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Infinite sum of the product of exponential and logarithmic functions, its analytic continuation, and application

Yung, Yuk L. and Taketa, Cameron and Cheung, Ross and Shia, Run-Lie (2010) Infinite sum of the product of exponential and logarithmic functions, its analytic continuation, and application. Discreet and Continuous Dynamical Systems Series B, 13 (1). pp. 229-248. ISSN 1531-3492.

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We show that the function S_(1)(x) = ∑_(k=1)^∞ e^(-2πkx) log k can be expressed as the sum of a simple function and an infinite series, whose coefficients are related to the Riemann zeta function. Analytic continuation to the imaginary argument S_(1)(ix) = K_0(x) is made. For x = p/q where p and q are integers with p < q, closed finite sum expressions for K_0(p/q) and K_1(p/q) are derived. The latter results enable us to evaluate Ramanujan's function ψ(x) = ∑_(k=1)^∞ [(logk)/k - (log(k+x))/(k+x)] for x = -2/3, -3/4, and -5/6, confirming what Ramanujan claimed but did not explicitly reveal in his Notebooks. The interpretation of a pair of apparently inscrutable divergent series in the notebooks is discussed. They reveal hitherto unsuspected connections between Ramanujan's ψ(x), K_0(x), K_1(x), and the classical formulas of Gauss and Kummer for the digamma function

Item Type:Article
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URLURL TypeDescription
Yung, Yuk L.0000-0002-4263-2562
Shia, Run-Lie0000-0003-1911-3120
Additional Information:© AIMS 2009. Received: June 2009; revised: September 2009; published: October 2009. We thank Vijay Natraj, Tom Apostol, K. K. Tung, Marguerite Gerstell, Christopher Parkinson, and Tomasz Tyranowski for their valuable inputs. Special thanks are due to an anonymous referee of an unnamed journal for a careful reading of the manuscript and many useful suggestions for improving the paper. Cameron Taketa and Ross Cheung acknowledge support by the Summer Undergraduate Research Fellowship (SURF) program at the California Institute of Technology. One of us (YLY) thanks Professor Richard Goody for the class in 1972 on atmospheric radiation where he learned the beauty of the mathematics of radiative transfer. The research was supported in part by the Orbiting Carbon Observatory Project at the Jet Propulsion Laboratory, and the California Institute of Technology.
Funding AgencyGrant Number
Summer Undergraduate Research Fellowship (SURF), CaltechUNSPECIFIED
Subject Keywords:special function; radiation; exponential integral; series
Issue or Number:1
Classification Code:Mathematics Subject Classification: Primary: 11Yxx; Secondary: 44A15, 44A20
Record Number:CaltechAUTHORS:20091110-113936352
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:16642
Deposited By: Jason Perez
Deposited On:13 Nov 2009 22:59
Last Modified:09 Mar 2020 13:19

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