The Summation of One Class of Infinite Series
Author(s)
Jonathan D. Weiss
ABSTRACT
This paper presents closed-form expressions for the series,
, where the sum is from n = 1 to n = ∞. These expressions were obtained by recasting the series in a different form, followed by the use of certain relationships involving the elliptical nome. Among the values of x for which these expressions can be obtained are of the form: 
and 
, where l is an integer between –∞ and ∞. The values of λ include 1,
,
and 3. Examples of closed-form expressions obtained in this manner are first presented for 
, 
, 
, and 
. Additional examples are then presented for 
, 
, 
, and 
. This undertaking was prompted by the author’s work on an electrostatics boundary-value problem related to the van der Pauw measurement technique of electrical resistivity. The presence of this series for x = 
in the solution of that problem and its absence from any compendium of infinite series that he consulted led to this work.
This paper presents closed-form expressions for the series,
, where the sum is from n = 1 to n = ∞. These expressions were obtained by recasting the series in a different form, followed by the use of certain relationships involving the elliptical nome. Among the values of x for which these expressions can be obtained are of the form: and , where l is an integer between –∞ and ∞. The values of λ include 1,,and 3. Examples of closed-form expressions obtained in this manner are first presented for , , , and . Additional examples are then presented for , , , and . This undertaking was prompted by the author’s work on an electrostatics boundary-value problem related to the van der Pauw measurement technique of electrical resistivity. The presence of this series for x = in the solution of that problem and its absence from any compendium of infinite series that he consulted led to this work.
Cite this paper
Weiss, J. (2014) The Summation of One Class of Infinite Series. Applied Mathematics, 5, 2815-2822. doi: 10.4236/am.2014.517269.
Weiss, J. (2014) The Summation of One Class of Infinite Series. Applied Mathematics, 5, 2815-2822. doi: 10.4236/am.2014.517269.
References
[1] Hansen, E.H. (1970) A Table of Series and Products. Prentice Hall, Englewood Cliffs. [2] Prudnikov, A.P., Brychkov, Yu.A. and Marichev, O.I. (1986) Integrals and Series. Elementary Functions Vol. 1. Gordon and Breach, New York. [3] Gradshteyn, I.S. and Ryzhik, I.M. (1965) Table of Integrals, Series, and Products. Academic Press, New York. [4] Dieckmann, A. (2000) Collection of Infinite Products and Series.
http://pi.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html [5] Weiss, J.D. (2014) A Comparison of Two Van-der-Pauw Measurement Configurations. Materials Science in Semiconductor Processing, Submitted. [6] Van der Pauw, L.J. (1958) A Method of Measuring the Resistivity and Hall Coefficient on Lamellae of Arbitrary Shape. Philips Technical Review, 20, 220-224. [7] Abramowitz, A. and Stegun, I.A. (1972) Handbook of Mathematical Functions. Dover Publications, New York. [8] Whittaker, E.T. and Watson, G.N. (1996) A Course of Modern Analysis. 4th Edition, Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511608759 [9] Erdélyi, A., Ed. (1955) Higher Transcendental Functions (Vol. III). Mc Graw Hill, New York.
[1] Hansen, E.H. (1970) A Table of Series and Products. Prentice Hall, Englewood Cliffs. [2] Prudnikov, A.P., Brychkov, Yu.A. and Marichev, O.I. (1986) Integrals and Series. Elementary Functions Vol. 1. Gordon and Breach, New York. [3] Gradshteyn, I.S. and Ryzhik, I.M. (1965) Table of Integrals, Series, and Products. Academic Press, New York. [4] Dieckmann, A. (2000) Collection of Infinite Products and Series.
http://pi.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html [5] Weiss, J.D. (2014) A Comparison of Two Van-der-Pauw Measurement Configurations. Materials Science in Semiconductor Processing, Submitted. [6] Van der Pauw, L.J. (1958) A Method of Measuring the Resistivity and Hall Coefficient on Lamellae of Arbitrary Shape. Philips Technical Review, 20, 220-224. [7] Abramowitz, A. and Stegun, I.A. (1972) Handbook of Mathematical Functions. Dover Publications, New York. [8] Whittaker, E.T. and Watson, G.N. (1996) A Course of Modern Analysis. 4th Edition, Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511608759 [9] Erdélyi, A., Ed. (1955) Higher Transcendental Functions (Vol. III). Mc Graw Hill, New York.