Bernstein, Pick, Poisson and related integral expressions for Lambert W

References

• Anastasselou , E. G. and Ioakimidis , N. I. 1984 . A generalization of the Siewert–Burniston method for the determination of zeros of analytic functions . J. Math. Phys , 25 : 2422 – 2425 .
   Web of Science ®Google Scholar
• Anastasselou , E. G. and Ioakimidis , N. I. 1984 . A new method for obtaining exact analytical formulae for the roots of transcendental functions . Lett. Math. Phys , 8 : 135 – 143 .
   Web of Science ®Google Scholar
• Berg , C. 2005 . “ On a generalized gamma convolution related to the q-calculus ” . In Theory and Applications of Special Functions , Edited by: Ismail , M. E.H. and Koelink , E. 61 – 76 . New York : Springer . Developmental Mathematics Vol. 13
   Google Scholar
• Berg , C. 2008 . “ Stieltjes–Pick–Bernstein–Schoenberg and their connection to complete monotonicity ” . In Positive Definite Functions. From Schoenberg to Space-Time Challenges , Edited by: Mateu , S. and Porcu , E. 15 – 45 . Castellón de la Plana , , Spain : Department of Mathematics, University Jaume I .
   Google Scholar
• Bouwkamp , C. J. 1986 . A conjectured definite integral. Problem 85-16 . SIAM Rev , 28 : 568 – 569 .
   Web of Science ®Google Scholar
• Burniston , E. E. and Siewert , C. E. 1973 . The use of Riemann problems in solving a class of transcendental equations . Proc. Cambridge Philos. Soc , 73 : 111 – 118 .
   Google Scholar
• Caillol , J.-M. 2003 . Some applications of the Lambert W function to classical statistical mechanics . J. Phys. A: Math. Gen , 36 : 10431 – 10442 .
   Google Scholar
• Carathéodory , C. 1958 . “ Theory of Functions of a Complex Variable ” . Vol. 1 , New York : Chelsea .
   Google Scholar
• Cauer , W. 1932 . The Poisson integral for functions with positive real part . Bull. Amer. Math. Soc , 38 : 713 – 717 .
   Google Scholar
• Corless , R. M. , Gonnet , G. H. , Hare , D. E.G. , Jeffrey , D. J. and Knuth , D. E. 1996 . On the Lambert W function . Adv. Comput. Math , 5 : 329 – 359 .
   Web of Science ®Google Scholar
• Flajolet , P. and Sedgewick , R. 2009 . “ Analytic Combinatorics ” . London : Cambridge University Press, Cambridge .
   Google Scholar
• Henrici , P. 1977 . “ Applied and Computational Complex Analysis ” . Vol. 2 , New York : Wiley .
   Google Scholar
• Henrici , P. 1986 . “ Applied and Computational Complex Analysis ” . Vol. 3 , New York : Wiley .
   Google Scholar
• Jeffrey , D. J. , Hare , D. E.G. and Corless , R. M. 1996 . Unwinding the branches of the Lambert W function . Math. Sci , 21 : 1 – 7 .
   Google Scholar
• Kalugin , G. A. , Jeffrey , D. J. , Corless , R. M. and Borwein , P. B. 2011 . Stieltjes and other integral representations for functions of Lambert W . Integral Transforms Spec. Funct , available at http://dx.doi.org/10.1080/10652469.2011.613830
   Web of Science ®Google Scholar
• Kheyfits , A. I. 2004 . Closed-form representations of the Lambert W function . Fract. Calc. Appl. Anal , 7 : 177 – 190 .
   Google Scholar
• Levin , B. Ya. 1996 . “ Lectures on Entire Functions ” . In Translations of Mathematical Monographs , Vol. 150 , Providence , RI : American Mathematical Society .
   Google Scholar
• Markushevich , A. I. 1965 . “ Theory of Functions of a Complex Variable ” . Vol. II , Englewood Cliffs , NJ : Prentice-Hall .
   Google Scholar
• Markushevich , A. I. 1967 . “ Theory of Functions of a Complex Variable ” . Vol. III , Englewood Cliffs , NJ : Prentice-Hall .
   Google Scholar
• Nuttall , A. H. 1985 . A conjectured definite integral. Problem 85-16 . SIAM Rev , 27 : 573
   Google Scholar
• Pakes , A. G. 2011 . Lambert's W, infinite divisibility and Poisson mixtures . J. Math. Anal. Appl , 378 : 480 – 492 .
   Web of Science ®Google Scholar
• Poisson , S.-D. 1823 . Suite du mémoire sur les intégrales définies et sur la sommation des séries . J. École Roy. Polytech , 12 : 404 – 509 .
   Google Scholar
• Schilling , R. L. , Song , R. and Vondraček , Z. 2010 . “ Bernstein Functions. Theory and Applications ” . Berlin : De Gruyter .
   Google Scholar
• Siewert , C. E. and Burniston , E. E. 1973 . Exact analytical solutions of zez=a . J. Math. Anal. Appl , 43 : 626 – 632 .
   Web of Science ®Google Scholar
• A.D. Sokal, Another question about the Lambert W, Private email, 16 October 2008
   Google Scholar
• Corless , R. M. and Jeffrey , D. J. “ The poster ” . In The Lambert W Function Available at http://www.orcca.on.ca/LambertW/
   Google Scholar
• Weideman , J. A.C. 2002 . Numerical integration of periodic functions: A few examples . Amer. Math. Monthly , 109 : 21 – 36 .
   Web of Science ®Google Scholar
• Whittaker , E. T. and Watson , G. N. 1927 . “ A Course of Modern Analysis ” . London : Cambridge University Press, Cambridge .
   Google Scholar
• Wolfram Research, Available at http://functions.wolfram.com/01.31.07.0001.01, Wolfram Research, 2001
   Google Scholar