Geodesic
On Euler's differential methods for continued fractions
Electronic transactions on numerical analysis, Tome 25 (2006), pp. 178-200
A differential method discovered by Euler is justified and applied to give simple proofs to formulas relating important continued fractions with Laplace transforms. They include Stieltjes formulas and some Ramanujan formulas. A representation for the remainder of Leibniz's series as a continued fraction is given. We also recover the original Euler's proof for the continued fraction of hyperbolic cotangent.
Classification : 30B70
Keywords: continued fractions, Ramanujan formulas, Laplace transform 
@article{ETNA_2006__25__a19,
     author = {Khrushchev,  Sergey},
     title = {On {Euler's} differential methods for continued fractions},
     journal = {Electronic transactions on numerical analysis},
     pages = {178--200},
     year = {2006},
     volume = {25},
     zbl = {1107.30005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2006__25__a19/}
}
TY  - JOUR
AU  - Khrushchev,  Sergey
TI  - On Euler's differential methods for continued fractions
JO  - Electronic transactions on numerical analysis
PY  - 2006
SP  - 178
EP  - 200
VL  - 25
UR  - http://geodesic.mathdoc.fr/item/ETNA_2006__25__a19/
LA  - en
ID  - ETNA_2006__25__a19
ER  - 
%0 Journal Article
%A Khrushchev,  Sergey
%T On Euler's differential methods for continued fractions
%J Electronic transactions on numerical analysis
%D 2006
%P 178-200
%V 25
%U http://geodesic.mathdoc.fr/item/ETNA_2006__25__a19/
%G en
%F ETNA_2006__25__a19
Khrushchev,  Sergey. On Euler's differential methods for continued fractions. Electronic transactions on numerical analysis, Tome 25 (2006), pp. 178-200. http://geodesic.mathdoc.fr/item/ETNA_2006__25__a19/