4. Let h(2) be a holomorphic function defined on a neighborhood of the origin such h()...
Consider the generalized integrator function (2) discussed in class, defined by its proper- ties: | dr 8(x) = 1, Ve > 0, | dx 8(x) = 12+ = ſo if r* <0 11 if x* 20' dx 8(2 – c)f(x) = f(c), VER, where dc 8() is understood as a slight abuse of notation and f(x) in the last formula is a suitably well-behaved (at least bounded and continuous - and perhaps even smoother- in a neighborhood of x=c) function...
Let f be a function which is holomorphic on the unit disk D (0). We define the quantity R= sup \f (2) - f(w) ) ED (0) ( (which can be infinite). 1. Prove that Vr € (0,1), 2f'(0) = 2 1 f6)-f(-) .0) (2 where the circle C,(0) is traversed in the counterclockwise direction 2. Deduce from the previous result that 28' (O SR.
Please give clear detailed explanation. Let a 0 and suppose that the function f is Riemann integrable on [0, a]. Prove that f(a-x) dx = 2S0[f(x) + f(a- 1 ca f(x) dx = x)j dx. Prove that f' in(1 + tan(a) tan(x)) dx = a ln(sec(a)) (0<a<T/2) Let f: [0, 1] → R be defined by f(x) = VX , 0 1 , and let x 2 n-1,2 be a partition of [0, 1]. Calculate lRll and show that lim...
5.72. Let A = A(0,1) and let g: A → be an analytic function sat- isfying 9(0) = 0 and 1g'(0) = 1 whose derivative is a bounded function in A. Show that w > (4m)-1 for every point w of C ~ g(A), where m = sup{]g'(x): z E A}; i.e., show that the range of g contains the disk A(0,(4m)–?). (Hint. Fix w belonging to C ~ g(A). Then w # 0. The function h defined by h(z)...
Suppose f : B(0.1) C is holomorphic, with irg:) 1 for every z є B(0,1). Suppose also that f(0)-0, so f(z)g(2) for some holomorphic function g: B(0,1)C. (a) By applying the Maximum Principle to g on B(0, r) where 0 < r < 1 , deduce that If( S for every 2E (0, 1) . (b) Show also that |f'(0) S1 (c) Show that if lf(z)- for some z B(0,1)\(0), or if If,(0)| = 1 , then there is a...
(2) Let f : Rn → R be a C2 function. Suppose a sequence (zk) converges to x*, where the Hessian Hf(z.) is positive definite. Let ▽ := ▽f(xk)メ0, Hfk := H f(zk), dkー-Bİigfe, and :=-[Hfel-ı▽fk for each k, where each matrix Bk is ll(Be-Hfe)del = 0 if and only if ei adtive lim lidt dall =0. (11 points) (2) Let f : Rn → R be a C2 function. Suppose a sequence (zk) converges to x*, where the Hessian...
(8) 2 points Let f be a function defined and continuous, with continuous first partial derivative at the origin (0,0). A unit vector u for which D.f (0,0) is the maximum is: maximum a 1 (0,0)), A. /(0,0)x,0),y (0 af B. (0,0) 8x0,0),(0,0)), af 1 ((0,0),-y C. (0,0), /(0,0) D. None of the above. (8) 2 points Let f be a function defined and continuous, with continuous first partial derivative at the origin (0,0). A unit vector u for which...
Problem 4: Let f: [0, 1] → R be an integrable function that is continuous at 0. Prove that lim f(") dx = f(0). n+Jo [ Hint: there are several approaches. It might help to first show that for a fixed 0 <b< 1, we have limn700 Sº f(x) dx = b. f(0). ]
Please answer it step by step and Question 2. uniformly converge is defined by *f=0* clear handwritten, please, also, beware that for the x you have 2 conditions , such as x>n and 0<=x<=n 1- For all n > 1 define fn: [0, 1] → R as follows: (i if n!x is an integer 10 otherwise Prove that fn + f pointwise where f:[0,1] → R is defined by ſo if x is irrational f(x) = 3 11 if x...
Let f : [0,∞) → R be the function defined by f ( x ) = 2 ⌊ x ⌋ − x? where x? = x − ⌊x⌋ is the decimal part of x. Prove that f is injective. Let f: 0,00) + R be the function defined by f(3) = 212) where ã = x — [x] is the decimal part of x. Prove that f is injective.