Manuscript XVIII-B@2 · XVIII-B-v2.tex · section · 27ed4f4b

Explicit evaluation of the reduced integrals

The once-subtracted retarded prescription › Explicit evaluation of the reduced integrals · lines 725–779

XVIII-B@2#eq:F2closed, XVIII-B@2#eq:F0closed, XVIII-B@2#eq:Jrecursion, XVIII-B@2#eq:J0below, XVIII-B@2#eq:J0above

Put $w = z / a$ and

J_{n} (w) = \int_{0}^{1} \frac{v ^{2 n}}{1 - w + w v ^{2}} d v .

These functions are analytic on the same slit plane. The change $u = a / (1 - v^{2})$ in (eq:subtracted) gives

F_{2} (z) F_{0} (z) = \frac{w}{480 π ^{2}} J_{3} (w), = \frac{w}{192 π ^{2}} (9 J_{1} (w) - 6 J_{2} (w) + J_{3} (w)) .

For $n \geq 1$ ,

w J_{n} (w) = \frac{1}{2 n - 1} - (1 - w) J_{n - 1} (w) .

The regular value at $w = 0$ is $J_{n} (0) = 1/ (2 n + 1)$ .

For real $w < 1$ , $w \neq = 0$ ,

J_{0} (w) = ⎩ ⎨ ⎧ \frac{artanh - w / ( 1 - w )}{- w ( 1 - w )}, \frac{arctan w / ( 1 - w )}{w ( 1 - w )}, w < 0, 0 < w < 1.

For $w > 1$ , set $v_{*} = 1 - 1/ w$ . Then

J_{0} (w + i 0 sgn p^{0}) = \frac{lo g ( ( 1 - v _{*} ) / ( 1 + v _{*} ) ) + i π sgn p ^{0}}{2 w v _{*}} .

The apparent singularities of the intermediate $J_{0}$ representation cancel in the combinations (eq:F2closed)–(eq:F0closed) at $w = 0, 1$ .

Displayed formulas in this passage

XVIII-B@2#eq:F2closed · line 735
$F_{2} (z) F_{0} (z) = \frac{w}{480 π ^{2}} J_{3} (w), = \frac{w}{192 π ^{2}} (9 J_{1} (w) - 6 J_{2} (w) + J_{3} (w)) .$
XVIII-B@2#eq:F0closed · line 735
$F_{2} (z) F_{0} (z) = \frac{w}{480 π ^{2}} J_{3} (w), = \frac{w}{192 π ^{2}} (9 J_{1} (w) - 6 J_{2} (w) + J_{3} (w)) .$
XVIII-B@2#eq:Jrecursion · line 745
$w J_{n} (w) = \frac{1}{2 n - 1} - (1 - w) J_{n - 1} (w) .$
XVIII-B@2#eq:J0below · line 753
$J_{0} (w) = ⎩ ⎨ ⎧ \frac{artanh - w / ( 1 - w )}{- w ( 1 - w )}, \frac{arctan w / ( 1 - w )}{w ( 1 - w )}, w < 0, 0 < w < 1.$
XVIII-B@2#eq:J0above · line 767
$J_{0} (w + i 0 sgn p^{0}) = \frac{lo g ( ( 1 - v _{*} ) / ( 1 + v _{*} ) ) + i π sgn p ^{0}}{2 w v _{*}} .$
unlabeled · line 728
$J_{n} (w) = \int_{0}^{1} \frac{v ^{2 n}}{1 - w + w v ^{2}} d v .$