arXiv subjects
Publications and source records attributed to Robert Reynolds.
The Watson-Harkins sum involving the product of the cosine and cosecant functions is extended to derive the finite sum of generalized Hurwitz-Lerch Zeta functions is derived in terms of the Hurwitz-Lerch Zeta function. A transformation formula arises for various finite values of the parameters involved. The finite product of trigonometric functions are also derived. All the results in this work are new.
Generating functions and functional equations of Dickson polynomials of the first and second kind are derived and continued analytically. These formulae are expressed in terms of the incomplete gamma function over complex variables of the parameters involved. Special cases are evaluated in terms of composite incomplete gamma functions and mathematical constants.
In this paper we use a contour integral method to derive a generating function in the form of a double series involving the product of two Chebyshev polynomials over generalized independent indices expressed in terms of the incomplete gamma function. The generating function represents a more generalized form relative to current literature. A possible application of this function to solving partial differential equations is discussed and some special cases of this generating function are derived. The work involved in the computation of this generating function is easier relative to previous methods as we have a closed form solution as opposed to numerical methods.
For the function $f(m,p,q,n)$, where $k,s,a$ general complex numbers and $q$ any positive integer, we establish the sum of values of the Hurwitz-Lerch zeta function $Φ(f(m,p,q,n),k,a)$ taken at prime numbers $n$. Special cases of this sum are evaluated in terms of products of trigonometric functions and Catalan's constant $K$.
In this present paper we derive a six dimensional integral containing the product of the Associated Legendre Polynomials $P_v^u(x) P_{ν}^{μ}(y)$ where the indices are different and general. Included in the kernel of this integral is the generalized logarithmic function and coefficient logarithmic functions. The derivation of this integral is written in terms of the Hurwitz-Lerch zeta function and constant coefficients raised to a power. Special cases of this integral are derived in terms of fundamental constants and other special functions. All the results in this work are new.
We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan's constant and $π$.
In this work the authors use their contour integral method to derive a double integral connected to the modified Bessel function of the second kind and express it in terms of the Lerch function. There are some useful results relating double integrals of certain kinds of functions to ordinary integrals for which we know no general reference. Thus a table of integral pairs is given for interested readers. The majority of the results in this work are new.
In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta. These derivations are then expressed in terms of fundamental constants, elementary and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.
In this manuscript, the author derived a definite integral involving the logarithmic function, function of powers and polynomials in terms of the Lerch function. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.
In 1826 Cauchy presented an Integral over the real line. Al and I thought a derivation would be mighty fine. So we packed our contour integral bags that day, and we now present an analytic continuation this time.
We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan's constant and $π$.
Enabling efficient injection of light into single-mode fibers (SMFs) is a key requirement in realizing diffraction-limited astronomical spectroscopy on ground-based telescopes. SMF-fed spectrographs, facilitated by the use of adaptive optics (AO), offer distinct advantages over comparable seeing-limited designs, including higher spectral resolution within a compact and stable instrument volume, and a telescope independent spectrograph design. iLocater is an extremely precise radial velocity (EPRV) spectrograph being built for the Large Binocular Telescope (LBT). We have designed and built the front-end fiber injection system, or acquisition camera, for the SX (left) primary mirror of the LBT. The instrument was installed in 2019 and underwent on-sky commissioning and performance assessment. In this paper, we present the instrument requirements, acquisition camera design, as well as results from first-light measurements. Broadband single-mode fiber coupling in excess of 35% (absolute) in the near-infrared (0.97-1.31μm) was achieved across a range of target magnitudes, spectral types, and observing conditions. Successful demonstration of on-sky performance represents both a major milestone in the development of iLocater and in making efficient ground-based SMF-fed astronomical instruments a reality.
We are developing a stable and precise spectrograph for the Large Binocular Telescope (LBT) named "iLocater." The instrument comprises three principal components: a cross-dispersed echelle spectrograph that operates in the YJ-bands (0.97-1.30 microns), a fiber-injection acquisition camera system, and a wavelength calibration unit. iLocater will deliver high spectral resolution (R~150,000-240,000) measurements that permit novel studies of stellar and substellar objects in the solar neighborhood including extrasolar planets. Unlike previous planet-finding instruments, which are seeing-limited, iLocater operates at the diffraction limit and uses single mode fibers to eliminate the effects of modal noise entirely. By receiving starlight from two 8.4m diameter telescopes that each use "extreme" adaptive optics (AO), iLocater shows promise to overcome the limitations that prevent existing instruments from generating sub-meter-per-second radial velocity (RV) precision. Although optimized for the characterization of low-mass planets using the Doppler technique, iLocater will also advance areas of research that involve crowded fields, line-blanketing, and weak absorption lines.
Existing planet-finding spectrometers are limited by systematic errors that result from their seeing-limited design. Of particular concern is the use of multi-mode fibers (MMFs), which introduce modal noise and accept significant amounts of background radiation from the sky. We present the design of a single-mode fiber-based acquisition camera for a diffraction-limited spectrometer named "iLocater." By using the "extreme" adaptive optics (AO) system of the Large Binocular Telescope (LBT), iLocater will overcome the limitations that prevent Doppler instruments from reaching their full potential, allowing precise radial velocity (RV) measurements of terrestrial planets around nearby bright stars. The instrument presented in this paper, which we refer to as the acquisition camera "demonstration system," will measure on-sky single-mode fiber (SMF) coupling efficiency using one of the 8.4m primaries of the LBT in fall 2015.