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George Mason UniversityConor Philip NelsonSource: Home > Classes > Math 400 > Final Project > HistoryMath 400: History of MathematicsThe early origins of the Gamma Function go back to the ideas of 1660s with interpolating the sequences of products. In particular, interpolating the sequences of factorials. Wallis first began the leadway in the 1660s when he determined the reciprical of the quadrature of a circle can be represented as an infinite product, ie \( \frac{4}{\pi}=\frac{3}{2} \frac{3}{4} \frac{5}{4} \frac{5}{6} ...\). The issue here is that these infinite product were considered an unfavorable representation for various sequences due to their strenuous computational nature; therefore, other ways to represent a sequence of values (like some analytical function!) were being explored. When looking for some general function that could interpolate the sequence of factorials, it a more clever approach was considered than simply finding something that would lead to \( f(x)=x!\). The idea was if one could find a function that would satisfy the property \( f(x+1)=xf(x) \), this by induction could lead to at least a representation of \( (n-1)! \). 70 years later, Euler gains an interest in this problem through his colleague Christian Goldbach's papers and possibly Goldbach's letters to Daniel Bernoulli. Simultaneously, Stirling also gains an interest in find efficient methods of computing this sequence of factorials; however, Stirling naturally deviates from Euler's goal of looking for a general case because of his main interest in the computations. Additionally, after Daniel Bernoulli's attempt to describe Goldbach's interpolation in his own way using \( x!= (A+\frac{x}{2}))^{x-1}(\frac{2}{1+x})(\frac{3}{2+x})...(\frac{A}{A-1+x}) \) where \(A\) is some infinite number that brings a level of accuracy to the computation, he generally became uninterested in pure mathematics since he was more of a physicist in occupation. This left Euler essentially by himself on figuring out this problem, and in the 1729, he introduced the general case we know today as the gamma function.
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