R f (ρ, θ) = \int_{R} F (τ^{'}, ρ, θ) d τ^{'}, H h (ρ) = \frac{1}{π} PV \int_{R} \frac{h ( ρ ^{'} )}{ρ - ρ ^{'}} d ρ^{'} .

P^{+} = (I + i H) /2

and

P^{-} = (- I + i H) /2

. Thus

P^{+}

and

- P^{-}

are the complementary idempotent projections;

P^{-}

itself carries the signed lower-boundary convention. The PDF defines the outside limit

λ^{-} = (1 + ε) e^{i θ}

but omits the definition of the inside limit in its displayed formula. We take

λ^{+} = (1 - ε) e^{i θ}

, as required by that formula.

r \to 1

, apply the Sokhotski–Plemelj formula to the kernel from part (b):

\frac{1}{u + i 0 v} = PV \frac{1}{u} - iπ sgn (v) δ (u) .

The principal value contribution is

i sgn (b_{r}) H R f /2

. The delta contribution is the same from either side and equals

\frac{1}{2} \int_{R} sgn (τ - τ^{'}) F (τ^{'}, ρ, θ) d τ^{'} = \frac{1}{2} R f - \int_{τ}^{\infty} F (τ^{'}, ρ, θ) d τ^{'} .

Since

b_{r} < 0

for the inside limit, the resulting boundary value is

μ (λ^{+}) = - P^{-} R f (ρ, θ) - \int_{τ}^{\infty} F (τ^{'}, ρ, θ) d τ^{'} .

The outside limit similarly has

P^{+} R f

in place of

- P^{-} R f

. Naming the signed convention explicitly prevents a sign error from treating

P^{-}

as a positive projection.

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