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The Prabhakar Function is a mathematical concept named after the Indian mathematician T. R. Prabhakar. It was first introduced by Prabhakar in his research work in the field of fractional calculus. The Prabhakar function is a special type of function that arises in the study of fractional calculus and has numerous applications in various fields such as physics, engineering, finance, and others.

Fractional Calculus is a branch of mathematics that deals with the study of derivatives and integrals of fractional order. Unlike traditional calculus, which only deals with derivatives and integrals of integer orders, fractional calculus extends the notion of derivatives and integrals to fractional orders. The Prabhakar function is an important tool in fractional calculus, as it helps to generalize the concepts of derivatives and integrals to fractional orders.

Definition

The Prabhakar function is defined as a generalization of the exponential function and has a unique property that it can be used to solve fractional differential equations. This makes the Prabhakar function particularly useful in modeling real-world problems that involve fractional derivatives, such as in the study of viscoelastic materials, anomalous diffusion, and finance.

Applications

In finance, the Prabhakar function has been used to model the pricing of options and derivatives. By using the Prabhakar function, researchers have been able to accurately model the pricing of financial products that are sensitive to the volatility of underlying assets.

In physics, the Prabhakar function has been used in the study of viscoelastic materials. By modeling the behavior of these materials with the Prabhakar function, researchers have been able to gain insights into the mechanical properties of these materials and their response to external loads. The Prabhakar function has also been used in the context of the Havriliak–Negami model, which is a widely used model for describing the dielectric properties of polymers and other complex materials.

In conclusion, the Prabhakar function is an important concept in the field of fractional calculus that has numerous applications in various fields. It was first introduced by T. R. Prabhakar and continues to be an active area of research, with new applications being discovered and developed.

References:

Prabhakar, T. R. (1971). A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Mathematical Journal, 19(1), 7-15.

Havriliak, S., & Negami, S. (1967). A complex plane analysis of dielectric and mechanical relaxations in some polymers. Journal of Polymer Science Part C: Polymer Symposia, 16(64), 99-117.