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The logarithm with values in the extended reals

Lax606786.ExtendedLog · concepts/Lax606786/ExtendedLog.lean · lax-606786

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    Natural Language Statement

    Definition

    For r≥0r \ge 0 the logarithm log⁡r∈[−∞,∞)\log r \in [-\infty, \infty), with log⁡0=−∞\log 0 = -\infty. Growth rates lim⁡1nlog⁡an\lim \frac1n \log a_n are taken with this logarithm, so that a sequence which vanishes has rate −∞-\infty. (Arguments r<0r < 0 do not occur; they are also sent to −∞-\infty.)

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    Lean source view on GitLab

    1import Mathlib.Analysis.SpecialFunctions.Log.Basic
    2
    3/-!
    4---
    5title: The logarithm with values in the extended reals
    6type: definition
    7---
    8For r≥0r \ge 0 the logarithm log⁡r∈[−∞,∞)\log r \in [-\infty, \infty), with log⁡0=−∞\log 0 = -\infty. Growth rates
    9lim⁡1nlog⁡an\lim \frac1n \log a_n are taken with this logarithm, so that a sequence which vanishes has rate
    10−∞-\infty. (Arguments r<0r < 0 do not occur; they are also sent to −∞-\infty.)
    11-/
    12
    13namespace Lax606786.ExtendedLog
    14
    15/-- `log r` for `r > 0`, and `-∞` for `r ≤ 0`. -/
    16noncomputable def logEReal (r : ℝ) : EReal := if r ≤ 0 then ⊥ else (Real.log r : EReal)
    17
    18end Lax606786.ExtendedLog
    19
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