Indefinite Integration of the Gamma Integral
and Related Statistical Applications
1.1 Research Motive
Most beginning students of statistics are bewildered by the many statistical tables in their textbook appendices. They are taught that these tables are indispensable for evaluating the probability of certain distributions, such as a $Z$ distribution, $F$ distribution, or ${{\chi }^{2}}$ distribution. As those students become more knowledgeable, they realize that the true reason for using statistical tables stems from the lack of antiderivatives for certain integral functions such as the gamma, exponential, and Gaussian integrals (Risch, 1969, 1970; Rosenlicht, 1972). Historically, the three integrals were proposed in Euler (1729), Mascheronio (1790), and de Moivre (1733). The first tables, however, were not available until Pearson (1922), Glaisher (1870), and Kramp (1799), respectively created them. Since the cumulative distribution functions of many distributions are related to the three integrals, the use of prepared tables was the only way to evaluate a probability before the computer era.
In modern calculus, the indefinite integral is called "antiderivative" because it is an inverse operator of derivative. According to the fundamental theorem of calculus (Stewart, 2003, pp.284-290), all continuous functions have antiderivatives, but only some of them possess antiderivatives that can be expressed by elementary functions. Mathematicians today explain this result using the differential Galois theory, for which Liouville's theorem provides the basic argument: if an elementary antiderivative exists, it must be in the form of an elementary function constructed via simple arithmetic operations in a finite number of steps (Conrad, 2005; Fitt & Hoare, 1993; Kasper, 1980). These elementary functions include polynomial, trigonometric, exponential, and logarithmic functions; the simple operations comprise addition, subtraction, multiplication, division, and root extractions.
Because they are based on Liouville's theorem, many indefinite integrals are believed to be unevaluable in finite terms of elementary functions, including the gamma, exponential, and Gaussian integrals. Nevertheless, a logical inconsistency exists between the definition of elementary functions and the concept of finite terms since trigonometric, exponential, and logarithmic functions by definition are all infinite series. They become "closed-formed'' because mathematicians define them as elementary functions even though they are infinite series in nature (Chow, 1999). In fact, many irrational numbers, which cannot be expressed by a simple fraction, can be regarded as infinite series (Manning, 1906). What Liouville's theorem proves is contingent upon the definition of elementary functions, and the acceptance of this definition largely prevents any efforts in discovering the antiderivatives of those "unsolvable" integrals.
This research is about solving the indefinite integration of the gamma, Gaussian, and exponential integrals. By proposing a new transcendtal number "$h$", we can solve those unsolvable integrals in closed-form expressions.
Welcome all math lovers!
