basic course content added, more TODO

posts/2026-02-16-wigner.md

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+---
+title: "Computing the Wigner 9j Symbols Efficiently"
+author: "Thomas (Tom) C. Gorordo"
+published: false
+
+references:
+- type: book
+  id: GrahamKnuthPatashnik1994ConcreteMathematics2e
+  author:
+  - family: Graham
+    given: Ronald L.
+  - family: Knuth
+    given: Donald E.
+  - family: Patashnik
+    given: Oren
+  title: 'Concrete Mathematics: A Foundation for Computer Science'
+  title-short: 'Concrete Mathematics'
+
+---
+
+As a test post, here's a transcription of some notes I have laying around from writing an [`add9j` PR](https://github.com/tgorordo/WignerSymbols.jl)
+for [`Jutho/WignerSymbols.jl`](https://github.com/Jutho/WignerSymbols).
+
+## Wigner Symbols Formulae
+
+The Wigner \(3x\text{j}\) symbols are all exactly representable in the form
+
+\[W_{3x} = \frac{n\sqrt{s}}{q}\hspace{1cm} n,s,q\in\mathbb{Z}\]
+
+So, if one can use exact integer arithmetic in computing the three components,
+a computer implementation of the symbols should be able to incur at-worst
+typical floating point root and ratio errors.
+
+This requires the usage of formulae that are amenible to such integer manipulations.
+
+### \(3j\) Formula
+
+Let \(\delta(j_1, j_2, j_3) = (j_1 \leq j_2 + j_3)\bigwedge (j_2 \leq j_1 + j_3)\bigwedge (j_3 \leq j_1 + j_2)\bigwedge(j_1 + j_2 + j_3\in\mathbb{Z})\)
+for \(j_i\in\frac{1}{2}\mathbb{Z}\) a half-integer be called the "triangle condition". If not satisfied, a symbol is zero.
+
+The \(3j\) symbol can be computed according to:
+
+
+### \(6j\) Formula
+The \(6j\) symbol, similarly, can be computed as:
+
+Define the Wei square brackets purely in terms of binomial coefficients by
+
+### \(9j\) Formula
+The \(9j\) formula has a few additional pieces.
+
+
+One might have some fun manipulating some of these forms using the tricks in
+[@GrahamKnuthPatashnik1994ConcreteMathematics2e].
+
+## Calculation and Tabulation
+
+For fast, exact evaluation, the main integer-arithmetic idea is to simplify the evaluation 
+of the various sums of ratios of (potentially large) factorials by using prime factorizations
+of those factorials to reduce ratios by an LCD ad extract common factors across the sums.
+Remaining arithmetic can be done via a big integer implementation.
+
+The core of this approach uses that prime factorizations of factorials are conveniently
+obtained recursively (and thus, to tabulate). The crux is:
+
+### Euclid's Lemma
+> *If \( p | ab\) for prime \(p\), then \(p|a\) or \(p|b\).*
+
+> *\(\implies\) If prime \(p|n!\) then \(p|k\) for \(k\leq n\) one of the factors.*
+
+More specifically, *Legendre's Theorem* says:
+
+\[\sharp_p(n!) = \sum_{k=1} \sharp_p(k) = \sum_{k=1}^{\lfloor\log_p(n)\rfloor} \lfloor\frac{n}{p^k}\rfloor\]
+
+where \(\sharp_p(k)\) gives the number of times $p$ divides \(k\).
+