By Hirzebruch F., Scharlau W.

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X lnx Jo dx. dx. 1)(eibX — 1) dx (a, b ER). 26. 27. 31. 33. q > 0). 32. 16a). prove that +o(1). With the help of the Stirling formula find the value of the Euler-Poisson integral dx. §2. 34. 35. k=O,1,2,3,4,5. 36. 37. 38. f°° ir(x)/(x3 — x) dx, where ir(x) is the number of prime numbers not exceeding x. §2. 7 it is required to compute the given integrals. 1. 2. — a) x)dx1 . . 3. 4. b) (e>O). dx. fR a) ffR2 lax + 0"2dx (a E R'1, fR l(x, > —1). lix — yll1 dy1 dy2 dy3 (x E R3). 5. dx dy du dv. 6.

And < +oo f(x) = cosnx g(x) E C([0, irj). Prove that the series cos nx sin nx can be integrated termwise on [0, irj. 20. a) b) < +oo if and Suppose that sin nx, = sinnx. Prove that (ni E N). Prove that for XE (0. it): > 0; —1. v. 21. is — 45 §6. 22. Let = IG = (the Gibbs phenomenon). 23. 24. Let 1(x) = = f(+0) dU> > 0. Prove that a (x,y€R). 25. 26. where A E (1, 3). Prove that I E sin but f Let 1(x) = Lips for /3 > Prove that I E Sin for any E (0, 1). Is it true that f E Lip1? 27. x). Prove that: a) I for any 1J; b) f is nondifferentiable at every point.

7. Prove that as p —, +oo: ir/2p; sin(x")dx a) b) f°°cos(x")dx—'l. 8. as e—'+O. 9. 10. 11. c,/e 0. Prove that f11°° ço(t)dt b) Isinxllcosxl dx (p > —1); d) j Suppose that the function I decreases (p> 1). to zero on [a, +oo). the dt =0. is continuous on R and has period T> 0, and fj function for A a. 12. Suppose that I decreases to zero on [a, +oo) , and I on R and has period T> 0. Prove that JA JA dt = f(t)dt + I + 0(1(A)), where = fj dt and I = f is nonnegative and monotone on [a, +oo), is continuous on R and has period T> 0, and C, = + f' ço(t)dt 0.