A BMO estimate for the multilinear singular integral by Zhang Q.

By Zhang Q.

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Study his method and explore the possibility of embodying it in a computer program. ("Calculus of Equivalent Statements," Proc. London Math. Soc. 9(1878), 177-186; "On the Limits of Multiple Integrals," Proc. London Math. Soc. 16(1884), 142-148). Show that f' dx _ 0 (1+x2n)t/n I 0 dx (1-X2n)t/n = Cos it Zti for n = 2, 3, ... (Klamkin). Hint: It is not necessary to evaluate the integrals individually if you can find the correct geometrical interpretation in terms of a circle and inscribed polygon.

Y,, can be reduced to the integral of a rational function of t if there is a parametric representation x = Ro(t) Y, = R, (t) y2 = R2(t) y = R,, (t) where the R's are rational functions.

Although we will usually omit the constant of integration, it is interesting to note that it can be used to show that the standard integration formula x dx = n+l ( 2 . 3) INTEGRATION FOR ENGINEERS AND SCIENTISTS 40 as a limiting case. This is done by writing xn+I_cn+1 x"dx = n+1 which becomes indeterminate (0/0) as n -+ - 1. 5) n--1 is obtained. It is assumed that the reader has had the usual course in calculus covering methods for evaluating various integrals involving algebraic and elementary transcendental functions.

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