Khrushchev, SergeyMathematicsMathematics2024-10-062024-10-0620061068-96132-s2.0-33846993356https://hdl.handle.net/20.500.14411/8739Khrushchev, Sergey/0000-0002-8854-5317A 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.eninfo:eu-repo/semantics/closedAccesscontinued fractionsRamanujan formulasLaplace transformConference ObjectQ3Q325178200WOS:0002470309000151