I have an integral of the form: I = \int\limits_{-1}^{1} f(x)*w(x) dx, The function f(x) has a singularity of the form 1/(x-a) inside the interval [-1, 1], so that the integral I exists only as a cauchy principal value. The weight function w(x) also has a singularity of the form log|x-b| at some p…

As explained here, you can do Cauchy principal-value integrals easily using any integration package (e.g. QuadGK.jl) using a simple transformation that eliminates the singularity: Cauchy principal value integrals by fmeirinhos · Pull Request #44 · JuliaMath/QuadGK.jl · GitHub

You can use SingularIntegralEquations.jl for Hilbert and log integrals: julia> using ApproxFun, SingularIntegralEquations julia> x = Fun() Fun(Chebyshev(),[0.0, 1.0]) julia> hilbert(exp(x), 0.1) # 1/π * ∫ exp(t)/(t-x) dt 0.6363129376196711 julia> logkernel(exp(x), 0.1) # 1/π * ∫ exp(t)*log(abs(t…

Can SingularIntegralEquations.jl deal with the situation when there are both 1/x and log|x| singularities present? Also, is it possible to get the points and weights of the quadrature separately (I know f(x) only numerically, although I have an analytical estimate for the singularity)?

It’s not really a “quadrature rule” as it first expands in a basis then evaluates the relevant singular integrals exactly. If you have data you can use a least squares fit: Frequently Asked Questions · ApproxFun.jl It might be possible to get a quadrature rule out of this a la Clenshaw–Curtis…

Thank you. In my case, I would prefer not to evaluate x*f(x) at singularity. I suppose, I should separate a symmetric interval centered on 1/x singularity and construct a symmetric quadrature on this interval. Then, the odd part of the integrand won’t contribute to the integral. However, there is…

[image] Gregstrq: In the situation when the singularities of f(x) and w(x) are well separated, I could treat them separately. What can be done in the situation when they are close to each other? In any case where you have multiple singularities, you must typically break the integral into piec…

Numerical integration of cauchy principal value

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Numerical integration of cauchy principal value

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