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The Minimum in the Gamma Function


IT is well known that the gamma function “(z) for real and positive values of z has a minimum between z = 1.46 and 1.47. In a number of texts on the theory of functions it is stated that the minimum occurs at z = 1.4616321 and that the corresponding value of “(z) is 0.8856032 Only one text that we have examined, namely Joseph Edwards's monumental work Integral Calculus, vol. 2, chap, xxiv (Macmillan, 1922), gives any indication of how the minimum points can be calculated. The method therein explained depends on certain properties of the gamma function and of the related logarithmic derivative d In “(z + 1)/dz, commonly written (z), following Gauss. At best, the problem is finally one in successive approximation.

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