By M. A. Evgrafov (auth.), R. V. Gamkrelidze (eds.)
Infinite sequence, and their analogues-integral representations, grew to become funda psychological instruments in mathematical research, beginning within the moment half the seven teenth century. they've got supplied the ability for introducing into research all o( the so-called transcendental services, together with these that are now referred to as hassle-free (the logarithm, exponential and trigonometric functions). With their aid the ideas of many differential equations, either usual and partial, were came upon. in reality the full improvement of mathematical research from Newton as much as the top of the 19th century was once within the closest method attached with the advance of the gear of sequence and critical representations. additionally, many summary divisions of arithmetic (for instance, sensible research) arose and have been constructed so as to examine sequence. within the improvement of the speculation of sequence easy instructions could be singled out. One is the justification of operations with infmite sequence, the opposite is the production oftechniques for utilizing sequence within the resolution of mathematical and utilized difficulties. either instructions have constructed in parallel first and foremost growth within the first course was once considerably smaller, yet, after all, development within the moment course has continuously became out to be of higher difficulty.
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Extra resources for Analysis I: Integral Representations and Asymptotic Methods
5) where at #- 0, 0< Cit < Ci 2 < ... First of all we try to find the leading term of the series, that is, the exponent Cit and the coefficient at. For this we use the following simple reasoning. A. 4) satisfying the condition x ....... o, with some C #- 0, it is necessary that the quantity k = 1,2, ... ,N, take its least value for more than one value of k. 1. Series and Integral Representations 31 In fact, in the contrary case, the largest term in the expression for P(x, y(x» will obviously be different to zero (the question concerns the size of the terms for small x).
The validity of the asymptotic formulae obtained from the Euler-Maclaurin series was proved not just for individual functions f(x), but also for many classes of functions. It is interesting to note that for the Stirling's series itself a modification has been found which contains only convergent series: 1 00 logr(x) = xlogx - x - -2 10gx + ~ Ck ( 1) ( k ) k-1 x + ... x + where Ck 1 = 2k Joex(x + 1) ... 18) 1)(2x - l)dx. , fo· xX-1/2e-x is valid not only for x --+ +00, but also for x --+ 00 in any sector largxl ~ 6, () < 11:.
The problem initially was to find a simple formula which was a good approximation to the gamma-function for large values of the variable. Its solution was the so-called Stirling's formula F(x) '" fo -x, XX-l/2 e x- +00. To refine this formula the series 1 1 1 1 10gF(x) = xlogx - x - -logx + -log21t - + - -3 - ... 2 2 12x 720x named Stirling's series, was used. A. Evgrafov 44 Euler posed and solved the more general problem of finding a formula which was a good approximation to the sum F(n) = n L f(k) k=l for large values of n.
Analysis I: Integral Representations and Asymptotic Methods by M. A. Evgrafov (auth.), R. V. Gamkrelidze (eds.)