Sokhotski–Plemelj Boundary Jump

The Sokhotski–Plemelj theorem connects the boundary values of analytic functions integrated over a real line to the Cauchy principal value and a singular jump term. When approaching a Cauchy singularity from opposite sides of the real axis, the imaginary component exhibits a discrete jump, proportional to the delta function.

Worked Example

Let us compute the regularized integral of 1/(t - x - i ε) over the domain [-1, 1] at x = 0.5 as ε approaches 0 from above.

The real part limits to the principal value: PV ∫ 1/(t-x) dt = ln | (1-x)/(1+x) |.
For x = 0.5, the principal value is ln(0.5 / 1.5) = ln(1/3) ≈ -1.0986.
The imaginary part is given by s π if x ∈ (-1, 1). For s = +1 (Upper approach), it limits to π ≈ 3.14159.
Hence, the limit boundary value converges to -1.0986 + i π.

Experiment

Using the interactive module Sokhotski-Plemelj Boundary Jump:

Select the Probe position along the real axis.
Select a value for the Width regularizer parameter.
Select the Approach side.
Observe the Real part and Imaginary part outputs, as well as the Imaginary boundary limit.

Limitations

The visualization uses a finite interval [-1, 1] instead of the entire real line, which causes the principal value to diverge logarithmically at the endpoints (± 1). Furthermore, the regularized kernels plotted represent the pointwise functions for a finite ε > 0; the strictly distributional limit (δ(x) and principal value) only acts rigorously when integrated against smooth test functions rather than as pointwise values at the singularity itself.

References

For a rigorous derivation using distributions, refer to:

Christian Remling, Lecture Notes on Distributions (Section on singular distributions)

Sokhotski–Plemelj Boundary Jump