LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – MATHEMATICS
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FIRST SEMESTER – NOV 2006
MT 1805 – REAL ANALYSIS
Date & Time : 28-10-2006/1.00-4.00 Dept. No. Max. : 100 Marks
- a)(1) When does the Riemann-Stieltjes integral reduce to Riemann integral. Explain with usual notations.
OR
(2) If a < s < b, f ÎÂ (a) on [a,b] and a (x) = I (x – s), the unit step function, then prove that = f (s). (5)
b)(1) Let f be a bounded function on [a,b] having finitely many points of discontinuity on [a,b]. Let a be continuous at every point at which f is discontinuous. Prove that f ÎÂ(a). (8)
(2) Suppose f is strictly increasing continuous function that maps an interval [A.B] onto [a,b]. Suppose a is monotonically increasing on [a,b] and f ÎÂ (a) on [a,b]. Define b and g on [A,B] by b (y) = a (f (y)), g (y) = f (f (y)). Then prove that g ÎÂ (b) and . (7)
OR
(3) Let a be monotonically increasing function on [a,b] and let a¢ Î R on [a,b]. If f is a bounded real function on [a,b] then prove that f ÎÂ (a) on [a,b] Û f a¢ ÎÂ (a) on [a,b].(8)
(4) Let f ÎÂ (a) on [a,b]. For a £ x £ b, define F(x) = , then prove that F is continuous on [a,b]. Also, if f is continuous at some x o Î (a,b) then prove that F is differentiable at x o and F¢ ( x o ) = f (x o ). (7)
- a) Let : [a,b] ® R m and let x Î (a,b). If the derivatives of exist at x then prove that it is unique.
OR
(2) Suppose that maps a convex open set E Í Rn into Rm, is differentiable on E and there exists a constant M such that M, ” x Î E, then prove that
ú (b) – (a)ú £ M ú b – aú , ” a, b Î E. (5)
- b) (1) Suppose E is an open set in R n ; maps R into R m ; is differentiable at x o Î E, maps an open set containing (E) into R k and is differentiable at f (xo). Then the mapping of E into R k, defined by is differentiable at xo and . (8)
(2) Suppose maps an open set EÍ Â n into  m. Let be differentiable at x Î E, then prove that the partial derivatives (Dj f i) (x) exist and , 1£ j £ m, where {e 1, e 2, e 3, …, e n} and {u 1, u 2, u 3, …, u m} are standard bases of R n and R m. (7)
(3) If X is a complete metric space and if f is a contraction of X into X, then prove that there exists one and only one x ÎX such that f (x) = x. (15)
III. a) (1) Prove: where {f n} converges uniformly to a function f on E and x is a limit point of a metric space E.
OR
(2) Suppose that {f n} is a sequence of functions defined on E and suppose that ½f n (x)½£ M n, x ÎE, n = 1,2,… Then prove that converges uniformly on E if converges. (5)
- b) (1) Suppose that K is a compact set and
* {f n} is a sequence of continuous functions on K
** {f n} converges point wise to a continuous function f on K
*** f n (x) ³ f n+1 (x), ” n ÎK, n= 1,2,… then prove that f n ® f uniformly on K. (7)
(2) State and prove Cauchy criterion for uniform convergence of complex functions defined on some set E. (8)
OR
(3) State and prove Stone-Weierstrass theorem. (15)
IV a) (1)Show that converges if and only if n >0.
OR
(2) Prove that G = . (5)
b)(1) Derive the relation between Beta and Gamma functions. (7)
(2) State and prove Stirling’s formula. (8)
OR
3) If f is a positive function on (0,¥) such that f (x+1) = x f (x); f (1) =1 and log f is convex then prove that f (x) = G (x). (8)
(4) If x >0 and y >0 then (7)
- a) (1)If f (x) has m continuous derivatives and no point occurs in the sequence x 0, x 1, ..,x n more than (m+1) times then prove that there exists exactly one polynomial Pn (x) of degree £ n which agrees with f (x) at x 0, x 1, …, x n.
OR
2) Show that the error estimation for sine or cosine function f in linear interpolation is given by the formula ½f(x)-P(x)½£ . (5)
b)(1) Let x0, x1, …, xn be n+1 distinct points in the domain of a function f and let P be the interpolation polynomial of degree £ n, that agrees with f at these points. Choose a point x in the domain of f and let [a,b] be any closed interval containing the points x 0, x 1, …, x n and x. If f has a derivative of order n+1 in the interval [a,b], then prove that there is at least one point c in the open interval (a,b) such that where A (x) = (x – x0) (x – x1)…(x – x n). (7)
(2) Let P n+1 (x)= x n+1 +Q(x) where Q is a polynomial of degree £ n and let maximum of ½P n+1 (x)½, -1 £ x £ 1. Then prove that we get the inequality . Moreover , prove that if and only if , where T n+1 is the Chebyshev polynomial of degree n+1. (8)
OR
3) Let f be a continuous function on [a,b] and assume that T is a polynomial of degree £ n that best approximates f on [a,b] relative to the maximum norm. Let R(x) = f (x) –T(x) denote the error in the approximation and let D = . Then prove that
(i) If D= 0 the function R is identically zero on [a,b].
(ii) If D >0, the function R has at least (n+1) changes of sign on [a,b]. (15).