Let f: C→ Cbe a holomorphic function. Assume there is a ro > 0 such that |f(2)| < |z|? for |2| > ro- (i) Show that f(3) (2) = 0 for all z E C. (ii) Show that f is a quadratic polynomial.

Q: Q2: (a) Determine which of these functions are injective, surjective, and bijective. The function f…

A: a:Given are the following functions:i fx=2x+1ii fx=exiii fx=x3-3xTo find that the following…

Q: Q7; Functions (a) Prove that the function f : Z4 → Z12 given by f(T) = [3x] is well %3D defined.

A: we need to prove homomorphism

Q: 3) Let f(T) e R[T] be a polynomial of degree > 3. Show that there exists polynomials g(T) and h(T)…

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Q: Let f(x) = sin x. Find a polynomial P(x) of degree at most 2 such that P(1) = f(1), P(1.1) = f(1.1),…

A: So we have to do a lagrange interpolation to approximate the function f(x) = sin(x) about the…

Q: - Prove that the function f : R² → R defined by f(x, y) = x² + 4y² − 1 is neither injective nor…

A: A function is said to be injective if we have unique pre-image for every value of the function. If…

Q: (a) Illustrate in the complex plane those complex numbers z =1+ iy E C for which m (s + ) >. Im (z+

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

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Q: The function

A: Given, the function f : ℂ→ℂ defined by                   f(z)=ez+e-zhas

Q: Let f(z) be an entire function satisfying |ƒ (²1 +22)| ≤ |ƒ (2₁)| + + + |ƒ (2₂) | = n for all 2₁, 22…

A: As per the question we are given an entire function which satisfies the following inequality :And…

Q: c) Find the second degree polynomial approximation to f at (a, b) = (1, – 2) T2(x, y)

A: Given :- The second degree polynomial approximation to f at a,b = 1,-2 f x,y = -x2y2x2 +y2

Q: 6. Let 1 f (x) - cos x + sin(2x), x E [0, 1] 2 Find the least squares polynomial approximation of…

A: Let fx=12cos x-sin2x, x∈0,1. To find quadratic fitting  fx≈P2x=a0+a1x+a2x2 Consider, s0=∫011dx=1

Q: let f(z) be polynomial of degree at least two. show that |f(z)|>|z| for large |Z|.

A: We prove this result by contradiction.

Q: f(x,y) = x²y + 2xy² + 5y³

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Q: Solve the nonlinear inequality. Express the solution using interval notation. 3 + x ≥ 1 3-x Graph…

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Q: Let f(r) be a polynomial function such that f"(x) = xª – 5z + 6z². Find the æ values for the…

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Q: 2) Let f(T) e Q[T] be an irreducible polynomial of degree n > 1. Show that f(T) has n different…

A: We will use the basic knowledge of ring theory to answer this question properly, in detail.

Q: Find the (natural) domain of the function x + 1 2х — 1 f (x) = Select one: a. (-0, –1] U (1/2, ∞) b.…

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Q: of A, B, and C.

A: We are given that the polynomial function is of the form f(x) = Ax2+Bx+C. We need to find the values…

Q: Let f be some continuous function on an open interval I containing x0, x1 and x2, where xo <x1 <x2.…

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Q: Consider the quadratic function f: R3 →R defined by f(X) = xỉ – x1x2 + x – x2x3 + x3 + 1 A: Express…

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Q: 3. Let x0, x₁,...,xn be distinct points on the interval [a, b], where Xo <x₁ < x₂ <... < xn and let…

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Q: Consider the function g : Zx Z → Z × Z defined by g(x, y) = (4xy, 9x + 5y). (a) Is g injective? (b)…

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Q: f(2) = e=².

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Q: 20 x=0 3. Define f: R→ R by f(x) = and g: R → R by g(x) = -{ 1² r <0 0 r20 (a) Compute (fg)(-1) (b)…

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Q: 1 1+ 12,2 over [-1, 1]. SET-UP the Lagrange (-1, 1). 2. Consider the function defined by f(x)…

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Q: In linear space, consider all true polynomial functions with internal multiplication x(t)y(t)dt for…

A: We will use GRAM-SCHMIDT ORTHOGONALIZATION PROCESS to solve this problem For a basis y0,y1,y2 to…

Q: ƒ(X) e F[X] be a non-constant polynomial of degree d. Prove that f(a) has degree at least over F.

A: ANSWER :

Q: Example: Show that the function Az) = z² is analytic. z = x+iy S(2) = z

A: