James Stirling (mathematician)

See also: James Stirling, Stirling

James Stirling is a British Mathématicien , born in May 1692 with Garden close to Stirling, dead the December 5th 1770 with Leadhills.

He was discovered by Newton. In 1717, Stirling taught with Venice and publishes its first work with Rome, “ Lineae Tertii Ordinis Neutonianae ”, which develops the theory of Newton on the plane curves of degree 3, adding a new level of curves to the 72 data by Newton. Its work was published in Oxford and Newton itself accepted a copy from it. “ Lineae Tertii Ordinis Neutonianae ” contains other results that Stirling had obtained. They are results on the curves with fast descent, the sequences (in particular, these problems are relating to the placement of spheres in a vault), and on the orthogonal trajectories. The problem of the orthogonal trajectories was raised by Leibniz and of many mathematicians other than Stirling worked on the problem, thus Jean Bernoulli, Nicolas (I) Bernoulli, Nicolas (II) Bernoulli, and Leonhard Euler. It is known that Stirling solved this problem beginning 1716.

In London, Stirling published its principal work “ Methodus Differentialis ” in 1730. This book relates to the infinite series, the square addition, sum, interpolation and powers. At that time, Stirling was in correspondence with of Moivre, Cramer and Euler. The asymptotic equivalent of N!, for which Stirling is known the most, appears with Example 2 of Proposal 28 of “ Methodus Differentialis ”. One of the main objectives of this work was to study methods to accelerate the convergence of the series. Besides Stirling notes in its foreword that Newton had studied this problem. Many examples of its methods are given, of which the problem of Leibniz of $\ pi/4 = 1 - 1/3 + 1/4 - 1/5 + 1/6 -$ … It also applies its processes of acceleration to the sum of the series $1 + 1/4 + 1/9 + 1/16 +…$ whose exact value was still unknown at the time. It obtains (Prop.11, example 1) the relation $\ sum_ {n=1} ^ \ infty \ frac {1} {n^2} = \ sum_ {n=1} ^ \ infty \ frac {3} {n^2 {2n \ choose N}}$ which enables him to obtain the approximate value 1.64493406684822643, but $\ frac {\ pi^2} {6}$ does not recognize, which will be made by Euler few years afterwards. It also gives a theorem in connection with the convergence of an infinite product. In its work on the acceleration of the convergence of the series a discussion of the methods of Moivre is. The work contains other results on the Fonction Gamma of Euler and the hypergeometric function.

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