Problem (3) A function f(z) is analytic in the disk -1 where the modulus satisfies the bound Here b 2 a > 0 Find an...
Problem 5. This problem asks you to find the supply function of a firm that only uses one input (z) to produce one output (q). In addition, it highlights how the choices we make to represent an economic situation can influence the conclusions we arrive at. Suppose there is a firm operating in a competitive environment. The firm has a technology given by f(z)Az, where z 20 is the amount of labor the firm hires at a wage of w...
Can you help solve Problem 5 part c? Problem 5: . (a) Find the Taylor polynomials Τη ( ) for f(z)-1. centered at a-1 b) Compute the error bound for |f (2) Tn (2) (c) Find |f(2)-i00p(2) Problem 5: . (a) Find the Taylor polynomials Τη ( ) for f(z)-1. centered at a-1 b) Compute the error bound for |f (2) Tn (2) (c) Find |f(2)-i00p(2)
Problem 3: Consider the function f(2) = e2/ . (a) Determine the solutions to the equation f(2) =1 and sketch the locations of these points in the complex plane. (3 points) (b) Consider a circle in the complex plane described by |2 = 1 (unit circle). How many points satisfying f()1 are within the unit circle? Suppose you had considered a much smaller circle, say, described by 10-15. Now how many points are within this smaller circle? (3 points) Points...
(20 points) Consider the function f(z) z in the interval [0, 2π). (a) Derive the Fourier coefficients ck fork = 0, 1,土2, (b) Derive the Fourier coefficients ao, ak, bk for k 1,2,... .. (c) Plot the partial Fourier series, along with the function f, by retaining 1, 10, 50, 100 terms in the summation (use the second form involving cosines and sines) d) Comment on the convergence of the partial Fourier series. Note: you should submit only 1 plot...
Problem #3: Find the residues of the following functions at z = 0 a) f(3) = 2* cos () b) f(3) = 1-cosa; c) f(3) = CS2 f(3) = 25(1 – 22) COS 2 COS 2 e) f(3) = 15e *e*1 f) f(3) = cosz - 1 9) f(3) = (sin 2)23 W f(z) = (eš – 1)2
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0 if r <0 θ(z) = 2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z) 28?u(t,z), te (0,00), z (0,3); with initial condition u(0, z)fx), where f(0) 0 and f (3) 0 and with boundary conditions u(t,0)-0, r 30 Using separation of variables, the solution of this problem is 4X with the normalization conditions un(m3ī)-. n@) : ї, a. (5/10) Find the functions wn with index n1. Wnlz) b. (5/10) Find the functions vn with index n 1. n(t)...
25. The point (0, 0) is the only critical point of the function f(,y)(2+y)++3 in the interior of the disk D +(+1 s 9/4). The following graph is of z .) restricted to the boundary of D, which is parameterized by -cost, y1n t, for 0s t s 2. Find the absolute minimum value of f(x, y) on D. (A) 0 (B) 2 (C) 3 25 3.25 (E) 2.9 (F) 3 (G) 3.25 (H) 4 25.. -8.25 (J) 9.25 3.75...
The graph of f is shown to the right. The function F(z) is defined by F(z) = f f(t) dt for 0 x 4. a) Find F(0) and F(3). 2 b) Find F (1). c) For what value of z does F(z) have its maximum value? What is this maximum value? d) Sketch a possible graph of F. Do not attempt to find a formula for F. (You could, but it is more work than neces- sary.) -1 The graph...
using discrete structures 3. Consider the function F(x, y, z) for x, y, z z 0 defined as follows: a. F(x, y, 0)-y+1 b. F(x, 0, 1)-x c, F(x, 0, 2) = 0 d. F(x, 0, z+ 3)-1 e. F(x, y, z)-F(x, F(x, y-1, z), z-1) Using Induction, prove the following a. F(x, y, 1)-x +y b, F(x, y, 2) = xy c. F(x, y, 3)-xy 3. Consider the function F(x, y, z) for x, y, z z 0 defined...