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Unformatted text preview: Appendix D Asymptotic series Asymptotic series play a crucial role in understanding quantum field theory, as Feynman diagram expansions are typically asymptotic series expansions. As I will occasionally refer to asymptotic series, I have included in this appendix some basic information on the subject. See [AW] sections 5.9, 5.10, 7.3 (7.4 in 5th edition), 8.3 (10.3 in 5th edition) for some of the material below. D.1 Definition By now as graduate students you have seen infinite series appear many times. However, in most of those appearances, you have probably made the assumption that the series converged, or that the series is only useful when convergent. Asymptotic series are non-convergent series, that nevertheless can be made useful, and play an important role in physics. The infinite series one gets in quantum field theory by summing Feynman diagrams, for example, are asymptotic series. To be precise, consider a function f ( z ) with an expansion as f ( z ) = A + A 1 z + A 2 z 2 + where the A i are numbers. We can think of the series A i /z i as approximating f ( z ) / ( z ) for large values of z . We say that the series A i /z i represents f ( z ) asymptotically in direction e i if, for a given n , the first n terms of the series may be made as close as desired to f ( z ) by making | z | large enough with arg z fixed to , i.e. lim | z | z n f ( z ) n summationdisplay p =0 A p z p = 0 1 APPENDIX D. ASYMPTOTIC SERIES 2 (Put another way, write z = re i , then take the limit as r but hold fixed.) We shall see later that as one varies the direction e i , one can get different asymptotic series expansions for the same function this is known as Stokes phenomenon, and we shall study it in section *** CITE ***. Asymptotic series need not converge; in fact, in typical cases of interest, an asymptotic series will never converge. (Nonconvergence is sometimes added to the definition of asymptotic series, so that, in that alternate definition, an asymptotic series can never converge. In our definition here, convergence is allowed, albeit it is unusual.) It is important to note that asymptotic series are distinct from convergent series: a convergent series need not be asymptotic. For example, consider the Taylor series for exp( z ). This is a convergent power series, but the same power series does not define an asymptotic series for exp( z ). After all, lim z z n exp( z ) n summationdisplay p =0 z p n ! and so the series is not asymptotic to exp( z ), though it does converge to exp( z ). Not all functions have an asymptotic expansion; exp( z ) is one such function. If a function does have an asymptotic expansion, then that asymptotic expansion is unique. However, several different functions can have the same asymptotic expansion; the map from functions to asymptotic expansions is many-to-one, when it is well-defined....
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