Problem 3: Consider the function f(2) = e2/ . (a) Determine the solutions to the equation...
Log(2+5) 1. Consider function f(z) sin 2 (a) Determine all singular point (s) of f enclosed in the circle C4(0) (b) Are they isolated singularities? If so, which kind of isolated singularity are they (remov- able, pole, essential)? (c) Compute the residue of f at each of these singularities (d) Evaluate the integral f f(2)dz where y is the circle Ca(0) oriented counterclockwise 1.0 0.5 -0.5 Answer key 1. (а) z0,-T, T (b) Yes. Each is a pole of order...
Consider the following complex-variable function cosh a < T f(z) la! cosh πχ, a) Find all its singularities, state their nature and compute the residues b) Consider the rectangular contour y with vertices at tR and tRi. Evaluate 6 6 dz cosh πχ c) Using the previous result take the limit R-to prove that cosh ax (10] 2 cos (g Hint: remember that cosh(a + b) -cosh a cosh b + sinh a sinh b d) Why is the above...
3. (18pt) Consider the following problems. (31) Suppose that f is a real-valued continuous function on [a, b], and z, i s n are points satisfying a < xi < x2 . . . < zn < b. Then there is a c E [n, such that rn
Consider the function below. y z= 2+x+ (a) Match the function with its graph (labeled A-F). (b) Match the function with its contour map (labeled 1-VI). O V VI Give reasons for your choices. Select- Also, the values of z approach 0 as we use This function is not periodic, ruling out the graphs in (b) Match the function with its contour map (labeled I-VI) II III IV V VI Give reasons for your choices. This function is not periodic,...
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
Problem 3 Use series expansion to find the slope of the following function at 0. f(x) = V2 + x - V2 - x Problem 5 Solve the equation in the complex space: z-i|z| = Re{z} and represent the solution in the complex plane. Problem 6 What is the locus of points that satisfy the following equation in the complex plane Re[7z - 8z* – 31m{z} + z2 + zz*] = 0
8 pts Question 3 Consider the function f(x,y, 2)(x 1)3(y2)3 ( 1)2(y2)2(z 3)2 (a) Compute the increment Af if (r,y, z) changes from (1,2,3 (b) Compute the differential df for the corresponding change in position. What does (2,3,4) to this say about the point (1, 2,3)? ( 13y2)3 ( 1)2(y 2)2(z 3)2 with C (c) Consider the contour C = a constant. Use implicit differentiation to compute dz/Ox. Your answer should be a function of z. (d) Find the unit...
5. Consider the function f(z)-1. (a) Sketch the horizontal line y 1/2 together with its image under f b) Verify that the image of line y- b>0 is a circle. What are its center and radius? c) What is the image of the half-plane : y>1/2 under f? 5. Consider the function f(z)-1. (a) Sketch the horizontal line y 1/2 together with its image under f b) Verify that the image of line y- b>0 is a circle. What are...
Problem 3: Consider an IIR filter described by the difference equation (a) What is the system function H(a) of this fiter? [5 points) (b) Determine the zeros and poles of the system and sketch the zero-pole plot in z-plane. 5 points (c) Plot the block diagram of this IIR filter. [10 points (d) Given the input zfn-cos(mn/3) + 2δ[n] + 5in-11, determine the output yln. 15 points
4 Consider the autonomous differential equation y f(v) a) (3 points) Find all the equilibrium solutions (critical points). b) (3 points) Use the sign of y f(z) to determine where solutions are increasing / decreasing. Sketch several solution curves in each region determined by the critical points in c) (3 points) the ty-plane. d) (3 points) Classify each equilibrium point as asymptotically stable, unstable, or semi-stable and draw the corresponding phase line. 4 Consider the autonomous differential equation y f(v)...