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Djurdje Cvijović

  1. Closed-form formulae for the derivatives of trigonometric functions at rational multiples of $\pi$.

    Ср, 11/25/2009 - 16:01 — SomNambul
    Authors: Djurdje Cvijović
    Subjects: Classical Analysis and ODEs
    Abstract

    In this sequel to our recent note it is shown, in a unified manner, by making
    use of some basic properties of certain special functions, such as the Hurwitz
    zeta function, Lerch zeta function and Legendre chi function, that the values
    of all derivatives of four trigonometric functions at rational multiples of
    $\pi$ can be expressed in closed form as simple finite sums involving the
    Bernoulli and Euler polynomials. In addition, some particular cases are
    considered.

  2. New integral representations of the polylogarithm function.

    Втр, 11/24/2009 - 16:01 — SomNambul
    Authors: Djurdje Cvijović
    Subjects: Classical Analysis and ODEs
    Abstract

    Maximon has recently given an excellent summary of the properties of the
    Euler dilogarithm function and the frequently used generalizations of the
    dilogarithm, the most important among them being the polylogarithm function
    $Li_(z)$. The polylogarithm function appears in several fields of mathematics
    and in many physical problems. We, by making use of elementary arguments,
    deduce several new integral representations of the polylogarithm for any
    complex z for which $|z|$ < 1. Two are valid for all complex s, whenever
    $\Re(s)>1$ .

  3. A dilogarithmic integral arising in quantum field theory.

    Пт, 11/20/2009 - 16:01 — SomNambul
    Authors: Djurdje Cvijovi&#x107;
    Subjects: Classical Analysis and ODEs
    Abstract

    Recently, an interesting dilogarithmic integral arising in quantum field
    theory has been closed-form evaluated in terms of the Clausen function
    $\text{Cl}_2(\theta)$ by Coffey [J. Math. Phys.} 49 (2008), 093508]. It
    represents the volume of an ideal tetrahedron in hyperbolic space and is
    involved in two intriguing equivalent conjectures of Borwein and Broadhurst. It
    is shown here, by simple and direct arguments, that this integral can be
    expressed by the triplet of the Clausen function values which are involved in
    one of the two above-mentioned conjectures.

  4. Derivative Polynomials and Closed-Form Higher Derivative Formulae.

    Пт, 11/20/2009 - 16:01 — SomNambul
    Authors: Djurdje Cvijovi&#x107;
    Subjects: Classical Analysis and ODEs
    Abstract

    In a recent paper, Adamchik [V.S. Adamchik, On the Hurwitz function for
    rational arguments, Appl. Math. Comp. 187 (2007) 3--12] expressed in a closed
    form symbolic derivatives of four functions belonging to the class of functions
    whose derivatives are polynomials in terms of the same functions. In this
    sequel, simple closed-form higher derivative formulae which involve the
    Carlitz-Scoville higher order tangent and secant numbers are derived for eight
    trigonometric and hyperbolic functions.

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