Manuscript I-B-VP-2@1 · I-B-VP-2-v1.tex · section · f4bd2b11

An exact summed kernel and its boundary values

An exact summed kernel and its boundary values · lines 441–460

I-B-VP-2@1#eq:Qeta

Use the differences

T = t - t^{'}, Z = z - z^{'}, δ = φ - φ^{'}

in a chosen angular lift. For $ε > 0$ , define

Q_{ε} = \frac{r ^{2} + r ^{'2} + Z ^{2} - ( T - i ε ) ^{2}}{2 r r ^{'}}, η_{ε} = arcosh Q_{ε},

where the branch has $Re η_{ε} > 0$ . For real $T \neq = 0$ , $Q_{ε}$ approaches the real axis from the side $sgn T$ .

Displayed formulas in this passage

I-B-VP-2@1#eq:Qeta · line 450
$Q_{ε} = \frac{r ^{2} + r ^{'2} + Z ^{2} - ( T - i ε ) ^{2}}{2 r r ^{'}}, η_{ε} = arcosh Q_{ε},$
unlabeled · line 445
$T = t - t^{'}, Z = z - z^{'}, δ = φ - φ^{'}$