(a) Find functions u, v: R² → R such that g(x+iy) = u(x, y) + iv(x, y) for all x, y EE (b) State the Cauchy-Riemann equations, and show that u and us tions everywhere in R2. (c) Use the Cauchy-Riemann equations to show that gʻ(z) = −g(z)

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A: The function given to us is  F(x,y)=yx2+y2i-xx2+y2j

Q: 1. Let f(x+iy) = u(x, y) + iv(x, y), where u : R² → R and v : R² → R are defined by u(x, y) = x +…

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A: See the attachment

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Q: Find the real and imaginary parts u and v (respectively) of the given complex function f as…

A: Solution :- Given :-fz=z3-6z+9Let z = x+iyfz=fx+iy=x+iy3-6x+iy+9Expand it :-x+iy3-6x+iy+9

Q: [#1] Exercise 7.46. Prove that if y = y' then y = ky' for some scalar k.

A: Given that γ≡γ' The y can be written as,  γ≡r mod n The y' can be written as, γ'≡ r mod n

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A: According to the question,

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A: To answer the given question we shall use the following Proposition- Proposition: If f=u+jv , fx and…

Q: 1. Chapter 14 Review 43: Let g(u, v) = f(u³-v³, v³ - u³). Prove that v209 ,მყ dv +4² 0.

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Q: 1. Let f(x+iy) = u(x, y) + iv(x, y), where u : R² → R and v : R² → R are defined by u(x, y) = x +…

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Q: . Let f(x+iy) = u(x, y) + iv(x, y), where u : R² → R and v : R² → R are defined by u(x, y) = x +…

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Q: 1 Justify the Cauchy-Riemann equations for f(2) = e² where f(z) is holomorphic.

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A: Hi! Please see attached file. Thank you!Explanation:

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