On the decomposition of generalized incomplete gamma functions with applications to Fourier transforms

AbstractIn this paper we introduce decomposition functions CΓ(α,x;ω), SΓ(α,x;ω), Cγ(α,x;ω) and Sγ(α,x;ω) of the generalized gamma functions. These functions are found useful in the analytic study of the temperature distribution of a semi-infinite solid with periodic boundary conditions and to the theory of Fourier transforms. Several new identities involving the Fourier transforms are investigated and some of the classical ones are recovered as special cases. For numerical and scientific computations, tabular and graphical representations of the functions CΓ(α,x;ω) and SΓ(α,x;ω) are also given

Operator Representation of Fermi-Dirac and Bose-Einstein Integral Functions with Applications

Fermi-Dirac and Bose-Einstein functions arise as quantum statistical distributions. The Riemann zeta function and its extension, the polylogarithm function, arise in the theory of numbers. Though it might not have been expected, these two sets of functions belong to a wider class of functions whose members have operator representations. In particular, we show that the Fermi-Dirac and Bose-Einstein integral functions are expressible as operator representations in terms of themselves. Simpler derivations of previously known results of these functions are obtained by their operator representations

A probablistic proof of the series representation of the MacDonald function with applications

A series representation of the Macdonald function is obtained using the properties of a probability density function and its moment generating function. Some applications of the result are discussed and an open problem is posed

Higher Education Capacity Building in Water Resources Engineering and Management to Support Achieving the Sustainable Development Goal for Water in Pakistan

Achieving the Sustainable Development Goals requires a multi‐pronged approach, with a key element being the development of a trained Community of Practice to sustain the advances in the relevant sectors. The engagement of higher education as a catalyst in the development and capacity building of the next generation of professionals and citizens comprising the Community of Practice is essential to meet the challenges of poverty, climate change, and clean water and to sustain those advances past 2030. This paper describes a capacity building program funded by the United States Agency for International Development to partner the University of Utah, in the United States, with Mehran University of Engineering and Technology, in Pakistan, to create the U.S.‐Pakistan Center for Advanced Studies in Water (USPCASW). The USPCASW program includes six core components of Curriculum Reform, Applied Research, Exchanges and Training, Governance, Gender Equity, and Sustainability. This paper describes the project, the activities for each component, and the multi‐level assessment of the program, activities, and impact. The paper also highlights the overarching impact of the program and its alignment with achieving the Sustainable Development Goal for Water. Following the description of the program components and assessment, the paper concludes with a discussion of challenges and lessons learned