American Institute of Mathematical Sciences

Advanced
January  2010, 13(1): 229-248. doi: 10.3934/dcdsb.2010.13.229

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

Yuk L. Yung 1, , Cameron Taketa 2, , Ross Cheung 1, and  Run-Lie Shia 1,

1. 

Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, 91125, United States, United States, United States
2. 

Hawaii Baptist Academy, 2429 Pali Highway, Honolulu, HI, 96817, United States

Received  June 2009 Revised  September 2009 Published  October 2009

We show that the function $S_1(x)=\sum_{k=1}^\infty e^{-2\pi kx} \log k$ 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 $S_1(ix)=K_0(x) - iK_1(x)$ is made. For $x=\frac{p}{q}$ where $p$ and $q$ are integers with $p$<$q$, closed finite sum expressions for $K_0 (\frac{p}{q} )$ and $K_1 ( \frac{p}{q}) $ are derived. The latter results enable us to evaluate Ramanujan's function $\varphi (x)=\sum_{k=1}^\infty (\frac{\log k}{k}-\frac{\log(k+x)}{k+x})$ for $x=-\frac{2}{3}, -\frac{3}{4},$ and $-\frac{5}{6},$ 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 $\varphi(x), K_0(x), K_1(x),$ and the classical formulas of Gauss and Kummer for the digamma function.
Keywords: series., exponential integral, radiation, special function.
Mathematics Subject Classification: Primary: 11Yxx; Secondary: 44A15, 44A2.
Citation: Yuk L. Yung, Cameron Taketa, Ross Cheung, Run-Lie Shia. Infinite sum of the product of exponential and logarithmic functions, its analytic continuation, and application. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 229-248. doi: 10.3934/dcdsb.2010.13.229
[1]

Y. T. Li, R. Wong. Integral and series representations of the dirac delta function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 229-247. doi: 10.3934/cpaa.2008.7.229
[2]

Bai-Ni Guo, Feng Qi. Properties and applications of a function involving exponential functions. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1231-1249. doi: 10.3934/cpaa.2009.8.1231
[3]

Robert Stephen Cantrell, Suzanne Lenhart, Yuan Lou, Shigui Ruan. Preface on the special issue of Discrete and Continuous Dynamical Systems- Series B in honor of Chris Cosner on the occasion of his 60th birthday. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : i-ii. doi: 10.3934/dcdsb.2014.19.1i
[4]

Carlos Conca, Luis Friz, Jaime H. Ortega. Direct integral decomposition for periodic function spaces and application to Bloch waves. Networks & Heterogeneous Media, 2008, 3 (3) : 555-566. doi: 10.3934/nhm.2008.3.555
[5]

Josef Diblík, Zdeněk Svoboda. Asymptotic properties of delayed matrix exponential functions via Lambert function. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 123-144. doi: 10.3934/dcdsb.2018008
[6]

Armengol Gasull, Francesc Mañosas. Subseries and signed series. Communications on Pure & Applied Analysis, 2019, 18 (1) : 479-492. doi: 10.3934/cpaa.2019024
[7]

Leon Ehrenpreis. Special functions. Inverse Problems & Imaging, 2010, 4 (4) : 639-647. doi: 10.3934/ipi.2010.4.639
[8]

Yachun Li, Shengguo Zhu. Existence results for compressible radiation hydrodynamic equations with vacuum. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1023-1052. doi: 10.3934/cpaa.2015.14.1023
[9]

Danthai Thongphiew, Vira Chankong, Fang-Fang Yin, Q. Jackie Wu. An on-line adaptive radiation therapy system for intensity modulated radiation therapy: An application of multi-objective optimization. Journal of Industrial & Management Optimization, 2008, 4 (3) : 453-475. doi: 10.3934/jimo.2008.4.453
[10]

Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39.

[11]

Genni Fragnelli, Luca Lorenzi, Alain Miranville. Preface to this special issue. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : ⅰ-ⅲ. doi: 10.3934/dcdss.2020080
[12]

Genni Fragnelli, Dimitri Mugnai, Maria Cesarina Salvatori. Preface to this special issue. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020147
[13]

Ferenc A. Bartha, Hans Z. Munthe-Kaas. Computing of B-series by automatic differentiation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 903-914. doi: 10.3934/dcds.2014.34.903
[14]

Nikita Kalinin, Mikhail Shkolnikov. Introduction to tropical series and wave dynamic on them. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2827-2849. doi: 10.3934/dcds.2018120
[15]

Chuang Peng. Minimum degrees of polynomial models on time series. Conference Publications, 2005, 2005 (Special) : 720-729. doi: 10.3934/proc.2005.2005.720
[16]

Ruiqi Li, Yifan Chen, Xiang Zhao, Yanli Hu, Weidong Xiao. Time series based urban air quality predication. Big Data & Information Analytics, 2016, 1 (2&3) : 171-183. doi: 10.3934/bdia.2016003
[17]

Ricardo García López. A note on L-series and Hodge spectrum of polynomials. Electronic Research Announcements, 2009, 16: 56-62. doi: 10.3934/era.2009.16.56
[18]

G. Gentile, V. Mastropietro. Convergence of Lindstedt series for the non linear wave equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 509-514. doi: 10.3934/cpaa.2004.3.509
[19]

Guangwei Yuan, Yanzhong Yao. Parallelization methods for solving three-temperature radiation-hydrodynamic problems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1651-1669. doi: 10.3934/dcdsb.2016016
[20]

Sebastian Bauer. A non-relativistic model of plasma physics containing a radiation reaction term. Kinetic & Related Models, 2018, 11 (1) : 25-42. doi: 10.3934/krm.2018002

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences