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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2508.07939 |
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| _version_ | 1866916891191148544 |
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| author | Pant, Prakash Dhungana, Hem Lal Rokaya, Sudip |
| author_facet | Pant, Prakash Dhungana, Hem Lal Rokaya, Sudip |
| contents | The Gaussian integral, denoted as \( \int_{-\infty}^{\infty} e^{-x^2} dx \), plays a significant role in mathematical literature. In this paper, we explore a family of integrals related to Gaussian functions. Specifically, we introduce generalized Gaussian integrals, represented as \( \int_{0}^{\infty} e^{-x^n} dx \), and two distinct types of Gaussian-like integrals:
1. Type I: \( \int_{0}^{\infty} e^{-f(x)^2} dx \), and 2. Type II: \( \int_{0}^{\infty} e^{-x^2} f(x) dx \),
where f(x) is a continuous function. The study of integrals related to Gaussian-like functions has been explored in the work of Huang and Dominy \cite{Dnd}. Our approach to evaluating these integrals relies on specialized functions, including error functions, complementary error functions, imaginary error functions, and Basel functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07939 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Analysis of the Generalized Gaussian Integrals and Gaussian Like Integrals of Type I and II Pant, Prakash Dhungana, Hem Lal Rokaya, Sudip Complex Variables The Gaussian integral, denoted as \( \int_{-\infty}^{\infty} e^{-x^2} dx \), plays a significant role in mathematical literature. In this paper, we explore a family of integrals related to Gaussian functions. Specifically, we introduce generalized Gaussian integrals, represented as \( \int_{0}^{\infty} e^{-x^n} dx \), and two distinct types of Gaussian-like integrals: 1. Type I: \( \int_{0}^{\infty} e^{-f(x)^2} dx \), and 2. Type II: \( \int_{0}^{\infty} e^{-x^2} f(x) dx \), where f(x) is a continuous function. The study of integrals related to Gaussian-like functions has been explored in the work of Huang and Dominy \cite{Dnd}. Our approach to evaluating these integrals relies on specialized functions, including error functions, complementary error functions, imaginary error functions, and Basel functions. |
| title | An Analysis of the Generalized Gaussian Integrals and Gaussian Like Integrals of Type I and II |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2508.07939 |