Number Theory

The general theory of Dirichlet's series by G. H. Hardy

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By G. H. Hardy

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16) 2 In this work he also showed (Vol. I, Sects. 17–22) how to calculate values of F by showing how to give accurate approximations when c is nearly 0 or 1, and how to reduce the general case to this one by a transformation. He had done this earlier, in his (1788a, b) and his (1792). By analogy with the trigonometric case, he defined φn to be an amplitude such that F(φn ) = nF(φ ) and sought to find sin(φn ) and cos(φn ) in terms of sin φ and cos φ . He pointed out that this was easy when the modulus was 0 or 1 because then the elliptic integral can be evaluated explicitly.

As for the topic of elliptic integrals, it had a long-established place in contemporary astronomy. Since Kepler’s second law asserts that an elliptical orbit is parameterised by a satellite sweeping out equal areas in equal times, mathematicians were led straight away into questions involving the rectification of an ellipse and so to elliptic integrals. Newtonian theory then said that the orbit would be an ellipse only if the problem was a two-body one. For a 3- or n-body problem, the question was to compute the additional variation of that ellipse.

19) from which is followed that it was often enough to use just the first two terms. In the Exercises he gave a table of values of elliptic integrals to 14 decimal places. Legendre also sought to show how useful his new functions would be in various parts of mathematics. 20) 1 − c2 sin2 ψ where l is the length of the pendulum, c2 = 2lh where h is the height of the pendulum due to its speed at its lowest point (in units where the acceleration due to gravity =1) and the angle ψ is related to the angle of displacement from the vertical by the formula sin( φ2 ) = c sin ψ .

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