arXiv subjects
Publications and source records attributed to Mark B. Villarino.
We apply an integral inequality to obtain a rigorous \emph{a priori} estimate of the accuracy of the partial sum to the power series solution of the Ricatti-Bernoulli differential equation.
We present two new proofs of the monotonicity of the correction term $θ_n$ in Ramanujan's refinement of Stirling's formula.
An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number into negative powers of the nth triangular number. We also discuss the history of the Ramanujan expansion for the nth harmonic number as well as sharp estimates of its accuracy, with complete proofs, and we compare it with other approximative formulas.
We give a new, elementary, purely analytical development of \textsc{Descartes}' theorem that a smooth connected surface is a perfect focusing lens if and only if it is a connected subset of the ovoid obtained by revolving a cartesian oval around its axis of symmetry.
We prove the explicit formula for the probability of a run of r successes in n trials.
An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number.
We obtain best possible upper and lower bounds for the Lodge-Ramanujan and DeTemple-Wang approximations to the nth Harmonic Number
We present an elementary self-contained detailed computation of Ramanujan's most famous singular modulus, k210, based on the Kronecker Limit Formula.
We prove Landen's transformation by a direct elementary transformation of infinite series
We present a detailed error analysis of Ramanujan's most accurate approximation to the perimeter of an ellipse.
We suggest a continued fraction origin to Ramanujan's approximation to {(a-b)/(a+b)}^2 in terms of the arc length of an ellipse with semiaxes a and b. Moreover, we discuss the asymptotic accuracy of the approximation.
A simple integration by parts and telescopic cancellation leads to a rigorous derivation of the first 2 terms for the error in Ramanujan's asymptotic series for the nth partial sum of the harmonic series. Then Kummer's transformation gives three more terms and a rigorous error estimate. Finally best-possible estimates of Lodge's approximations.
We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler's polyhedral formula.
We present a self-contained elementary and detailed exposition of Mertens' own proof of his theorem on the divergence of the series of the reciprocals of the primes and compare it with the modern proofs. His proof contains explicit numerical error estimates and computations of the Mertens "constant," both of which are missing from modern expositions. In as much as his strategy and techniques are quite different from today's methods, his original methods deserve to be better known.
We prove that n independent abelian functions admit an algebraic addition theorem, with no appeal to theta functions.
A proper or singular abelian mapping from $C^n$ to $\bar\C^n$ is parametrized by $n$ meromorphic functions with at most $2n$ periods. We develop the existence and structure theorems of the classical theory of an abelian mapping purely on the basis of its defining functional equation, the so-called algebraic addition theorem (AAT), with no appeal to any representation as quotients of theta functions. We offer two new proofs of the periodicity of a nonrational mapping admitting an AAT. We also prove by new arguments the existence of a rational group law on an associated algebraic variety, and that all proper and singular abelian mappings do admit an AAT.