S lim f(x) log 1 -t- x ldx f(x) log ll x dx S, (log(x- 1)dx- (x- 1) log(x- 1) (x- 1)which tends to a limit as e - 0-t-.Hence, putting y -x in the second integral below and using (b) in thisintegral to obtain f(x) f( y) -f(y), we see that Also, the integral con-verges at 1 since, by (), the integrnd is essentially log Ix 1 in a neighbor-hood of. ![]() X - 1 logl-t- 1Ix 2/x-i-O(1/x3)x- 1 1- 1/xso that the integral I converges at oo by assumption (d). Uniformly in the closed upper half-plane.Under these conditions, we prove that Number of poles located at zl, z2, z.(d) For some c we have f(-x) -f(x) for all real x(c) f(z) is analytic in the open upper half-plane with the exception of finite ![]() ![]() (a) f(z) is real and continuous on the real xis(b) f(x) is an odd function i.e. THE ABOVE INTEGRAL, I, CAN BE EVALUATED by means of a contour integrationunder the following assumptions"
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