1873_solutions

Test that 1 n a n is convergent the interval of

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Unformatted text preview: e that x and y are real numbers that are not both zero and that   arccos x x2  y2 . We see that sin   1 cos 2   1 cos 2 arccos x2 x  y2 2  We define r  x x2  y2 1 x 2  y 2 and we observe that 380  y x2  y2 . x  r cos  and y  r sin . In the event that the latter equation says that y  r sin  we define    and if the equation y  r sin  is false (in which case y  r sin ) we define     . In either event we have the two equations x  r cos  y  r sin  Finally we observe that since 0   we must have 0  2. However, if    then the equation y  r sin  is true and     . Thus the case   2 cannot occur and we conclude that 0   2. Exercises on Analytic Functions Prove that the function arctan is analytic on R. Suppose that c is any number. Using the fact that the rational function whose value at every number x is 1 2 is analytic, choose   0 and a sequence a n such that 1 x 1  1  x2 whenever |x Ý an x Ý arctan c n0 for |x n n0 c |  . If we now define f x  arctan x c an x n1 c n 1 c |  1 then we see at once that f is the constant 0 and so Ý arctan x  arctan c  n0 whenever |x an x n1 c n 1 c |  . 1. Given that f x  3x for every number x, prove that f is analytic on R 0 but is not analytic on R. Suppose that c 0. The equation f x  cx c 1/3  c 1/3 1  x c c 1/3 Ý  c 1/3 n0 1 cn n x c n holds whenever |x c |  |c |. Therefore f is analytic on the set R 0 . Since f is not differentiable at 0, the function f can’t be analytic on any open interval that contains 0. 2. Prove that the function tan is analytic in some neighborhood of the number 0. The function tan, being the quotient of the two analytic functions sin and cos, is analytic on the  interval , . 22 3. Given that fx  exp 0 1 x2 if x 0 , if x  0 prove that f is analytic on R 0 but is not analytic on R. In spite of the fact that this function f has derivatives of all orders at every number, we saw in an 381 earlier example that f can’t be expressed as the sum of it’s Taylor series center 0 in any neighborhood of 0. On the other hand, the composition theorem for analytic functions guarantees that f is analytic on the set R 0. 4. Prove that a rational function is analytic on any open set in which its denominator does not vanish. This fact follows at once from the fact that the quotient of two analytic functions is analytic as long as the denominator is not zero. 5. Given that 1x fx  1/x if x 0 if x  0 e prove that f is analytic on the interval 1, 1 . Use Scientific Notebook to work out the first few terms of some of the series expansions of this function. To do so, point at the expression 1  x 1/x , open the Maple menu and click on the Power Series option. We begin by observing that Ý 1 n n 1 x n1 log 1  x  n0 whenever |x |  1. We now define Ý 1n n x n1 gx  n0 and we observe that, since g 0  1 and gx  for every x 1, 1 log 1  x x 0 we have f x  exp g x whenever |x |  1. Since g is analytic on the interval funct...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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