Skip to main content
×
×
Home

On the Horizontal Monotonicity of |Γ(s)|

  • Gopala Krishna Srinivasan (a1) and P. Zvengrowski (a2)
Abstract

Writing s = σ + it for a complex variable, it is proved that the modulus of the gamma function, |Γ(s)|, is strictly monotone increasing with respect to σ whenever |t| > 5/4. It is also shown that this result is false for |t| ≤ 1.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On the Horizontal Monotonicity of |Γ(s)|
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On the Horizontal Monotonicity of |Γ(s)|
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On the Horizontal Monotonicity of |Γ(s)|
      Available formats
      ×
Copyright
References
Hide All
[1] Alzer, H., On some inequalities for the gamma and psi functions. Math. Comp. 66(1997), no. 217, 373389. doi:10.1090/S0025-5718-97-00807-7
[2] Alzer, H., Monotonicity properties of the Hurwitz zeta function. Canad. Math. Bull. 48(2005), no. 3, 333339.
[3] Andrews, G. E., Askey, R., and Roy, R., Special functions. Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999.
[4] Davis, P. J., Leonhard Euler's integral: A historical profile of the gamma function. Amer. Math. Monthly 66(1959), 849869. doi:10.2307/2309786
[5] Edwards, H. M., Riemann's zeta function. Pure and Applied Mathematics, 58, Academic Press, New York-London, 1974.
[6] Euler, L., De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt. In: Opera Omnia Series 1, 14, B. G. Teubner, Berlin, 1925, pp. 124.
[7] Gauss, C. F., Disquisitiones generales circa seriem infinitam . In: Werke, 3, Königlichen Gesellschaft der Wissenschaften, Göttingen, 1866.
[8] Godefroy, M., La fonction gamma: théorie, histoire, bibliographie. Gauthier-Villars, Paris, 1901.
[9] Jahnke, E. and Emde, F., Tables of functions with formulae and curves. 4th Ed., Dover, New York, 1945.
[10] Landau, E., Handbuch der Lehre von der Verteilung der Primzahlen. 2nd ed., Chelsea Publishing Co., New York, 1953.
[11] Remmert, R., Classical topics in complex function theory. Graduate Texts in Mathematics, 172, Springer-Verlag, New York, 1998.
[12] Saidak, F. and Zvengrowski, P., On the modulus of the Riemann zeta function in the critical strip. Math. Slovaca 53(2003), no. 2, 145172.
[13] Srinivasan, G. K., The gamma function: an eclectic tour. Amer. Math. Monthly 114(2007), no. 4, 297315.
[14] Stieltjes, T.-J., Sur le développement de log Γ(a) . J. Math. Pures Appl. (9) 5(1889), 425444.
[15] Whittaker, E. T. and Watson, G. N., A course of modern analysis, an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. Cambridge University Press, Cambridge, 1996.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed