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George Mason UniversityConor Philip NelsonSource: Home > Classes > Math 400 > Final Project > ConclusionMath 400: History of MathematicsOne thing noted is that referring back to Daniel Bernoulli's infinite product representation that involves some infinite number \( A \), when getting the \( (\frac{3}{2})! \) through this method, he set his level of accuracy to \( A=8 \) and got \( 1.3005 \). Note that utilizing the gamma function, it can be shown that \( \Gamma (\frac{5}{2})=(\frac{3}{2})!\approx 1.329 \). It may seem obvious that these numbers are clearly wrong and sparks a question of correctness. In this case, it is safe to note that the gamma function is correct and Bernoulli was wrong. He had actually sent a letter to Goldbach two weeks later after sending the one where he attempted to determine \( (\frac{3}{2})! \) stating that he made a compuational error. The Gamma Function \( \Gamma (n) \) is a insightful analytical function that creates a deeper inquiry into understanding the abstract concept of the factorial operation and if there is an even more general computational process than just understanding the relation for \( (n \in\) N \( )!=(n)(n-1)(n-2)... \); in particularly for all real numbers. Although we notice \( \exists \) a transformation \(T: \Gamma ( \) R \ Z\( ^{-} ) \ \rightarrow\) R, we still cannot get whether or not there exists values for the factorial operation on negative integers. It is also interesting to note that there is specific definition for what a "special function" is as defined by the Mathematical Concepts (290)- "In practice, a special function is any function that, like the logarithm and the gamma function, has been extensively studied and has turned out to be useful. Some authors use the phrase special functions ina more restricted sense, meaning somethin glike "funciton that turn up in the solution of physical problems" or "functions other than those generally provided by a pocket calculator", but these restrictions do not seem to be very useful." Works Cited Artin, E. (1964). The Euler Integrals and the Gauss Product Formula. In M. Butler, The Gamma Function (p. 11). Leipzig: Verlag B. G. Teubner. Gowers, T. (2008). Special Functions. In T. Gowers, The Princeton Companion to Mathematics (pp. 290-291). Princeton: Princeton University Press. Ross, S. (2014). Other Continuous Distributions. In S. Ross, First Course in Probability (pp. 203-264). Boston: Pearson Education, Inc. . Roy, R. (2011). The Gamma Function. In R. Roy, Sources in the Development of Mathematics (pp. 444-450). New York: Cambridge University Press. Sokolnikoff, E., & Sokolnikoff, I. (1934). Ordinary Equation Equations . In E. Sokolnikoff, & I. Sokolnikoff, Higher Mathematics for Engineers and Physicists (pp. 243-249). York: Maple Press Company.
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