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nLab > Latest Changes: Cauchy principal value

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 14th 2017
    • (edited Nov 14th 2017)

    I want to be adding some details to Cauchy principal value. What’s a good reference? Say for the proof that up to addition of a delta-distribution, f(x)=pv(1x)f(x) = pv\left( \frac{1}{x}\right) is the unique distributional solution to xf=1x f = 1?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 14th 2017
    • (edited Nov 14th 2017)

    okay, I have added some content to Cauchy principal value.

    (Still lacking, though, the proof that PV(1/x)+cδ(x)PV(1/x) + c \delta(x) is the most general solution to xf(x)=1x f(x) = 1.)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 20th 2017

    Still lacking, though, the proof that

    Ah, that follows of course immediately with the characterization of extension of distributions to the point.

    I have also cross-linked now with step function, in fact I copied over the computation from there to Cauchy principal value: here

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2017
    • (edited Nov 27th 2017)

    I have made explicit the last step in the proof of PV(1x)=1x+i0 ++iπδPV\left(\frac{1}{x}\right) = \frac{1}{x + i 0^+} + i \pi \delta (here).

    Then I also complemented the Fourier expression of the Heaviside distribution with the converse statement (here) which is the “complex Schwinger parameterization”.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2017
    • (edited Nov 27th 2017)

    recorded a result from Gelfand-Shilov 66 on the Fourier transform of the principal value of powers of real quadratic forms (here).

    Also added as an example the observartion that this result immediately implies that the singular support of the Feynman propagator in any dimension is the light cone.

    Thanks to Igor Khavkine.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2017

    recorded another result from Gelfand-Shilov of this form, now for the Fourier transform of the delta distribution applied to the “mass shell” of a quadratic form (here)

    Also added as an example the observation that this result immediately implies that the singular support of the causal propagator in any dimension is the light cone.

  7. Corrected statement and proof about the wave front set of the distributions PV(1/x)+c \delta(x). The previous statement was wrong for c=±πic=\pm \pi i.

    Benjamin Delarue

    diff, v12, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 25th 2023

    I see, thanks.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeSep 29th 2023

    (Plemelj-Sochocki formula)

    diff, v13, current

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