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|Full account of Euler's work on continued fractions and orthogonal polynomials; illustrates the significance of his work on mathematics today.|
|10.8 Euler's constant, symbol γ Euler's constant was first introduced by Euler in |
1734 as the limit [Eul48] γ := lim n→∞ ( n∑k=1 1 k − ln(n) ) = 0.57721566490153
.... (10.8.1) It is also known as the Euler-Mascheroni constant. It is closely related
|fractions. For example, with a0 = 0, a1 = z, a2 = −z2/3, a3 = −3z2/5, ... , we find |
that arctanz = z − z3 3 + z5 5 − z7 7 + z9 9 −··· (9.7) with |z| ≤ 1, can be
expressed as an irregular continued fraction due to Euler: arctanz =z1+ z2 3−z2 +
9z2 5−3z2 ...
|We usually see continued fractions in which all the numerators are 1. As Euler |
will note soon, for such continued fractions, the rule is a good deal simpler. Euler
does a few explicit calculations to find the difference between consecutive terms,
|Lorentzen  has an alternative approach to Ramanujan's continued fractions. |
Ramanujan was a master of manipulatorics in the class of Euler himself. Thus it is
appropriate that the continued fraction formulas of Ramanujan here are all ...
|Our purpose in this month's column is to look at what Euler did, and to see just |
how rigorous Euler' s results were. Euler and Lambert both used the tools of
continued fractions to produce their results. Euler's 1737 article that MacTutor
|[Khru06a] S. Khrushchev, A recovery of Brouncker's proof of the quadrature |
continued fraction, Publications Mathematiques 50(1) (2006), 3–42. [Khru06b] S.
Khrushchev, On Euler's differential methods for continued fractions, Electr. Trans.
|Based on earlier work of his predecessors, Euler began his research on |
continued fractions and published many new ideas and results in his first paper
entitled, “De Fractionibus Continuis” in 1737. He also proved that any rational
number can ...
|we have U0 + Ul + ' ' ' + un = ^ (1.50) 1- unu 0"2 u + u - "n-2"n This is Euler's |
famous transformation of a series into a continued fraction which we shall use to
derive a most convenient algorithm for the efficient evaluation of a continued
|Continued fractions are part of the "lost mathematics," the mathematics now |
considered too advanced for high school and too elementary for college. (Petr
Beckman)1 ... epitomized by Euler's continued fractions for e. Whorled figures are