E101 -- Introductio in analysin infinitorum, volume 1

(Introduction to the Analysis of the Infinite, volume 1)


Summary:

In E101, together with E102, Euler lays the foundations of modern mathematical analysis. He summarizes his numerous discoveries in infinite series, infinite products, and continued fractions, including the summation of the series 1/1k + 1/2k + 1/3k + ... for all even values of k between 2 and 26, inclusive. Perhaps more importantly, the Introductio makes the function the central concept of analysis; moreover, Euler introduces the notation f(x) for a function and uses it for implicit as well as explicit functions and for both continuous and discontinuous functions. In addition, he calls attention to the central role of e and ex in analysis, and he gives ex and ln x the independent definitions
ex = limn → ∞ (1 + x/n)n,
ln x = limn → ∞ n(x1/n - 1),

putting them on an equal basis for the first time.

Euler also proves that every rational number can be written as a finite continued fraction and that the continued fraction of an irrational number is infinite. He also shows how infinite series correspond to infinite continued fractions; in particular, Euler derives continued fraction expansions for e and √e.

The book contains an Epistola dedicatoria by Bousquet and Euler’s own Praefatio.
The main body of the work is divided into 18 chapters:
  1. De functionibus in genere.
  2. De transformatione functionum.
  3. De transformatione functionum per substitutionem.
  4. De explicatione functionum per series infinitas.
  5. De functionibus duarum pluriumve variabilium.
  6. De quantitatibus exponentialibus ac logarithmis.
  7. De quantitatum exponentialium ac logarithmorum per series explicatione.
  8. De quantitatibus transcendentibus ex circulo ortis.
  9. De investigatione factorum trinomialium.
  10. De usu factorum inventorum in definiendis summis serierum infinitarum.
  11. De aliis arcuum atque sinuum expressionibus infinitis.
  12. De reali functionum fractarum evolutione.
  13. De seriebus recurrentibus.
  14. De multiplicatione ac divisione angulorum.
  15. De seriebus ex evolutione factorum ortis.
  16. De partitione numerorum.
  17. De usu serierum recurrentium in radicibus aequationum indagandis.
  18. De fractionibus continuis.
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