A Note on the Lambert W Function: Bernstein and Stieltjes Properties for a Creep Model in Linear Viscoelasticity
The purpose of this note is to propose an application of the Lambert <i>W</i> function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch, <inline-formula><math xmlns="http://ww...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-08-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/9/1654 |
| _version_ | 1797576621837778944 |
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| author | Francesco Mainardi Enrico Masina Juan Luis González-Santander |
| author_facet | Francesco Mainardi Enrico Masina Juan Luis González-Santander |
| author_sort | Francesco Mainardi |
| collection | DOAJ |
| description | The purpose of this note is to propose an application of the Lambert <i>W</i> function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, in a peculiar model of creep with two spectral functions in frequency that completely characterize the creep model. In order to calculate these spectral functions, it turns out that the conjugate symmetry property of the Lambert <i>W</i> function along its branch cut on the negative real axis is essential. We supplement our analysis by computing the corresponding relaxation function and providing the plots of all computed functions. |
| first_indexed | 2024-03-10T21:54:33Z |
| format | Article |
| id | doaj.art-6b9fb82769b94de8a6866ae058c2c3c1 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T21:54:33Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-6b9fb82769b94de8a6866ae058c2c3c12023-11-19T13:10:47ZengMDPI AGSymmetry2073-89942023-08-01159165410.3390/sym15091654A Note on the Lambert W Function: Bernstein and Stieltjes Properties for a Creep Model in Linear ViscoelasticityFrancesco Mainardi0Enrico Masina1Juan Luis González-Santander2Department of Physics, University of Bologna, and INFN, Via Irnerio 46, I-40126 Bologna, ItalyDepartment of Mathematics, University of Bologna, Piazza Porta San Donato, I-40127 Bologna, ItalyDepartment de Mathematics, University of Oviedo, C Leopoldo Calvo Sotelo 18, 33007 Oviedo, SpainThe purpose of this note is to propose an application of the Lambert <i>W</i> function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, in a peculiar model of creep with two spectral functions in frequency that completely characterize the creep model. In order to calculate these spectral functions, it turns out that the conjugate symmetry property of the Lambert <i>W</i> function along its branch cut on the negative real axis is essential. We supplement our analysis by computing the corresponding relaxation function and providing the plots of all computed functions.https://www.mdpi.com/2073-8994/15/9/1654Lambert functioncompletely monotonic functionsBernstein functionsStieltjes functionsLaplace transformStieltjes transform |
| spellingShingle | Francesco Mainardi Enrico Masina Juan Luis González-Santander A Note on the Lambert W Function: Bernstein and Stieltjes Properties for a Creep Model in Linear Viscoelasticity Symmetry Lambert function completely monotonic functions Bernstein functions Stieltjes functions Laplace transform Stieltjes transform |
| title | A Note on the Lambert W Function: Bernstein and Stieltjes Properties for a Creep Model in Linear Viscoelasticity |
| title_full | A Note on the Lambert W Function: Bernstein and Stieltjes Properties for a Creep Model in Linear Viscoelasticity |
| title_fullStr | A Note on the Lambert W Function: Bernstein and Stieltjes Properties for a Creep Model in Linear Viscoelasticity |
| title_full_unstemmed | A Note on the Lambert W Function: Bernstein and Stieltjes Properties for a Creep Model in Linear Viscoelasticity |
| title_short | A Note on the Lambert W Function: Bernstein and Stieltjes Properties for a Creep Model in Linear Viscoelasticity |
| title_sort | note on the lambert w function bernstein and stieltjes properties for a creep model in linear viscoelasticity |
| topic | Lambert function completely monotonic functions Bernstein functions Stieltjes functions Laplace transform Stieltjes transform |
| url | https://www.mdpi.com/2073-8994/15/9/1654 |
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