Closed-form formulae for the derivatives of trigonometric functions at rational multiples of π\pi

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.Comment: 5 page

Derivative Polynomials and Closed-Form Higher Derivative Formulae

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.Comment: 7 page

Polypseudologarithms revisited

Lee, in a series of papers, described a unified formulation of the statistical thermodynamics of ideal quantum gases in terms of the polylogarithm functions, $\textup{Li}_{s} (z)$. It is aimed here to investigate the functions $\textup{Li}_{s} (z),$ for $s = 0, -1, -2, ...,$ which are, following Lee, referred to as the polypseudologarithms (or polypseudologs) of order $n$. Various known results regarding polypseudologs, mainly obtained in widely differing contexts and currently scattered throughout the literature, have been brought together along with many new results and insights and they all have been proved in a simple and unified manner. In addition, a new general explicit closed-form formula for these functions involving the Carlitz--Scoville higher tangent numbers has been established.Comment: 10 page

A dilogarithmic integral arising in quantum field theory

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.Comment: 6 page

Another two families of integer-valued polynomials associated with finite trigonometric sums

As a sequel to our recent paper, its general approach was here extended to finite alternating trigonometric sums giving rise to polynomials which were systematically examined in full detail as well as in a unified manner using simple arguments. Two new general families of integer-valued polynomials (along with four other families derived from them, also integer-valued, including two already known) were deduced. Also, these polynomials enable closed-form summation of a great deal of general families of finite sums

New integral representations of the polylogarithm function

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 $\Re(s)>1$ . The other two involve the Bernoulli polynomials and are valid in the important special case where the parameter s is an positive integer. Our earlier established results on the integral representations for the Riemann zeta function $\zeta(2n+1)$ ,$n\in\mathbb{N}$, follow directly as corollaries of these representations.Comment: 15 page

The impact of the transition radius lower flange-web on local stress of monorail crane girder

Širokopojasne profile sa paralelnom konturom pojaseva, koji se danas dominantno koriste za izradu jednošinskih nosača dizalica, karakteriše relativno veliki radijus tranzicije konture pojasa u konturu rebra. Upravo zbog toga, uticaj pomenutog radijusa na naponsko stanje izazvano lokalnim savijanjem usled dejstva točkova dizaličnih kolica znatno je izraženiji kod širokopojasnih (IPB) profila, u odnosu na klasične I profile i srednje široke (IPE) profile. U radu su prezentirani rezultati numeričko-analitičkog i eksperimentalnog istraživanja lokalnih napona u zoni tranzicije donji pojas/rebro kod širokopojasnih profila. Utvrđeno je da se najveće vrednosti razmatranih napona javljaju na početku tranzicije konture, a ne u tački fiktivnog preseka konture rebra i donje konture donjeg pojasa, kako se navodi u relevantnoj literaturi i aktuelnoj tehničkoj regulativi (standard EN 15011:2014). Osim toga, rezultati istraživanja dokazuju da apsolutne vrednosti lokalnih napona na donjoj i gornjoj konturi nisu jednake.Wide flange I-beams with parallel flange contours, which are now predominantly used in production of monorail crane girders, are characterized by a relatively large radial transition between flange contour and rib contour. Therefore, the influence of the radius on the stress state, due to the local bending caused by the action of crane trolley wheels, is more pronounced in wide-flange I-beams (IPB) than in conventional (I) and mid-wide (IPE) I-beams. This paper presents the results of numerical-analytical and experimental research of local stresses in the lower flange-rib transition zone at wide flange I-beams. It was found that the highest values of the considered stresses occur at the start of the transition contour, and not in the fictive point of intersection of the rib contour and the upper contour of the lower flange, as stated in relevant literature and current technical regulations (standard SRPS EN 15011: 2014). In addition, research results show that the absolute values of local stresses on the lower and upper contours of the lower flange are not equal

Gas corrosion damage in Ti-stabilized interstitial free steel

The selective oxidation damage in the Ti-stabilized interstitial free steel during 60 s of recrystallization annealing at 820 °C under the different compositions of protective H2-N2 atmosphere at low dew point (–40 °C) was investigated using various experimental techniques. It was found that Mn, Al and Si oxide particles are the main products of external and internal oxidation. Increase of the H2 content in gas atmosphere favors external oxidation and leads to appearance of greater nonwetted surface areas.Physical chemistry 2006 : 8th international conference on fundamental and applied aspects of physical chemistry; Belgrade (Serbia); 26-29 September 200