Question

Another math problem: It is often convenient to replacea sum by an integral. However, this is an approximation and it is sometimes useful, or needed, to find the "leading corrections" to this approximation. Specifically, one has where j is an integer, and thus Δ-1 formally fu) signifies some function f evaluated for a given value of the integer label j and a is the smallest value j takes. The goal of this problem is to compute the leading terms that make the... in Eq. (Il.1). Specifically, one has where The goal of this problem can sow be restated as "determining once and for all the C, coefficients in Eq. (TL2)". A formal proof of this is involved, but we can use a trick. We take a trial" function fU) that is "arbitrary enough", but that we "like" and that allows to compute both sides of Eq. (IL.1), and thus find the C To begin, take fU) expi-ju) where w is a real number (constant) and, for simplicity, we shall take a 0 above. a) Compute the integral (IL.3) b) Compute next (114) Hint: toshow this, recall that Σ=r:r, forld < 1. We will use this sum most frequently in this course. You should remember it! Recalling the Taylor series expansion of ecal+x+ r/2+/6+x/24+.... combine all the results above to show E--, _ т + 2 + ' 12 720 30240 可.in + (115) Hence, show that Co,c-.C-o.C- C-o, Csand so on

Answer 1