The Oxford, Cambridge, and Dublin Messenger of Mathematics Volume 1

The Oxford, Cambridge, and Dublin Messenger of Mathematics Volume 1

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1862 edition. Excerpt: ...evident. it is evident that f (ac) 1-f (: c"') can be expressed in terms of trigonometrical functions of 9. Hence by equating the two values of f (av) + f (9: "), we have, on reduction, Similarly it may be shewn that by equating the two values of f (x)--f (a: "), we obtain By giving to 0 different values within the proper limits, various important summations of series will he effected. Those limits are in I. and II. from 0=--rr to 0=+-yr and 7. By assigning different forms to _f(x), we can obtain by the above method numerous trigonometrical formulae, some of which appear to be remarkable. I subjoin some examples which the intelligent young student, for whom this paper is chiefly intended, may work out for himself without much difliculty. Ex. 5. Let _f(a: ) = cosa: and f = sinae be used, then the following elegant formulae may be obtained: These two values are nearly identical, the difference being 483 _ _6i __ 1 884 119 " 6188 ' 8. The preceding formulas exhibit the summations of various infinite trigonometrical series, which, being periodic, possess only a limited range of application. It is manifestly important therefore to assign the limits within which such summations hold good. The following rule of limits, which I suppose to be new, and which is applicable to a large class of formulae, I have found in every instance to be correct. The proof of it will be subsequently given. lst. If all the terms of the series are positive, the range of application in the summation of that series is from 2nd. If the terms are alternately positive and negative, the range of application is from It may be observed that the limits themselves are ordina1-ily (I am not sure that they are universally) included within the more

Product details

  • Paperback | 46 pages
  • 189 x 246 x 3mm | 100g
  • United States
  • English
  • black & white illustrations
  • 1236916840
  • 9781236916846