9. Let f(z) z4 +6z2 13. Find the residue of z2/f(z) at the zeros f(2)=0 which lie in the upper half-plane fwEC: Rew> 0}. of = 9. Let f(z) z4 +6z2 13. Find the residue of z2/f(z) at the zeros f(2)=0 which lie in the upper half-plane fwEC: Rew> 0}. of =
Problem 8 (15). Let f(2) 2+1)(22 + 2) (a) Find the residue of f(z) at the point 2 = 1. (b) Find the residue of f(z) at the point (c) Evaluate the integral P.V (r21)(r22) Problem 8 (15). Let f(2) 2+1)(22 + 2) (a) Find the residue of f(z) at the point 2 = 1. (b) Find the residue of f(z) at the point (c) Evaluate the integral P.V (r21)(r22)
Problem 3. Evaluate the integral co sinx dx. Hint: Apply residue theorem to the function f(z) = and the contour y of the following shape:
2. Find three different Laurent series representations (about 0) for the function 3 f(z) 2. Find three different Laurent series representations (about 0) for the function 3 f(z)
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...
1. (20pts=7+5+8) (a) Find the order of the zero z = 0 of the function f(3) = ** (e*- 1). (b) Let 2 denote the principal branch of z3. Can in power of z in the annular domain be expanded in Laurent's series ann (0;0, R) = {2 € C:0< |2|< R} for some R >0? (c) Find the Laurent series in powers of 2 (i.e., Zo=0) that represents the function f(3) = in the annular domain 1 < 121...
7. Find a holomorphic function F(z) on Ω-z I Izl < r} such that for any a E Ω, F(a) F(0)-da. 0 7. Find a holomorphic function F(z) on Ω-z I Izl
Fourier Series MA 441 1 An Opening Example: Consider the function f defined as follows: f(z +2n)-f(z) Below is the graph of the function f(x): 1. Find the Taylor series for f(z) ontered atェ 2. For what values of z is that series a good approximation? 3. Find the Taylor series for this function centered at . 4. For what values ofェis that series a good approximation? 5, Can you find a Taylor series for this function atェ-0? Fourier Series...
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 optimal bound on |f'(0) in terms of a and b. Complete arguments required By optimal it is meant that (1) the bound holds for all functions with the stated property and (2) there actually is a function with the stated property such that the bound holds as an equality. The second part of this problem...
Consider the function z(z-3) f (z) = - (z+1)2 (22+16) Syntax notes: • When entering lists in the questions below, use commas to separate elements of the list. Order does not matter. • The complex number i is entered as I (capital i). (a) List all the poles of f(z). -1,4-1,-4*1 BD (b) Enter the residue of the second-order pole. -1/4 OD