 2◦, respectively, and the function K(x) the property 3◦, with τ+ > τ-. Then, by applying the Fourier transform to Eq. (1.29) and using the convolution theorem (see the previous section), we arrive at the following relation:
L(s) Φ+(s) = F+(s) + F-(s), τ-< τ < τ+, (1.31) where L(s) is the Fourier image of K(x).
