De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt

English Title

On transcendental progressions, that is, those whose general terms cannot be given algebraically

Authors

Leonhard Euler

Enestrom Number

19

Fuss Index

120

Original Language

Latin

Content Summary

One of Euler’s earliest papers, he begins with Wallis's "hypergeometric series" 1! + 2! + 3! + 4! + ... (he didn’t have the "!" factorial notation yet) to find the value of the function for a general value of x. He defines [x] to be ∫₀¹ (ln(1/t))^x dt, which after substituting t = e^-z gives [x] = ∫₀^∞ z^x e^-z dz, what we know as the Gamma function. He finds [1/2] to be √(π/2) and derives the recursive rule [x+1] = (x+1)∙[x]. He finds [p/q] to be a product of beta functions, and derives a differential quotient.

Published as

Journal article

Published Date

1738

Written Date

1729

Original Source Citation

Commentarii academiae scientiarum Petropolitanae, Volume 5, pp. 36-57.

Opera Omnia Citation

Series 1, Volume 14, pp.1-24.

Record Created

2018-09-25