Harmonic Functions on Four Dimensions
Fuente:
arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915569831247872 |
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| author | Johnston, William Moore, Sara Wahl, Rebecca G. |
| author_facet | Johnston, William Moore, Sara Wahl, Rebecca G. |
| contents | This paper develops theory for a newly-defined bicomplex hyperbolic harmonic function with four real-dimensional inputs, in a way that generalizes the connection between real harmonic functions with two real-dimensional inputs and complex analytic functions. For example, every bicomplex hyperbolic harmonic function appears as this paper's newly-defined hyperbolic real part of a bicomplex analytic function, just as every real harmonic function with two real-dimensional inputs is the real part of a complex analytic function. In addition, this connection produces a unique (up to additive constant) and newly-defined hyperbolic harmonic conjugate function, just as every real harmonic function has a unique (up to additive constant) real harmonic conjugate. Finally, the paper determines a bicomplex Poisson kernel function that produces a corresponding integral representation for bicomplex harmonic functions, one that generalizes the complex harmonic function Poisson integral representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_17033 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Harmonic Functions on Four Dimensions Johnston, William Moore, Sara Wahl, Rebecca G. Complex Variables Functional Analysis 31A05, 31A30 This paper develops theory for a newly-defined bicomplex hyperbolic harmonic function with four real-dimensional inputs, in a way that generalizes the connection between real harmonic functions with two real-dimensional inputs and complex analytic functions. For example, every bicomplex hyperbolic harmonic function appears as this paper's newly-defined hyperbolic real part of a bicomplex analytic function, just as every real harmonic function with two real-dimensional inputs is the real part of a complex analytic function. In addition, this connection produces a unique (up to additive constant) and newly-defined hyperbolic harmonic conjugate function, just as every real harmonic function has a unique (up to additive constant) real harmonic conjugate. Finally, the paper determines a bicomplex Poisson kernel function that produces a corresponding integral representation for bicomplex harmonic functions, one that generalizes the complex harmonic function Poisson integral representation. |
| title | Harmonic Functions on Four Dimensions |
| topic | Complex Variables Functional Analysis 31A05, 31A30 |
| url | https://arxiv.org/abs/2311.17033 |