Should higher order singular integral operators be implemented as (normal) derivatives of log|y-x|?

[MikaelSlevinsky]

One could then resolve the fundamental solution as A*log|y-x| + B, take the normal derivative wrt y, and this would be ∂A * log|y-x| + A * ∂(log|y-x|) + ∂B. If CauchyWeight(1) were ∂(log|y-x|), then this would be simplest. (Otherwise some other combinations and re-sampling would be required.)

[dlfivefifty]

CauchyWeight should always be 1/(-2pi_im_(x-z)) to avoid confusion. I could see adding an order to LogWeight.

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On 1 Dec 2015, at 12:58 AM, Richard Mikael Slevinsky [email protected] wrote:

One could then resolve the fundamental solution as A*log|y-x| + B, take the normal derivative wrt y, and this would be ∂A * log|y-x| + A * ∂(log|y-x|) + ∂B. If CauchyWeight(1) were ∂(log|y-x|), then this would be simplest. (Otherwise some other combinations and re-sampling would be required.)

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[MikaelSlevinsky]

LogWeight subsequently appeared in ApproxFun for logarithmic endpoint weights. It could be SingularWeight/DiagonalSingularity... and CauchyWeight a type alias.