Formal calculus MOC

Formal logarithm

Let 𝕂 be a field with [[Characteristic|char⁡𝕂 =0]]. The formal logarithm is defined as¹ fcalc

ln⁡(1+𝑎𝑥)=−∑𝑘∈ℕ(−𝑎)𝑘𝑘𝑥𝑘∈𝕂[[𝑧]]≤𝕂{𝑧}

Properties

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

ln⁡(exp⁡𝑥)=𝑥exp⁡(ln⁡(1+𝑎𝑥))=1+𝑎𝑥ln⁡((1+𝑎𝑥)(1+𝑏𝑥))=ln⁡(1+𝑎𝑥)+ln⁡(1+𝑏𝑥)ln⁡(1+𝑎𝑥)𝑏=𝑏ln⁡(1+𝑎𝑥)

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

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Footnotes

1. 1988. Vertex operator algebras and the Monster, §3.4, pp. 76–77 ↩