(a) Let R be a simply connected region and let f : R → C be a holomorphic functio Fix a point z in R and define F: R → C via F(z) = √( f (6) ds, where C(z) is any contour in R that starts at z and ends at z. i. Show that = √(20.20+41 5 (5) ds, [20,zo+h] F (²0 + h) − F (zo) = √ - 20 for all zo ER and h E C, where h is small enough so that the line segme [zo, zo+h] is contained in R. You may assume without proof that integra of holomorphic functions satisfy contour independence on simply connect regions.

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: Construct bijective maps (a) from [0, 1) to (0, 2]. (b) from R − {3} to R − {1} (c) from Z to Z −…

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Q: (a) Prove that f is a one-to-one correspondence. (b) Prove that ray AF is between rays AB and AE if…

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Q: Q: Show that the functi on f:R →R given by f (x) = ax +b, where ab e R, a #0 is a bijection.

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Q: z=x+iy, consider the entire function f(z)=u(x,y) + iv(x,y) where u(x,y)=3x2y+2x2-y3-2y2. Prove that…

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Q: (c) (x+1)=-y+15 dx 00000000 ✔separable ✔exact 0000000 €0.000000 linear dy 1 (d) m dx x(x-y)…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: (a) Let M = R. Give the radius r and the center c of B(-2,5) n B(6, 7). ne

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Parametrize the directed line segment which joins the point −2 + 3i to the point 5 − 7i, that is,…

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Q: Let f:R → R such that f(x) = x² – 2x + 1. Verify whether f is bijective ? explain your answer.

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Q: 4. Show whether or not f is bijective. Letf : R → R_be defined as f(x) = I피+1 %3D

A: A function f is said to bijective if it is one-one and onto

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Q: IS a function.. (b) Let R= {(x,y) € R × R : 1 < ¤ < 4 or x = 6}, %3D and S = {(x, y) E R × R : –1 <…

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Q: Show that f: R\{2} → R\{5} defined by f(x) = (5x+1)/(x - 2) is bijective.

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

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Q: If f has a simple pole at zo and g is analytic at zo, show that Res (fg, z = 20) = g(zo) Res (f, z =…

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Q: Let f: R → [1, ), f(x) = x² + 1 g : [1, 0) → [0, ∞), g(x) = Vx – 1 i) Derive g • f and show your…

A: Composition of two functions

Q: Let R be the relation on R given by R= {(x, y) : xy > 0}. Is R reflexive? symmetric? antisymmetric?…

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Q: Let the function f: R R be defined by f(r) = 3r -5. Prove that the function is bijective.

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Q: Let 7,V be a projection af the Vect or space. Accordling to this: Show that every Xe Im (T) for…

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Q: Let f(z) = E (z – 5)" n4" (n + 2)2 n=1 i) ii) iii) Find the region where the function f(2) is…

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Q: Let f: R2 R2 be defined f(x, y) = (3x + y,x+y). (a) Prove that f is a bijection. (b) Keep f the same…

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Q: Show that  ℝ2 = span([1 1], [1 -1])

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Q: Sketch the domain D in the (x, y)-plane satisfying ≤ y ≤ 2e- and e ≤ y ≤ 3e. In your sketch, label D…

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Q: 3. Construct a bijection f: [0, 1] → (0, 1) UN.

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Q: Exc.: Let c[a, b] = (f:f: [a, b] → R be a cont. function), define d: c[a, b] * c[a, b] → R as d(f,g)…

A: Let Ca,b=f:a,b→ℝ be continious function  define d: Ca,b∗Ca,b→ℝ as df,g=∫abfx-gxdx

Q: f: X → Y and g: Y→ V be bijective functions. Show the following: ) gof: X → V is injective. gof: X→V…

A: It is given that f:X→Y and g:Y→V are bijective functions.

Q: Show that f : R → R is bijective, where: f(x) %3D필

A: A function is injective if f(x)=f(y)⇒x=y. A function is onto if for every y in Y there exists a x in…

Q: Let f: R\{3} → R\{5} be defined by f(x) = 5x/(x-3). Show that f is bijective and find f-¹.

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Q: ii).Now consider a function space of the set of functions f(x) of a real variable x E(-00, +00),…

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Q: If f(z) u * 1V is an analytic function of z then prove that 2B. Əx?

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Q: Let f(z) = (z-₁)(z-3₂)... (zn), where ₁, 2,... E C. Ify is a positively oriented simple close…

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Q: Let f : [0, ∞) → [0, ∞) be the function defined by f(x) = √x. Show that f is a bijection.

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Q: B. Show that the space V = {f : R → R} of functions from R to R is not finite- dimensional.

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Q: Define T: R2 R² by T(x1, x2) = (x2 +3, x₁ + 4) (a) Find u € R² and L = O(R²) such that T = T₂ o L.…

A: Given: T:ℝ2→ℝ2 by  Tx1,x2=-x2+3,x1+4  , x1,x2∈ℝ2 To find:  (a) u∈ℝ2 and  L∈Oℝ2 such that T=Tu∘L.…