Lambert W function (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Lambert W function" in English language version.

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Chow, Timothy Y. (1999), "What is a closed-form number?", American Mathematical Monthly, 106 (5): 440–448, arXiv:math/9805045, doi:10.2307/2589148, JSTOR 2589148, MR 1699262.
Kalugin, German A.; Jeffrey, David J.; Corless, Robert M. (2012). "Bernstein, Pick, Poisson and related integral expressions for Lambert $W$ " (PDF). Integral Transforms and Special Functions. 23 (11): 817–829. doi:10.1080/10652469.2011.640327. MR 2989751. See Theorem 3.4, p. 821 of published version (p. 5 of preprint).

arxiv.org

Chow, Timothy Y. (1999), "What is a closed-form number?", American Mathematical Monthly, 106 (5): 440–448, arXiv:math/9805045, doi:10.2307/2589148, JSTOR 2589148, MR 1699262.
Chatzigeorgiou, I. (2013). "Bounds on the Lambert function and their Application to the Outage Analysis of User Cooperation". IEEE Communications Letters. 17 (8): 1505–1508. arXiv:1601.04895. doi:10.1109/LCOMM.2013.070113.130972. S2CID 10062685.
https://arxiv.org/abs/2005.03051 J. Batson et al., "A COMPARISON OF GROUP TESTING ARCHITECTURES FOR COVID-19 TESTING".
A.Z. Broder, "A Note on Double Pooling Tests".
A.M. Ishkhanyan, "The Lambert W barrier – an exactly solvable confluent hypergeometric potential".
Deur, Alexandre; Brodsky, Stanley J.; De Téramond, Guy F. (2016). "The QCD running coupling". Progress in Particle and Nuclear Physics. 90: 1–74. arXiv:1604.08082. Bibcode:2016PrPNP..90....1D. doi:10.1016/j.ppnp.2016.04.003. S2CID 118854278.
de la Madrid, R. (2017). "Numerical calculation of the decay widths, the decay constants, and the decay energy spectra of the resonances of the delta-shell potential". Nucl. Phys. A. 962: 24–45. arXiv:1704.00047. Bibcode:2017NuPhA.962...24D. doi:10.1016/j.nuclphysa.2017.03.006. S2CID 119218907.
Floratos, Emmanuel; Georgiou, George; Linardopoulos, Georgios (2014). "Large-Spin Expansions of GKP Strings". JHEP. 2014 (3): 0180. arXiv:1311.5800. Bibcode:2014JHEP...03..018F. doi:10.1007/JHEP03(2014)018. S2CID 53355961.
Floratos, Emmanuel; Linardopoulos, Georgios (2015). "Large-Spin and Large-Winding Expansions of Giant Magnons and Single Spikes". Nucl. Phys. B. 897: 229–275. arXiv:1406.0796. Bibcode:2015NuPhB.897..229F. doi:10.1016/j.nuclphysb.2015.05.021. S2CID 118526569.
Mendonça, J. R. G. (2019). "Electromagnetic surface wave propagation in a metallic wire and the Lambert $W$ function". American Journal of Physics. 87 (6): 476–484. arXiv:1812.07456. Bibcode:2019AmJPh..87..476M. doi:10.1119/1.5100943. S2CID 119661071.
Scott, T. C.; Mann, R. B.; Martinez Ii, Roberto E. (2006). "General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function". AAECC (Applicable Algebra in Engineering, Communication and Computing). 17 (1): 41–47. arXiv:math-ph/0607011. Bibcode:2006math.ph...7011S. doi:10.1007/s00200-006-0196-1. S2CID 14664985.
Farrugia, P. S.; Mann, R. B.; Scott, T. C. (2007). "N-body Gravity and the Schrödinger Equation". Class. Quantum Grav. 24 (18): 4647–4659. arXiv:gr-qc/0611144. Bibcode:2007CQGra..24.4647F. doi:10.1088/0264-9381/24/18/006. S2CID 119365501.
Scott, T. C.; Aubert-Frécon, M.; Grotendorst, J. (2006). "New Approach for the Electronic Energies of the Hydrogen Molecular Ion". Chem. Phys. 324 (2–3): 323–338. arXiv:physics/0607081. Bibcode:2006CP....324..323S. CiteSeerX 10.1.1.261.9067. doi:10.1016/j.chemphys.2005.10.031. S2CID 623114.

bath.ac.uk

opus.bath.ac.uk

Bronstein, Manuel; Corless, Robert M.; Davenport, James H.; Jeffrey, D. J. (2008). "Algebraic properties of the Lambert ⁠ $W$ ⁠ function from a result of Rosenlicht and of Liouville" (PDF). Integral Transforms and Special Functions. 19 (10): 709–712. doi:10.1080/10652460802332342. S2CID 120069437. Archived (PDF) from the original on 2015-12-11.

boost.org

"Lambert W function - Boost libraries".

crates.io

Sörngård, Johanna (2024-07-28), lambert_w, retrieved 2024-09-11

dartmouth.edu

math.dartmouth.edu

Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus". Acta Acad. Scient. Petropol. 2, 29–51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350–369, 1921.

doi.org

Lehtonen, Jussi (April 2016), Rees, Mark (ed.), "The Lambert W function in ecological and evolutionary models", Methods in Ecology and Evolution, 7 (9): 1110–1118, Bibcode:2016MEcEv...7.1110L, doi:10.1111/2041-210x.12568, S2CID 124111881
Chow, Timothy Y. (1999), "What is a closed-form number?", American Mathematical Monthly, 106 (5): 440–448, arXiv:math/9805045, doi:10.2307/2589148, JSTOR 2589148, MR 1699262.
Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. (1996). "On the LambertW function" (PDF). Advances in Computational Mathematics. 5: 329–359. doi:10.1007/BF02124750. S2CID 29028411.
Scott, TC; Babb, JF; Dalgarno, A; Morgan, John D (Aug 15, 1993). "The calculation of exchange forces: General results and specific models". J. Chem. Phys. 99 (4). American Institute of Physics: 2841–2854. Bibcode:1993JChPh..99.2841S. doi:10.1063/1.465193. ISSN 0021-9606.
Mező, István (2022). The Lambert W Function: Its Generalizations and Applications. doi:10.1201/9781003168102. ISBN 9781003168102. S2CID 247491347.
Bronstein, Manuel; Corless, Robert M.; Davenport, James H.; Jeffrey, D. J. (2008). "Algebraic properties of the Lambert ⁠ $W$ ⁠ function from a result of Rosenlicht and of Liouville" (PDF). Integral Transforms and Special Functions. 19 (10): 709–712. doi:10.1080/10652460802332342. S2CID 120069437. Archived (PDF) from the original on 2015-12-11.
Chatzigeorgiou, I. (2013). "Bounds on the Lambert function and their Application to the Outage Analysis of User Cooperation". IEEE Communications Letters. 17 (8): 1505–1508. arXiv:1601.04895. doi:10.1109/LCOMM.2013.070113.130972. S2CID 10062685.
Iacono, Roberto; Boyd, John P. (2017-12-01). "New approximations to the principal real-valued branch of the Lambert W-function". Advances in Computational Mathematics. 43 (6): 1403–1436. doi:10.1007/s10444-017-9530-3. ISSN 1572-9044. S2CID 254184098.
Kalugin, German A.; Jeffrey, David J.; Corless, Robert M. (2012). "Bernstein, Pick, Poisson and related integral expressions for Lambert $W$ " (PDF). Integral Transforms and Special Functions. 23 (11): 817–829. doi:10.1080/10652469.2011.640327. MR 2989751. See Theorem 3.4, p. 821 of published version (p. 5 of preprint).
Robert M., Corless; David J., Jeffrey; Donald E., Knuth (1997). "A sequence of series for the Lambert W function". Proceedings of the 1997 international symposium on Symbolic and algebraic computation - ISSAC '97. pp. 197–204. doi:10.1145/258726.258783. ISBN 978-0897918756. S2CID 6274712.
More, A. A. (2006). "Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes". Chemical Engineering Science. 61 (16): 5515–5519. Bibcode:2006ChEnS..61.5515M. doi:10.1016/j.ces.2006.04.003.
Pellegrini, C. C.; Zappi, G. A.; Vilalta-Alonso, G. (2022-05-12). "An Analytical Solution for the Time-Dependent Flow in Simple Branch Hydraulic Systems with Centrifugal Pumps". Arabian Journal for Science and Engineering. 47 (12): 16273–16287. doi:10.1007/s13369-022-06864-9. ISSN 2193-567X. S2CID 248762601.
Sotero, Roberto C.; Iturria-Medina, Yasser (2011). "From Blood oxygenation level dependent (BOLD) signals to brain temperature maps". Bull Math Biol (Submitted manuscript). 73 (11): 2731–47. doi:10.1007/s11538-011-9645-5. PMID 21409512. S2CID 12080132.
Braun, Artur; Wokaun, Alexander; Hermanns, Heinz-Guenter (2003). "Analytical Solution to a Growth Problem with Two Moving Boundaries". Appl Math Model. 27 (1): 47–52. doi:10.1016/S0307-904X(02)00085-9.
Braun, Artur; Baertsch, Martin; Schnyder, Bernhard; Koetz, Ruediger (2000). "A Model for the film growth in samples with two moving boundaries – An Application and Extension of the Unreacted-Core Model". Chem Eng Sci. 55 (22): 5273–5282. doi:10.1016/S0009-2509(00)00143-3.
Asadian, M; Saeedi, H; Yadegari, M; Shojaee, M (June 2014). "Determinations of equilibrium segregation, effective segregation and diffusion coefficients for Nd+3 doped in molten YAG". Journal of Crystal Growth. 396 (15): 61–65. Bibcode:2014JCrGr.396...61A. doi:10.1016/j.jcrysgro.2014.03.028. https://doi.org/10.1016/j.jcrysgro.2014.03.028
Asadian, M; Zabihi, F; Saeedi, H (March 2024). "Segregation and constitutional supercooling in Nd:YAG Czochralski crystal growth". Journal of Crystal Growth. 630: 127605. Bibcode:2024JCrGr.63027605A. doi:10.1016/j.jcrysgro.2024.127605. S2CID 267414096. https://doi.org/10.1016/j.jcrysgro.2024.127605
Braun, Artur; Briggs, Keith M.; Boeni, Peter (2003). "Analytical solution to Matthews' and Blakeslee's critical dislocation formation thickness of epitaxially grown thin films". J Cryst Growth. 241 (1–2): 231–234. Bibcode:2002JCrGr.241..231B. doi:10.1016/S0022-0248(02)00941-7.
Rudolf Hanel, Stefan Thurner (2020). "Boosting test-efficiency by pooled testing for SARS-CoV-2—Formula for optimal pool size". PLOS ONE. 15, 11 (11): e0240652. Bibcode:2020PLoSO..1540652H. doi:10.1371/journal.pone.0240652. PMC 7641378. PMID 33147228.
Deur, Alexandre; Brodsky, Stanley J.; De Téramond, Guy F. (2016). "The QCD running coupling". Progress in Particle and Nuclear Physics. 90: 1–74. arXiv:1604.08082. Bibcode:2016PrPNP..90....1D. doi:10.1016/j.ppnp.2016.04.003. S2CID 118854278.
de la Madrid, R. (2017). "Numerical calculation of the decay widths, the decay constants, and the decay energy spectra of the resonances of the delta-shell potential". Nucl. Phys. A. 962: 24–45. arXiv:1704.00047. Bibcode:2017NuPhA.962...24D. doi:10.1016/j.nuclphysa.2017.03.006. S2CID 119218907.
Bot, A.; Dewi, B.P.C.; Venema, P. (2021). "Phase-separating binary polymer mixtures: the degeneracy of the virial coefficients and their extraction from phase diagrams". ACS Omega. 6 (11): 7862–7878. doi:10.1021/acsomega.1c00450. PMC 7992149. PMID 33778298.
Cardoso, T. R.; de Castro, A. S. (2005). "The blackbody radiation in a $D$ -dimensional universe". Rev. Bras. Ens. Fis. 27 (4): 559–563. doi:10.1590/S1806-11172005000400007. hdl:11449/211894.
Floratos, Emmanuel; Georgiou, George; Linardopoulos, Georgios (2014). "Large-Spin Expansions of GKP Strings". JHEP. 2014 (3): 0180. arXiv:1311.5800. Bibcode:2014JHEP...03..018F. doi:10.1007/JHEP03(2014)018. S2CID 53355961.
Floratos, Emmanuel; Linardopoulos, Georgios (2015). "Large-Spin and Large-Winding Expansions of Giant Magnons and Single Spikes". Nucl. Phys. B. 897: 229–275. arXiv:1406.0796. Bibcode:2015NuPhB.897..229F. doi:10.1016/j.nuclphysb.2015.05.021. S2CID 118526569.
Mendonça, J. R. G. (2019). "Electromagnetic surface wave propagation in a metallic wire and the Lambert $W$ function". American Journal of Physics. 87 (6): 476–484. arXiv:1812.07456. Bibcode:2019AmJPh..87..476M. doi:10.1119/1.5100943. S2CID 119661071.
Scott, T. C.; Mann, R. B.; Martinez Ii, Roberto E. (2006). "General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function". AAECC (Applicable Algebra in Engineering, Communication and Computing). 17 (1): 41–47. arXiv:math-ph/0607011. Bibcode:2006math.ph...7011S. doi:10.1007/s00200-006-0196-1. S2CID 14664985.
Scott, T. C.; Fee, G.; Grotendorst, J. (2013). "Asymptotic series of Generalized Lambert W Function". SIGSAM (ACM Special Interest Group in Symbolic and Algebraic Manipulation). 47 (185): 75–83. doi:10.1145/2576802.2576804. S2CID 15370297.
Scott, T. C.; Fee, G.; Grotendorst, J.; Zhang, W.Z. (2014). "Numerics of the Generalized Lambert W Function". SIGSAM. 48 (1/2): 42–56. doi:10.1145/2644288.2644298. S2CID 15776321.
Farrugia, P. S.; Mann, R. B.; Scott, T. C. (2007). "N-body Gravity and the Schrödinger Equation". Class. Quantum Grav. 24 (18): 4647–4659. arXiv:gr-qc/0611144. Bibcode:2007CQGra..24.4647F. doi:10.1088/0264-9381/24/18/006. S2CID 119365501.
Scott, T. C.; Aubert-Frécon, M.; Grotendorst, J. (2006). "New Approach for the Electronic Energies of the Hydrogen Molecular Ion". Chem. Phys. 324 (2–3): 323–338. arXiv:physics/0607081. Bibcode:2006CP....324..323S. CiteSeerX 10.1.1.261.9067. doi:10.1016/j.chemphys.2005.10.031. S2CID 623114.
Maignan, Aude; Scott, T. C. (2016). "Fleshing out the Generalized Lambert W Function". SIGSAM. 50 (2): 45–60. doi:10.1145/2992274.2992275. S2CID 53222884.
Scott, T. C.; Lüchow, A.; Bressanini, D.; Morgan, J. D. III (2007). "The Nodal Surfaces of Helium Atom Eigenfunctions" (PDF). Phys. Rev. A. 75 (6): 060101. Bibcode:2007PhRvA..75f0101S. doi:10.1103/PhysRevA.75.060101. hdl:11383/1679348. Archived (PDF) from the original on 2017-09-22.
Lóczi, Lajos (2022-11-15). "Guaranteed- and high-precision evaluation of the Lambert W function". Applied Mathematics and Computation. 433: 127406. doi:10.1016/j.amc.2022.127406. hdl:10831/89771. ISSN 0096-3003.
Fukushima, Toshio (2020-11-25). "Precise and fast computation of Lambert W function by piecewise minimax rational function approximation with variable transformation". doi:10.13140/RG.2.2.30264.37128.

emis.de

A. Hoorfar, M. Hassani, Inequalities on the Lambert W Function and Hyperpower Function, JIPAM, Theorem 2.7, page 7, volume 9, issue 2, article 51. 2008.

gnu.org

"Lambert W Functions - GNU Scientific Library (GSL)".

handle.net

hdl.handle.net

Cardoso, T. R.; de Castro, A. S. (2005). "The blackbody radiation in a $D$ -dimensional universe". Rev. Bras. Ens. Fis. 27 (4): 559–563. doi:10.1590/S1806-11172005000400007. hdl:11449/211894.
Scott, T. C.; Lüchow, A.; Bressanini, D.; Morgan, J. D. III (2007). "The Nodal Surfaces of Helium Atom Eigenfunctions" (PDF). Phys. Rev. A. 75 (6): 060101. Bibcode:2007PhRvA..75f0101S. doi:10.1103/PhysRevA.75.060101. hdl:11383/1679348. Archived (PDF) from the original on 2017-09-22.
Lóczi, Lajos (2022-11-15). "Guaranteed- and high-precision evaluation of the Lambert W function". Applied Mathematics and Computation. 433: 127406. doi:10.1016/j.amc.2022.127406. hdl:10831/89771. ISSN 0096-3003.

harvard.edu

ui.adsabs.harvard.edu

Lehtonen, Jussi (April 2016), Rees, Mark (ed.), "The Lambert W function in ecological and evolutionary models", Methods in Ecology and Evolution, 7 (9): 1110–1118, Bibcode:2016MEcEv...7.1110L, doi:10.1111/2041-210x.12568, S2CID 124111881
Scott, TC; Babb, JF; Dalgarno, A; Morgan, John D (Aug 15, 1993). "The calculation of exchange forces: General results and specific models". J. Chem. Phys. 99 (4). American Institute of Physics: 2841–2854. Bibcode:1993JChPh..99.2841S. doi:10.1063/1.465193. ISSN 0021-9606.
More, A. A. (2006). "Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes". Chemical Engineering Science. 61 (16): 5515–5519. Bibcode:2006ChEnS..61.5515M. doi:10.1016/j.ces.2006.04.003.
Asadian, M; Saeedi, H; Yadegari, M; Shojaee, M (June 2014). "Determinations of equilibrium segregation, effective segregation and diffusion coefficients for Nd+3 doped in molten YAG". Journal of Crystal Growth. 396 (15): 61–65. Bibcode:2014JCrGr.396...61A. doi:10.1016/j.jcrysgro.2014.03.028. https://doi.org/10.1016/j.jcrysgro.2014.03.028
Asadian, M; Zabihi, F; Saeedi, H (March 2024). "Segregation and constitutional supercooling in Nd:YAG Czochralski crystal growth". Journal of Crystal Growth. 630: 127605. Bibcode:2024JCrGr.63027605A. doi:10.1016/j.jcrysgro.2024.127605. S2CID 267414096. https://doi.org/10.1016/j.jcrysgro.2024.127605
Braun, Artur; Briggs, Keith M.; Boeni, Peter (2003). "Analytical solution to Matthews' and Blakeslee's critical dislocation formation thickness of epitaxially grown thin films". J Cryst Growth. 241 (1–2): 231–234. Bibcode:2002JCrGr.241..231B. doi:10.1016/S0022-0248(02)00941-7.
Rudolf Hanel, Stefan Thurner (2020). "Boosting test-efficiency by pooled testing for SARS-CoV-2—Formula for optimal pool size". PLOS ONE. 15, 11 (11): e0240652. Bibcode:2020PLoSO..1540652H. doi:10.1371/journal.pone.0240652. PMC 7641378. PMID 33147228.
Deur, Alexandre; Brodsky, Stanley J.; De Téramond, Guy F. (2016). "The QCD running coupling". Progress in Particle and Nuclear Physics. 90: 1–74. arXiv:1604.08082. Bibcode:2016PrPNP..90....1D. doi:10.1016/j.ppnp.2016.04.003. S2CID 118854278.
de la Madrid, R. (2017). "Numerical calculation of the decay widths, the decay constants, and the decay energy spectra of the resonances of the delta-shell potential". Nucl. Phys. A. 962: 24–45. arXiv:1704.00047. Bibcode:2017NuPhA.962...24D. doi:10.1016/j.nuclphysa.2017.03.006. S2CID 119218907.
Floratos, Emmanuel; Georgiou, George; Linardopoulos, Georgios (2014). "Large-Spin Expansions of GKP Strings". JHEP. 2014 (3): 0180. arXiv:1311.5800. Bibcode:2014JHEP...03..018F. doi:10.1007/JHEP03(2014)018. S2CID 53355961.
Floratos, Emmanuel; Linardopoulos, Georgios (2015). "Large-Spin and Large-Winding Expansions of Giant Magnons and Single Spikes". Nucl. Phys. B. 897: 229–275. arXiv:1406.0796. Bibcode:2015NuPhB.897..229F. doi:10.1016/j.nuclphysb.2015.05.021. S2CID 118526569.
Mendonça, J. R. G. (2019). "Electromagnetic surface wave propagation in a metallic wire and the Lambert $W$ function". American Journal of Physics. 87 (6): 476–484. arXiv:1812.07456. Bibcode:2019AmJPh..87..476M. doi:10.1119/1.5100943. S2CID 119661071.
Scott, T. C.; Mann, R. B.; Martinez Ii, Roberto E. (2006). "General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function". AAECC (Applicable Algebra in Engineering, Communication and Computing). 17 (1): 41–47. arXiv:math-ph/0607011. Bibcode:2006math.ph...7011S. doi:10.1007/s00200-006-0196-1. S2CID 14664985.
Farrugia, P. S.; Mann, R. B.; Scott, T. C. (2007). "N-body Gravity and the Schrödinger Equation". Class. Quantum Grav. 24 (18): 4647–4659. arXiv:gr-qc/0611144. Bibcode:2007CQGra..24.4647F. doi:10.1088/0264-9381/24/18/006. S2CID 119365501.
Scott, T. C.; Aubert-Frécon, M.; Grotendorst, J. (2006). "New Approach for the Electronic Energies of the Hydrogen Molecular Ion". Chem. Phys. 324 (2–3): 323–338. arXiv:physics/0607081. Bibcode:2006CP....324..323S. CiteSeerX 10.1.1.261.9067. doi:10.1016/j.chemphys.2005.10.031. S2CID 623114.
Scott, T. C.; Lüchow, A.; Bressanini, D.; Morgan, J. D. III (2007). "The Nodal Surfaces of Helium Atom Eigenfunctions" (PDF). Phys. Rev. A. 75 (6): 060101. Bibcode:2007PhRvA..75f0101S. doi:10.1103/PhysRevA.75.060101. hdl:11383/1679348. Archived (PDF) from the original on 2017-09-22.

ieee.org

ieeexplore.ieee.org

F. Nielsen, "Jeffreys Centroids: A Closed-Form Expression for Positive Histograms and a Guaranteed Tight Approximation for Frequency Histograms"

isa-afp.org

https://isa-afp.org/entries/Lambert_W.html Note: although one of the assumptions of the relevant lemma states that x must be > 1/e, inspection of said lemma reveals that this assumption is unused. The lower bound is in fact x > 0. The reason for the branch switch at e is simple: for x > 1 there are always two solutions, −ln x and another one that you'd get from the x on the other side of e that would feed the same value to W; these must crossover at x = e: [1] W_n cannot distinguish a value of ln x/x from an x < e from the same value from the other x > e, so it cannot flip the order of its return values.

jstor.org

Chow, Timothy Y. (1999), "What is a closed-form number?", American Mathematical Monthly, 106 (5): 440–448, arXiv:math/9805045, doi:10.2307/2589148, JSTOR 2589148, MR 1699262.

kuttaka.org

Lambert J. H., "Observationes variae in mathesin puram", Acta Helveticae physico-mathematico-anatomico-botanico-medica, Band III, 128–168, 1758.

maplesoft.com

"LambertW - Maple Help".

mathworks.com.au

lambertw – MATLAB

metacpan.org

ntheory - MetaCPAN

nature.com

precedings.nature.com

Sotero, Roberto C.; Iturria-Medina, Yasser (2011). "From Blood oxygenation level dependent (BOLD) signals to brain temperature maps". Bull Math Biol (Submitted manuscript). 73 (11): 2731–47. doi:10.1007/s11538-011-9645-5. PMID 21409512. S2CID 12080132.

nih.gov

pubmed.ncbi.nlm.nih.gov

Sotero, Roberto C.; Iturria-Medina, Yasser (2011). "From Blood oxygenation level dependent (BOLD) signals to brain temperature maps". Bull Math Biol (Submitted manuscript). 73 (11): 2731–47. doi:10.1007/s11538-011-9645-5. PMID 21409512. S2CID 12080132.
Rudolf Hanel, Stefan Thurner (2020). "Boosting test-efficiency by pooled testing for SARS-CoV-2—Formula for optimal pool size". PLOS ONE. 15, 11 (11): e0240652. Bibcode:2020PLoSO..1540652H. doi:10.1371/journal.pone.0240652. PMC 7641378. PMID 33147228.
Bot, A.; Dewi, B.P.C.; Venema, P. (2021). "Phase-separating binary polymer mixtures: the degeneracy of the virial coefficients and their extraction from phase diagrams". ACS Omega. 6 (11): 7862–7878. doi:10.1021/acsomega.1c00450. PMC 7992149. PMID 33778298.

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https://isa-afp.org/entries/Lambert_W.html Note: although one of the assumptions of the relevant lemma states that x must be > 1/e, inspection of said lemma reveals that this assumption is unused. The lower bound is in fact x > 0. The reason for the branch switch at e is simple: for x > 1 there are always two solutions, −ln x and another one that you'd get from the x on the other side of e that would feed the same value to W; these must crossover at x = e: [1] W_n cannot distinguish a value of ln x/x from an x < e from the same value from the other x > e, so it cannot flip the order of its return values.

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Scott, TC; Babb, JF; Dalgarno, A; Morgan, John D (Aug 15, 1993). "The calculation of exchange forces: General results and specific models". J. Chem. Phys. 99 (4). American Institute of Physics: 2841–2854. Bibcode:1993JChPh..99.2841S. doi:10.1063/1.465193. ISSN 0021-9606.
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Lóczi, Lajos (2022-11-15). "Guaranteed- and high-precision evaluation of the Lambert W function". Applied Mathematics and Computation. 433: 127406. doi:10.1016/j.amc.2022.127406. hdl:10831/89771. ISSN 0096-3003.