A Brief History and Exposition of the Fundamental Theory of Fractional Calculus

This opening lecture is intended to serve as a propaedeutic for the papers to be presented at this conference whose nonhomogeneous audience includes scientists, mathematicians, engineers and educators. This expository and developmental lecture, a case study of mathematical growth, surveys the origin and development of a mathematical idea from its birth in intellectual curiosity to applications. The fundamental structure of fractional calculus is outlined. The possibilities for the use of fractional calculus in applicab]e mathematics is indicated. The lecture closes with a statement of the purpose of the conference. Fractional calculus has its origin in the question of the extension of meaning. A well known example is the extension of meaning of real numbers to complex numbers, and another is the extension of meaning of factorials of integers to factorials of complex numbers. In generalized integration and differentiation the question of the extension of meaning is: Can the meaning of derivatives of integral order dny/dx n be extended to have meaning where n is any number--irrational, fractional or complex? Leibnitz invented the above notation. Perhaps, it was naive play with symbols that prompted L'Hospital to ask Leibnitz about the possibility that n be a fraction. "What if n be 1⁄2?", asked L'Hospital. Leibnitz [i] in 1695 replied, "It will lead to a paradox." But he added prophetically, "From this apparent paradox, one day useful consequences will be drawn." In 1697, Leibnitz, referring to Wallis's infinite product for ~/2, used the notation d2y and stated that differential calculus might have been used to achieve the same result. In 1819 the first mention of a derivative of arbitrary order appears in a text. The French mathematician, S. F. Lacroix [2], From "Fractional Calculus and its Applications", Springer Lecture Notes in Mathematics, volume 57, 1975, pp.1-36.