2. Suppose f is holomorphic in an open set UC C and a € U. Show that for any integer n ≥ 0 there is a unique holomorphic function h : U → C such that ƒ(z) = f(a)+ƒ'(a)(z−a)+ƒ"(a). (z − a)² 2! (z − a)n n! ·+...+ f(n) (a). -+(z−a)"+¹h(z), z EU.

Q: Prove that If f is holomorphic on D(0, 2), f(0) = -i and |ƒ(z)| > 1 on the unit circle then f has at…

A: It is given that f is holomorphic in the disc with centre 0 and radius 2. So, the function f is…

Q: 5. Let f : R → [1, ∞) and g : [1, 0) → R be the functions defined by the following formulæ f (x) =…

A: Answer

Q: et [a, b] be a real interva b]→X is continuous, the is continuously differentia

A:

Q: Let f be a function from N to (0,1). Prove -h for any nEN, y = f(n). Using part (a), prove that…

A:

Q: Let f and g be the following functions. f: N→ P(N) - {0} defined by f(x) = {n: n € N and x ≤ n ≤…

A:

Q: Prove that for  1 < p < infinity   lp(F) is banach

A:

Q: (2.1) Let h: Z → Z be a function given by h = |r|, Va € Z. Show that (a) h is not one-to-one (b) h…

A:

Q: kx +5 for x ≤ 2 Let fx = = x-1 for x > 2 Find the value of k for which f(x) is continuous at x=2.…

A:

Q: If f is a continuous function satisfying for all x, then f "(1) = ? A) –1 B) 0 C) 1 D) 2 E) 3

A:

Q: 4. Let f: Z Z be defined by f(n) = 3n-1. Determine whether this function is 1-to-1 and/or onto.

A:

Q: 2. !. (n!)². > Show that there is no holomorphic function f (z) on any open set containing 0 such…

A: To Prove: There is no holomorphic function fz on any open set containing 0 such that fn0=n!2.

Q: Is there a continuous function such that - f: [1,2] U [3,4] → R - f(1) * f(4) < 0 - f(x) not egual 0…

A:

Q: 4. Let D be a subset of R. (a) A point xo in D is said to be an isolated point of D provided that…

A:

Q: Consider the function f : Z → Z, defined by f(n) = 2 [n/2]. Prove that f(n) is NEITHER one-to-one…

A: Please find the detailed solution in the next step where we show that the function is neither one…

Q: Let S be a closed n-cube with v(S) > 0 and suppose f : S –→ R is continuous on S (relative to S).…

A: let S be a closed n-cube with vS&gt;0 and suppose f:S→R is continuous on S (a) prove that infSf and…

Q: Let x0, x1, ..., Xn be distinct points on the interval [a, b], where Xo <x₂ < x₂ <--- < x₂ and let f…

A:

Q: Let f: Nx N → Z defined by f(n, m) = n − m² 1. Find f({(0, 0), (1,0), (0, 1), (1, 1)}) 2. find…

A:

Q: 3. Formulate an analogous statement on integrability on (a, b) under the assumptions of…

A:

Q: Find all the strictly monotonic functions f: R→R such that f(x + f(y)) = f(x) + y, for all z, y € R.…

A: a. Consider the given condition, fx+fy=fx+y, for all x,y∈ℝ. Here we need to find the strictly…

Q: 2. Let (r) be the function defined by V(x) = 1. Let 1+x. In(x) Vovo...(x) n-copies Note that…

A:

Q: 5. Prove that f has a zero of order m at zo if and only if f(zo), ƒ'(zo),...‚ ƒ(m-¹) (zo) are all…

A: I have explained below.Explanation:To prove that a function f has a zero of order m at z0​ if and…

Q: Let f:R → R be defined by: x - 1 x +1 if x s0 if x >0 f(x) : Then f]- 3,0[)= 1-2.oL

A:

Q: (a) Let f, 9, h : N → R²º. Prove or disprove that if f + g€ O(h), then f e O(h) and gE O(h). (b) For…

A:

Q: Let f:R → R given by f(x) = 2x² and A = [-3, 1] U [2, 4]. Then f(A) is

A:

Q: If f is a continuous function satisfying f(x) = x* - x+(1-x)[,f'f)dt + x[ (t -r°)f(@dt for all x,…

A:

Q: 6. Determine all functions f analytic in |2|< 1 and satisfying ƒ(²-) = k+k² 1+k2' k = 2,3,4,...

A: We have to determine all the analytic functions f  in  z&lt;1  such that  f1k = k+k21+k2 ,  k = 2,…

Q: Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f'…

A: A function f is convex in a domain a,b if for all 0≤λ≤1, fλx1+(1-λ)x2≤λfx1+1-λfx2  ∀x1,x2∈a,b. Also…

Q: = 1. Define the function f : Z x Z → Z by 2. Suppose a,b are natural numbers with ged(a, b) f(m, n)…

A:

Q: 4. Determine whether each of the following functions is a bijection and if not bijection, 1 explain…

A:

Q: f(x + a) dx = f(1) dæ = | f(x) d.r. -T+a

A: Proving of integral of periodic function

Q: Theorem 4.2.5 (Cauchy's mean value theorem). Let f : [a,b] –→R and o: [a,b] → R be continuous…

A:

Q: -B-Let f:( M, (Z).+-)-(M, (Z). +..) be a function define by such that g[]=[ EM₂(2). Is f is…

A:

Q: (a) Show (a, b) ∼ R for any interval (a, b). (b) Show that an unbounded interval like (a,∞) = {x : x…

A: Let          (a, b)⊂R                        be an open interval.   So,         We define a function…

Q: 4. (a) Let Q = : a, b e Z, b 0, gcd(a,b) = 1} be the set of rational numbers. An integer p is called…

A:

Q: prove If f:X→Y is a function and F is a non empty family of subsets of y, then o(f¹(I)) = f¯¹ (o(I))

A:

Q: Let f be a mapping of X into Y , A ⊆ X and B ⊆ Y . (a) Show that f(f−1(B)) ⊆ B and f−1(f(A)) ⊇ A.…

A:

Q: 4. (a) Let a : Z→ Z be the function defined by a(n) = 2n + 1, and let i: Z → Z be the identity…

A:

Q: (18) Let f: [0, ∞) → (0, ∞) be a surjection. Prove that f is not strictly monotone.

A: