On the Cauchy transform of complex powers of the identity function
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866917566764548096 |
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| author | Faktor, Benjamin Kuhn, Michael Shemy, Gahl |
| author_facet | Faktor, Benjamin Kuhn, Michael Shemy, Gahl |
| contents | The integral $\int_{|z|=1} \frac{z^β}{z-α} dz$ for $β=\frac{1}{2}$ has been comprehensively studied by Mortini and Rupp for pedagogical purposes. We write for a similar purpose, elaborating on their work with the more general consideration $β\in \mathbb{C}$. This culminates in an explicit solution in terms of the hypergeometric function for $|α| \neq 1$ and any $β\in \mathbb{C}$. For rational $β$, the integral is reduced to a finite sum. A differential equation in $α$ is derived for this integral, which we show has similar properties to the hypergeometric equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_07649 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Cauchy transform of complex powers of the identity function Faktor, Benjamin Kuhn, Michael Shemy, Gahl Complex Variables 30E20 (Primary) 33C05 (Secondary) The integral $\int_{|z|=1} \frac{z^β}{z-α} dz$ for $β=\frac{1}{2}$ has been comprehensively studied by Mortini and Rupp for pedagogical purposes. We write for a similar purpose, elaborating on their work with the more general consideration $β\in \mathbb{C}$. This culminates in an explicit solution in terms of the hypergeometric function for $|α| \neq 1$ and any $β\in \mathbb{C}$. For rational $β$, the integral is reduced to a finite sum. A differential equation in $α$ is derived for this integral, which we show has similar properties to the hypergeometric equation. |
| title | On the Cauchy transform of complex powers of the identity function |
| topic | Complex Variables 30E20 (Primary) 33C05 (Secondary) |
| url | https://arxiv.org/abs/2209.07649 |