Other Nineteenth Century Figures
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After Gauss there were a considerable number of excellent mathematicians who came along more or less in his footsteps and either continued his ideas on numerical analysis or who utilized their own discoveries to make more elegant what earlier mathematicians had done. Thus, for example, we find on the one hand Jacobi reconsidering some of Gauss’s work and on the other Cauchy using his Residue theorem to establish interpolation formulas.
KeywordsRemainder Term Interpolation Formula Bernoulli Number Bernoulli Polynomial Definite Integral
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- 2.Recall that the functions so of Jacobi are proportional to the Legendre polynomials on the interval 0, 1. Cf., e.g., Courant, Methoden, Vol. I, pp. 70–74.Google Scholar
- 4.Chebyshev , pp. 19–34 or Works, Vol. III, pp. 49–62. It is in this article that Chebyshev defines his functions as (1/2“-1) cos (n arc cos x). The remainder term in his formula appears in Milne—Thomson [ 1933 ], pp. 177–180.Google Scholar
- 5.Jordan , pp. 521–523. He notes, moreover, that for n = 8 and 10, e.g., the roots are not always real. This, of course, makes the utility of the method for n 7 questionable. One big advantage of the procedure is that all observations f(x,) receive equal weight. The remainder term is calculated in Milne—Thomson [ 1933 ], pp. 177–178.Google Scholar
- 6.Bronwin [ 1849 ]. What he does here is to find formulas for the coefficients of a finite Fourier series; cf. Gauss’s results in Chapter IV, pp. 246ff.Google Scholar
- 7.Lubbock , and Woolhouse . The original result was improved by de Morgan .Google Scholar
- 8.Steffensen , pp. 166–170; Whittaker, WR, pp. 150–151 or Milne-Thomson [ 1933 ], pp. 171–173.Google Scholar
- 9.Cf. Jacobi VI , pp. 64–75. The paper is entitled “De usu legitimo formulae summatoriae Maclaurinianae.” The subject was also treated by Poisson , pp. 571–602. In previous discussions we have seen only infinite series given with no suggestion of error terms. Jacobi now made a detailed investigation of the remainder term when the Euler-Maclaurin expansion is terminated. The interested reader may wish to consult the analysis of this error term in Milne-Thomson , pp. 187–190 or Nörlund , pp. 30–32.Google Scholar
- 12.The remainder term in the Euler—Maclaurin series is discussed in many texts such as Steffensen , pp. 131ff. A valuable list of references to papers on the subject appears in Nörlund , p. 30.Google Scholar
- 25.An elegant study of the Cauchy interpolation formula was made by Jacobi III , pp. 479–511. The paper is followed by G. Rosenhain , pp. 157–165, on this subject. Rosenhain uses Cauchy’s result to find conditions that f = 0, tp = 0 have a common root by considering the quotient fop when f and tp are polynomials. It is also relevant to mention the work of Padé on rational approximation, particularly since it is of interest today. Padé , pp. 1–93. We need also to mention Thiele’s important work on his reciprocal differences. Cf. Thiele  or Nörlund [ 1924 ], pp. 415–438.Google Scholar
- 26.Cauchy 1, VI , p. 71. This is the same as Gauss’s result (4.61) above.Google Scholar
- 27.Cauchy 1, V , pp. 455–473; 2, III , pp. 378–425; and 2, IV , pp. 573–609. (In Série 1, Vol. V of his Oeuvres, pp. 431–493, there are a number of papers by Cauchy on numerical methods. None is very important. They all are from the Comptes Rendus, Vol. XI (1840).)Google Scholar