Concrete mathematics : a foundation for computer science

58 Prove that
B,(Ix}) = -26 x cos(2xk;m- inrn) ,
kal
by using residue calculus, integrating
1
-f
2ni $nize dz
2ni - e2niz -_ 1 =rn-
9 EXERCISES 481
for m 3 2,
on the square contour z = xfiy, where max(lxl, IyI) = M+i, and letting
the integer M tend to 00.
59 Let o,(t) = tk e mik+t)‘/n, a periodic function of t. Show that the
expansion of O,,(t) as a Fourier series is
o,(t) = &Ei(l + 2eCnJn(cos27rt) + 2e 4XLn(cos4xt)
+2e~9X’n(cos6xt)+...).
(This formula gives a rapidly convergent series for the sum 0, = 0, (0)
in equation (g.g3).)
60 Explain why the coefficients in the asymptotic expansion
(?) = -&(I-&+j&+&-j&g++
all have denominators that are powers of 2.
61 Exercise 45 proves that the discrepancy D( 01, n) is o(n) for all irrational
numbers 01. Exhibit an irrational 01 such that D (01, n) is not 0 (n’ ’ ) for
any c > 0.
62 Given n, let { ,;‘,)} = ma& {E} be the largest entry in row n of Stirling's
subset triangle. Show that for all sufficiently large n, we have m(n) =
1777(n)] or m(n) = [m(n)], where
m(n)(%n) + 2) In@(n) + 2) = n(K(n) + 1)
Hint: This is difficult.
63 Prove that S.W. Golomb’s self-describing sequence of exercise 2.36 satisfies
f(n) = @2m+n@m1 + O(n+-‘/logn).
64 Find a proof of the identity
cos 2n7tx
x7-= 7T2 (x2 -x+ ;) for 0 6 x 6 1,
that uses only “Eulerian” (eighteenth-century) mathematics.