By David Bleecker, George Csordas (auth.)

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Let g(x) = coth(x) - x. For small x > 0, show that g(x) > 0, while g(x) < 0 for large x> O. Show that g(x) is strictly decreasing for x> 0, by computing g'(x). (c) Show that the tangent lines in part (a) must be of the form l' = ±sinh(a)·z, where a is defined in (b). Hence, regardless of the value of C, these lines are tangent to each of the curves r = C· cosh(z/C). (d) From Part (c) and the convexity of the curves r = C·cosh(z/C) (C > 0), deduce that all of these curves are contained in the wedge r ~ sinh( a)· 1z I.

Let u(x) be an arbitrary c1 function defined for ordinary di fferent ial operator d/dx x ~ 0 , such that u(O) = o. which assigns to each such function Consider the u the new continuous function u' (x). Show that the inverse operator, say B, assigns to each continuous function f(x), defined for x ~ 0 , the function r B[~(x) :: Jng(x,z)f(z) dz , where g(x,z) = o [10

For small x > 0, show that g(x) > 0, while g(x) < 0 for large x> O. Show that g(x) is strictly decreasing for x> 0, by computing g'(x). (c) Show that the tangent lines in part (a) must be of the form l' = ±sinh(a)·z, where a is defined in (b). Hence, regardless of the value of C, these lines are tangent to each of the curves r = C· cosh(z/C). (d) From Part (c) and the convexity of the curves r = C·cosh(z/C) (C > 0), deduce that all of these curves are contained in the wedge r ~ sinh( a)· 1z I. 3255· R.