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

Lemma: Evaluation of every required profile moment

Exact nonphysical eight-variable diagnostic › All diagnostic coefficients · lines 2179–2218 · proof included or linked

Em01-D@2#lem:diagnosticmoments, Em01-D@2#eq:diagnosticmoments

The moments are

M_{d}^{a, b} = ⎩ ⎨ ⎧ b! \int_{R} S^{a} p (S) p^{(d - b - 1)} (S) d S, d! \int_{R} S^{a} p (S) P (S) d S, b < d, b = d .

Every moment in (eq:diagnosticcoeff) is of this type, with $d = 3 - j$ , $a$ the power of $S$ , and $b$ the power of $T$ .

Proof. For $b = 0$ ,

\int_{0}^{\infty} p^{(d)} (S - T) d T = p^{(d - 1)} (S) .

For $b > 0$ , integration by parts gives the recursion

\int_{0}^{\infty} T^{b} p^{(d)} (S - T) d T = b \int_{0}^{\infty} T^{b - 1} p^{(d - 1)} (S - T) d T .

If $b < d$ , iterate $b$ times and use the first identity. If $b = d$ , iterate $d$ times, leaving $d! P (S)$ . Multiplication by $S^{a} p (S)$ gives (eq:diagnosticmoments).

In $L_{n}$ , the $T$ -power is at most $n$ , and $n \leq 3 - j = d$ , proving the final assertion.

Displayed formulas in this passage

Em01-D@2#eq:diagnosticmoments · line 2182
$M_{d}^{a, b} = ⎩ ⎨ ⎧ b! \int_{R} S^{a} p (S) p^{(d - b - 1)} (S) d S, d! \int_{R} S^{a} p (S) P (S) d S, b < d, b = d .$
unlabeled · line 2202
$\int_{0}^{\infty} p^{(d)} (S - T) d T = p^{(d - 1)} (S) .$
unlabeled · line 2206
$\int_{0}^{\infty} T^{b} p^{(d)} (S - T) d T = b \int_{0}^{\infty} T^{b - 1} p^{(d - 1)} (S - T) d T .$

Lemma: Evaluation of every required profile moment

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