Manuscript Em01-E@2 · Em01-E-v2.tex · section · 54f8f8c2

Angular Taylor expansion and profile moments

Evaluated complete degenerate specialization › Angular Taylor expansion and profile moments · lines 1891–1982

Em01-E@2#eq:appA, Em01-E@2#eq:appT

Put

Q_{p} = \int_{R} p (S)^{2} d S > 0, P (S) = \int_{- \infty}^{S} p (V) d V,

and

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

For the required moments, integration by parts gives

M_{d, a, b} = b! \int S^{a} p (S) p^{(d - b - 1)} (S) d S (b < d),

and

M_{d, a, d} = d! \int S^{a} p (S) P (S) d S .

The latter includes $d = 0$ .

Evenness and normalization evaluate all moments through finite order:

$M_{2, 0, 0}$	$M_{2, 0, 1}$	$M_{2, 1, 0}$	$M_{2, 0, 2}$	$M_{2, 1, 1}$	$M_{2, 2, 0}$
$0$	$Q_{p}$	$- Q_{p} /2$	$1$	$0$	$0$

$M_{1, 0, 0}$	$M_{1, 0, 1}$	$M_{1, 1, 0}$	$M_{0, 0, 0}$
$Q_{p}$	$1/2$	$0$	$1/2$

For example,

\int p p^{'} = 0, \int S p p^{'} = - Q_{p} /2, \int pP = 1/2.

The other zero moments are odd integrals. No two-variable moment needed for the expansion is left unevaluated.

Define

A (Φ) = - \frac{i Q _{p}}{2} \int d τ [\frac{1}{2} \partial_{u} F_{0} - \frac{1}{4} \partial_{v} F_{0} + \frac{B}{2} Δ_{X} F_{0} + F_{1}]_{0},

and

T (Φ) = - \frac{i}{2} \int d τ [(\frac{1}{8} \partial_{u}^{2} + \frac{B}{4} \partial_{u} Δ_{X} + \frac{B ^{2}}{16} Δ_{X}^{2}) F_{0} + (\frac{1}{4} \partial_{u} + \frac{B}{4} Δ_{X}) F_{1} + \frac{1}{2} F_{2}]_{0} .

Displayed formulas in this passage

Em01-E@2#eq:appA · line 1954
$A (Φ) = - \frac{i Q _{p}}{2} \int d τ [\frac{1}{2} \partial_{u} F_{0} - \frac{1}{4} \partial_{v} F_{0} + \frac{B}{2} Δ_{X} F_{0} + F_{1}]_{0},$
Em01-E@2#eq:appT · line 1967
$T (Φ) = - \frac{i}{2} \int d τ [(\frac{1}{8} \partial_{u}^{2} + \frac{B}{4} \partial_{u} Δ_{X} + \frac{B ^{2}}{16} Δ_{X}^{2}) F_{0} + (\frac{1}{4} \partial_{u} + \frac{B}{4} Δ_{X}) F_{1} + \frac{1}{2} F_{2}]_{0} .$
unlabeled · line 1894
$Q_{p} = \int_{R} p (S)^{2} d S > 0, P (S) = \int_{- \infty}^{S} p (V) d V,$
unlabeled · line 1899
$M_{d, a, b} = \int_{R} \int_{0}^{\infty} S^{a} T^{b} p (S) p^{(d)} (S - T) d T d S .$
unlabeled · line 1906
$M_{d, a, b} = b! \int S^{a} p (S) p^{(d - b - 1)} (S) d S (b < d),$
unlabeled · line 1913
$M_{d, a, d} = d! \int S^{a} p (S) P (S) d S .$
unlabeled · line 1944
$\int p p^{'} = 0, \int S p p^{'} = - Q_{p} /2, \int pP = 1/2.$