(4) Show that a subset X C C of the complex plane is open iff its...
Exercise 3. A subset A C R is said to be closed if A contains all of its limit points. A subset B CR is said to be open if its complement is a closed subset. (A) Let A CR be a closed set and let & A. Show that there is a positive 8 >0 such that A does not intersect the interval (-0,2+). (B) Let B CR be an open subset and let 3 € B. Show that...
2) If F Uc R2R is a 1-1 continuously differentiable map of an open subset U of the plane and A is a measurable subset of U, then the area of F(A) is given by area(F(A)) - A det Jrx, )dm(x, y) If F is the induced map of a holomorphic f, what is the resulting formula? 2) If F Uc R2R is a 1-1 continuously differentiable map of an open subset U of the plane and A is a...
. Let A, B and C be subset of a universal set U. (a) Prove that: Ac x Bc ⊂ (A × B)c (the universal set for A × B is U × U). So A compliment x B compliment = AxB Compliment
Question 1. In this question, for brevity we define an open sector in the complex plane to be a set β), where 0 < β-α < 2π. Consider : α < arg(z) < β} and a closed sector to be a set {z : α < arg(z) the following transcendental elementary functions: e* cos(z) f(z) = e:cos(z): g(z)= In which sectors, if any, do each of these functions decay to zero as 0o? Explain your answers and distinguish clearly between...
Question 1. In this question, for brevity we define an open sector in the complex plane to be a set {z : α < arg(z) < β} and a closed sector to be a set {z : α < arg(z) β), where 0 < β-a < 2π. Consider the following transcendental elementary functions f(z)=e: cos(z): g(z) =e*cos(z) cos(2); 92 2 In which sectors, if any, do each of these functions decay to zero as 00? Explain your answers and distinguish...
Question 1. In this question, for brevity we define an open sector in the complex plane to be a set {z : α < arg(z) < β} and a closed sector to be a set {z : α < arg(z) β), where 0 < β-a < 2π. Consider the following transcendental elementary functions f(z)=e: cos(z): g(z) =e*cos(z) cos(2); 92 2 In which sectors, if any, do each of these functions decay to zero as 00? Explain your answers and distinguish...
this question is about complex variables Exercise 2 (i If is open in C and Ac C any subset, then ON A is open in A (ii) If A C C is any subset and U is open in A, then there exists 2 open in C with 2NA=U. Hint. Use the definition and construct O as union of disks. Exercise 2 (i If is open in C and Ac C any subset, then ON A is open in A...
1 Let f: R R be a continuously differentiable map satisfying ilf(x)-FG) ll 리1x-vil, f Rn. Then fis onto 2. f(RT) is a closed subset of R'" 3, f(R") is an open subset of RT 4. f(0)0 or all x, y E 5) S= (xe(-1,4] Sin(x) > 0). Let of the following is true? I. inf (S).< 0 2. sup (S) does not exist Which . sup (S) π ,' inf (S) = π/2 1 Let f: R R be...
A. Consider complex plane C and identify it with a plane R2 in 3D-space Rº with basis vectors i, j, k, so the real line goes along i and imaginary line along j. Then a complex number z = x + yi is identified with a vector z = xi+ yj. Show that the inner (dot) product and vector product of z and w are given by z. = Re(zw), Ž x ū = Im(zw)k.
1) Show that if U is a non-empty open subset of the real numbers then m(U) > O. 2) Give an example of an unbounded open set with finite measure. Justify your answer, 3) If a is a single point on the number line show that m ( a ) = O. 4) Prove that if K is compact and U is open with K U then m(K) m(U). 5) show that the Cantor set C is compact and m(C)...