The Analyst Volume 1-4
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The Analyst Volume 1-4

The Analyst Volume 1-4


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About the Book

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1874 Excerpt: ...even if the expressions, given hereafter, are deemed too cumbrous for practical use, they may still possess sonic interest from a theoretical point of view. 2. It is known that if we have a function S of a variable, which is never infinite, and such that the relation function ( + 2(')--function ( ) is satisfied for all integral values of ' both positive and negative, it can be developed in a series of the form 1. (Kc) cos C-f "' sin i), in which i denotes a positive integer; and that, in the cuses where this series is infinite, it is convergent. In general the handling of periodic series is easier if we introduce imaginary exponentials in the place of the circular functions. Thus, e denoting the base of natural logarithms, we shall put 2-, whence 2 cos C = 2 + 2-', 2 COS /--2-(-2_i, 2! '(--1) Sin = Z-- "1, 2j/( 1) 8 C = 2(--2-', 2 = COS + j/( l)sin, 2' = COSt'C + j/( l)si'ni . The alove theorem then comes to the same thing as to say that S is developable in a series of the form where the summation is extended to negative as well as positive values oft. The coefficients are given in terms of the coefficients by the equations KP = Q + CU XP = (G-C_t)u-1, except the case where i = 0, when $ = C'0. It will be seen that when S is real, C is a complex number a + by/--1, and C_, ite conjugate a--bV--1, which renders the coefficients real, as they should be. The integral JVc/C = J"(cos iz + i/(--1) sin idc, taken between the limits 0 and 2n, vanishes in all cases except when = 0. when ite value is lit. Hence any function, capable of expansion in a series of positive and negative integral powers of 2, integrated wit...


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Product Details
  • ISBN-13: 9781158147731
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 182
  • Spine Width: 10 mm
  • Width: 189 mm
  • ISBN-10: 1158147732
  • Publisher Date: 01 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 336 gr


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