Manuscript Em01-D@2 · Em01-D-v2.tex · section · d3b49151

All diagnostic coefficients

Exact nonphysical eight-variable diagnostic › All diagnostic coefficients · lines 2151–2178

Em01-D@2#eq:diagnosticcoeff

For $e = - 3, - 2, - 1, 0$ , define

B_{e} (Φ) = \frac{i}{2} 0 \leq j \leq 2 n = e + 3 - j \geq 0 \sum c_{j} \int d τ \int_{R} d S \int_{0}^{\infty} d T P_{j} (S, T) [L_{n} (B, S, T) F_{j}]_{0} .

For these exponents, $n \leq 3 - j$ . Thus the formula is finite.

The profile integrals can all be reduced explicitly. Let

P (S) = \int_{- \infty}^{S} p (V) d V,

and for $d \geq 1$ , $0 \leq b \leq d$ , set

M_{d}^{a, b} = \int_{R} \int_{0}^{\infty} S^{a} T^{b} p (S) p^{(d)} (S - T) d T d S .

Displayed formulas in this passage

Em01-D@2#eq:diagnosticcoeff · line 2154
$B_{e} (Φ) = \frac{i}{2} 0 \leq j \leq 2 n = e + 3 - j \geq 0 \sum c_{j} \int d τ \int_{R} d S \int_{0}^{\infty} d T P_{j} (S, T) [L_{n} (B, S, T) F_{j}]_{0} .$
unlabeled · line 2168
$P (S) = \int_{- \infty}^{S} p (V) d V,$
unlabeled · line 2172
$M_{d}^{a, b} = \int_{R} \int_{0}^{\infty} S^{a} T^{b} p (S) p^{(d)} (S - T) d T d S .$