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

Lemma: Uniform angular Taylor formula

Exact nonphysical eight-variable diagnostic › All diagnostic coefficients · lines 2114–2151 · proof included or linked

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

For $M \geq 0$ ,

\frac{1}{2 π} \int_{0}^{2 π} F (\frac{δ T}{2}, \frac{δ S}{2}, 2 B δ T e_{θ}) d θ = n = 0 \sum^{M} δ^{n} [L_{n} (B, S, T) F]_{0} + O (δ^{M + 1} ∥ F ∥_{2 M + 2}),

uniformly on compact positive $B$ -ranges and compact $S, T$ -ranges with $T \geq 0$ .

Proof. Taylor-expand in $X$ through order $2 M + 1$ . Odd angular moments vanish. The averaged order- $2 c$ term is

\frac{( R ^{2} /4 ) ^{c}}{( c ! ) ^{2}} Δ_{X}^{c} F, R^{2} = 2 B δ T .

This follows by integrating the even monomials on the unit circle.

Expand each such coefficient in $(u, v)$ through total order $M - c$ . Collecting scale order $n = a + b + c$ gives (eq:diagnosticLn). The transverse remainder is $O (R^{2 M + 2})$ . The longitudinal remainder for the $c$ -term is $O (δ^{M - c + 1})$ multiplied by $δ^{c}$ . All errors are $O (δ^{M + 1})$ with the stated order.

Displayed formulas in this passage

Em01-D@2#eq:diagnosticTaylor · line 2117
$\frac{1}{2 π} \int_{0}^{2 π} F (\frac{δ T}{2}, \frac{δ S}{2}, 2 B δ T e_{θ}) d θ = n = 0 \sum^{M} δ^{n} [L_{n} (B, S, T) F]_{0} + O (δ^{M + 1} ∥ F ∥_{2 M + 2}),$
unlabeled · line 2137
$\frac{( R ^{2} /4 ) ^{c}}{( c ! ) ^{2}} Δ_{X}^{c} F, R^{2} = 2 B δ T .$

Lemma: Uniform angular Taylor formula

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