Q.1 If fi(z) and f2(z) are both analytic functions, show that fa(z)-fi(2).f2(z) is also analytic function using Cauchy-Riemann equations, and, hence, show that if f(z) is analytic then f"(z) is also analytic, where n is positive integer.

Q: b) Euler's theorem states that f(:) = "lax, X1 +°"/ax,X2 for functions that are homogenous of degree…

A: Homogeneous Function: A function f of two variables x and y is said to be homogeneous of degree n if…

Q: 5. Let z = and f(z) = 27 ,then simplify f(2) and indicate the real V2 – yi and imaginary part.

A: The given complex number is, z=1-2-yi Rationalize the term,…

Q: (A) Suppose that f(x) is a real analytic function such that: f(-4)= 0, f'(-4)= 9, f"(-4)= 8, f"(-4)=…

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: b) Find all the analytic functions for which Rez = Im f(z)

A: b) A function f(z) is said to be an analytic function at any point z=a if f(z) is differentiable not…

Q: B)Is w = Ze² analytical function or not?

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Q: Determine all analytic functions f on nEN. D(0, 1) satisfying f() = 1 for all

A: The identity theorem for analytic functions :The identity theorem states that if two analytic…

Q: 4. Verify Rolle's Theorem for f(x) = x² – 2x + 1 on [-1,3]

A: Verify Rolle's theorem

Q: Let f(2)= (2+1)|z|². (2.1) Use the Cauchy-Riemann equations to determine where f is differentiable.…

A: The given function fz=z+1z2 We have to find the points for which fz is differentiable and analytic.

Q: Does f(x) =x3 -1 on [-1,1] satisfy the conditions of Rolle's the Rolle's Theorem. Justify your…

A: Rolle’s Theorem: If a real valued function f(x) (i) is continuous on [a, b] (ii) is differentiable…

Q: We can find an analytic function f=u+vi such that 2u-3v+1 is not harmonic. False True

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Q: Show that the set of functions 1,cosx,sinx,cos2x,sin2x,…,cosmx,sinmx,… is orthogonal on the interval…

A: We are given the following functions: 1, cosx, sinx, cos2x, sin2x,…, cosmx, sinmx, …   Our aim is to…

Q: Show that the function f : [0, 1] →[0, 1] given by f(x) = (x −1)2 has a fixed point. Can you apply…

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Q: 6. Determine whether the following functions are harmonic. (a) u(x, y) = 2e* cos3y and v(x, y) = 2e"…

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Q: Suppose f(x) is a function with the following properties: • ƒ"(1) = ƒ"(2) = ƒ”(4) = 0, f"(x) > 0 for…

A: Let's solve for inflection points at which points.

Q: u+iv is analytic function, under what condition (if any) v+iu

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Q: What are the necessary and sufficient conditions for a complex-valued function to be analytic in a…

A: Analyticity is a fundamental concept in complex analysis. A function of a complex variable is said…

Q: 7. Let f(z) be an analytic function in a domain S. Show that (01² 982) 1 |ƒ(z)|² = 4|f'(z)|². + მ2…

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Q: Zez analytical function or not?

A: We check whether w=zez is analytic function or not.

Q: Any function of the form f(x, y) = ax*y" where a is a real number and k, n are positive integers…

A:

Q: 1. Either f(x) a graphically defined function.     A) Choose the point or points that are not roots…

A: Roots of a graphically defined function, also known as zeros, are the values of x for which the…

Q: (6) If f(z) is analytic function show that: [v] + [0] =

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Q: Let f(2)=u+iv be an analytic function. Given that u=v² show that f(z) is a constant.

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Q: 5- Verify that the given function u is harmonic in an appropriate domain D. Find v(x,y), the…

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Q: b) Prove that if f(z) and f(Z) are analytic in a domain I, then I is a constant function.

A: Given Data: We have to prove  if fz and fz¯ are analytic in a domain Ω, then f is a constant…

Q: -2 -1 0 -1- 1 2 3 4 5 6 7 8 9 10 X Find the domain of the function fo g. (Give your answer as an…

A: domain of   f∘g=domain of  g(x)domain of  g(x)=[0,6]domain of   f∘g=domain of  g(x)=[0,6]

Q: c) Which of the following functions are analytic in 121² 2 2: 42²26649 (ii) Z²-1 Z-√√Z²74

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Q: 1) Are the following functions analytic? a) f(z) = z z b) f(z) = z + z?

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Q: Note: N denotes the set of natural numbers, Z denotes the set of integers, and R denotes the Net of…

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Q: 7. Suppose that f = u + iv is analytic and not identically constant. a) Show that uv is the real…

A: