Differentiability & Harmonics State and prove the Cauchy-Riemann Theorem or Let ACC be an open set, 20 EA, and let f: AC where f = u + iv (u, v: R2R). Then f is analytic at zo if and only there is a neighbourhood NCA of zo for which: 1. Juu, u, and all exist and are continuous on N, მემოყმე 2. The equations ди дх av and u = ay ay მეს , called the Cauchy-Riemann Equations, are satisfied on all of N. Moreover we have that f'(x + y) = u(x, y) + ix (x, y) = y(x, y) – idy (x, y).

Function f is defined by the equation above for all pairs (x,y) of real numbers such that f(x,y) is a real number. The graph of the function f in the xyz space is which of the following? a) A sphere centered at (0,0,0) b) A hemisphere centered at (0,0,0) c) A circle centered at (0,0,0) d) A semicircle centered at (0,0,0)

Let ƒz = Əƒ/Əz and fz = əƒ/əz. Recall f = u + iv, the notation ƒ is used for the complex conjugate function u iv. Verify the chain rule (fog)z (fog)z = = (fz ○g)9z+(fz° g)āz (fzᵒg) gz+ (fz° g) Iz

3b written out on paper please, not typed out.

4. Show whether or not f is bijective. Letf : R → R be defined as f(x) = I피+1 °

1b written out on paper please, not typed out.

4a written out on paper please

Find the value of b so that the point x = (13, b) belongs to the boundary of the set U = {XER² : x₁ + x₂ < 3}

Let φ1 = 1, φ2 = x, φ3 = x2 on the interval 0 ≤x ≤1. Find the following inner products:(a) < φ1,φ2 >(b) < φ2,φ3 >(c) Distance between φ1 and φ2

2b c(iii)