Formal calculus MOC

Formal logarithm

Let $𝕂$ be a field with $c h a r 𝕂 = 0$ . The formal logarithm is defined as¹ #m/def/fcalc

l n (1 + 𝑎 𝑥) = - \sum 𝑘 \in ℕ (- 𝑎) 𝑘 𝑘 𝑥 𝑘 \in 𝕂 [[𝑧]] \leq 𝕂 {𝑧}

Properties

The following identities relate the formal logarithm with the formal exponential amd formal binomial expansion

l n (e x p 𝑥) = 𝑥 e x p (l n (1 + 𝑎 𝑥)) = 1 + 𝑎 𝑥 l n ((1 + 𝑎 𝑥) (1 + 𝑏 𝑥)) = l n (1 + 𝑎 𝑥) + l n (1 + 𝑏 𝑥) l n (1 + 𝑎 𝑥) 𝑏 = 𝑏 l n (1 + 𝑎 𝑥)

the proofs being given by the relations of the coëfficients in standard calculus.

#state/tidy | #lang/en | #SemBr

1988. Vertex operator algebras and the Monster, §3.4, pp. 76–77 ↩