By E. T. Whittaker, G. N. Watson
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Publish yr note: First released in 1927
This vintage textual content is understood to and utilized by hundreds of thousands of mathematicians and scholars of arithmetic thorughout the area. It provides an creation to the final idea of limitless strategies and of analytic features including an account of the primary transcendental functions.
Read or Download A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions (4th Edition) (Cambridge Mathematical Library) PDF
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Additional resources for A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions (4th Edition) (Cambridge Mathematical Library)
We shall follow the exposition given by H. von Koch, Ada Math. xvi. (1892), p. 217. +m the determinant formed of the numbers A& (i, k = — m, ... -f m); then if, as m -*- oo, the expression Dm tends to a determinate limit D, we shall say that the infinite determinant is convergent and has the value D. If the limit D does not exist, the determinant in question will be said to be divergent The elements An, (where i takes all values), are said to form the principal diagonal of the determinant D; the elements Aiky (where i is fixed and k takes all values), are said to form the row i; and the elements AH, (where k is fixed and i takes all values), are said to form the column k.
M««« of the row through the origin by the elements which satisfy the inequality I Mr I < /*, where fx is a positive number; and let the new values of Dm and D be denoted by Dm' and D'. Moreover, denote by Pm' and P' the products obtained by suppressing in Pm and P the factor corresponding to the index zero ; we see that no terms of Dm' can have a greater modulus than the corresponding term in the expansion of pPm'; and consequently, reasoning as in the last article, we have which is sufficient to establish the result stated.
In the series whose general term is un = qn-vx*¥(v+l\ (0
+ n\ the logarithm being taken in its arithmetic sense, is convergent for all values of «, when 1 (x) is positive, and is convergent for values of s whose real part is positive, when x is real and not an integer.
A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions (4th Edition) (Cambridge Mathematical Library) by E. T. Whittaker, G. N. Watson