SingularIntegrals.jl

julia> using SingularIntegrals, ClassicalOrthogonalPolynomials

julia> P = Legendre(); x = axes(P, 1); f = expand(P, exp); #  expand exp(x) in Legendre polynomials

julia> @time inv.(10 .- x') * f # Stieltjes: ∫₋₁¹ exp(x) / (10-x) dx
  0.000034 seconds (22 allocations: 2.266 KiB)
0.24332755428373515

julia> @time inv.(0.1+0im .- x') * f - inv.(0.1-0im .- x') * f ≈ -2π*im*exp(0.1) # example of Plemelj
  0.000052 seconds (49 allocations: 6.031 KiB)
true

julia> @time abs.(10 .- x') .^ 0.2 * f # Power law: ∫₋₁¹ (10-x)^0.2 * exp(x) dx
  0.000077 seconds (21 allocations: 1.875 KiB)
3.7006631248289135

julia> @time abs.(0.3 .- x') .^ 0.2 * f # ∫₋₁¹ abs(0.3-x)^0.2 * exp(x) dx
  0.000040 seconds (25 allocations: 2.172 KiB)
1.9044201526740234

julia> W = Weighted(ChebyshevU()); f = expand(W, x -> exp(x) * sqrt(1-x^2));

julia> @time log.(abs.(10 .- x')) * f # Log-kernel: ∫₋₁¹ log(10-x) * exp(x) * sqrt(1-x^2) dx
  0.000040 seconds (14 allocations: 400 bytes)
4.043032838853287

julia> @time log.(abs.(0.3 .- x')) * f # ∫₋₁¹ log(abs(0.3-x)) * exp(x) * sqrt(1-x^2) dx
  0.000035 seconds (116 allocations: 6.250 KiB)
-2.320391559008445