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Consider the function z(z-3) f (z) = - (z+1)2 (22+16) Syntax notes: • When entering lists...
Re -3 -2 -1 0 1 2 3 4 Note that C is not a simple...
Re -3 -2 -1 0 1 2 3 4 Note that C is not a simple curve, so Cauchy's integral formula does not directly apply. By breaking up C as needed, evaluate T z2+9 Jc (2+2-i)(2+1-i)z dz. 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). z2+9 (a) The poles of (z+2-i)(2+1-i)z that lie inside Care...
Log(2+5) 1. Consider function f(z) sin 2 (a) Determine all singular point (s) of f enclosed...
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...
Complex Analysis: 1 + COS Z Define the function 1 f(2)= (z + 1)2(23 +1) (a)...
Complex Analysis:
1 + COS Z Define the function 1 f(2)= (z + 1)2(23 +1) (a) Find all the singularities of f(z) and classify each one as either a removable singulatiry, a pole of order m (and find m), or an essential singularity. (b) Let I = 71+72, where yi and 72 are the directed smooth curves parameterized by TT zi(t) = 2i(1 – 2t), 0 < t < 1 z2(t) = 2eit, 277 < t < 2' respectively. Compute...
For this problem, we will consider the polynomial function f(z) = 4x4 - 1623 +2422 -...
For this problem, we will consider the polynomial function f(z) = 4x4 - 1623 +2422 - 322 +32 over the interval -3 <I<3 (a) The degree of f(x) is Number (b) Which of the following choices describe the end behavior of f(x)? The graph of f(x) acts O like 22 (ie both ends up) Olike -3 (ie both ends down) like 23 (ie left end down, right end up) Olike -23 -(e. left end up, right end down) O None...
Problem 3: Consider the function f(2) = e2/ . (a) Determine the solutions to the equation...
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...
Q3: 5 marks (A) Expand f(z) (2-1)(2-3) in a Laurent series valid for (i) Iz -...
Q3: 5 marks (A) Expand f(z) (2-1)(2-3) in a Laurent series valid for (i) Iz - 11 < 2, and (ii) Iz - 31 < 2. 1.5 marks each part (B) Use Laurent series to find the residue of f(2)= e (x - 2)-2 at its pole z = 2. 2 marks
Complex Analysis: . (a) Find a single function f(z) which has all of the following properties:...
Complex Analysis:
. (a) Find a single function f(z) which has all of the following properties: f(z) is discontinuous at the origin z = 0, at z = 1, and at all points z with Arg(z) = 7/4, but f(z) is continuous at all other points of C; • f(z) has a simple zero at z = :i; and f(z) has a pole of order 3 at z = n. Justify that your function f(x) has each of the properties...
1 1 + COS Z 8. Define the function f(x) = (2 + 1)2( 23 +1)...
1 1 + COS Z 8. Define the function f(x) = (2 + 1)2( 23 +1) (a) (6 points) Find all the singularities of f(z) and classify each one as either a removable singulatiry, a pole of order m (and find m), or an essential singularity. (b) (6 points) Let I = 71+72, where 71 and 42 are the directed smooth curves parameterized by -TT TT zi(t) = 2i(1 – 2t), 05t51 z2(t) = 2eit, sts 2' respectively. Compute Sr...
Problem #1. Topics: Z Transform Find the Z transform of: x[n]=-(0.9 )n-2u-n+5] X(Z) Problem #2. T...
Problem #1. Topics: Z Transform Find the Z transform of: x[n]=-(0.9 )n-2u-n+5] X(Z) Problem #2. Topics: Filter Design, Effective Time Constant Design a causal 2nd order, normalized, stable Peak Filter centered at fo 1000Hz. Use only two conjugate poles and two zeros at the origin. The system is to be sampled at Fs- 8000Hz. The duration of the transient should be as close as possible to teft 7.5 ms. The transient is assumed to end when the largest pole elevated...
Consider the function f(t) = 23 +22-22 Answer all parts: (a) ..(). (a) What is the...
Consider the function f(t) = 23 +22-22 Answer all parts: (a) ..(). (a) What is the degree of f(x)? The degree of f() is (b) Which of the following choices describe the end behavior of f(x)? The graph of f(x) acts O like 22 (ie, both ends up) Olike -22 (1.e. both ends down) Olike 23 (i e left end down, right end up) O like-3(e. left end up, right end down) None of the above (C) State the z-intercepts...
Re -3 -2 -1 0 1 2 3 4 Note that C is not a simple curve, so Cauchy's integral formula does not directly apply. By breaking up C as needed, evaluate T z2+9 Jc (2+2-i)(2+1-i)z dz. 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). z2+9 (a) The poles of (z+2-i)(2+1-i)z that lie inside Care...
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...
Complex Analysis:
1 + COS Z Define the function 1 f(2)= (z + 1)2(23 +1) (a) Find all the singularities of f(z) and classify each one as either a removable singulatiry, a pole of order m (and find m), or an essential singularity. (b) Let I = 71+72, where yi and 72 are the directed smooth curves parameterized by TT zi(t) = 2i(1 – 2t), 0 < t < 1 z2(t) = 2eit, 277 < t < 2' respectively. Compute...
For this problem, we will consider the polynomial function f(z) = 4x4 - 1623 +2422 - 322 +32 over the interval -3 <I<3 (a) The degree of f(x) is Number (b) Which of the following choices describe the end behavior of f(x)? The graph of f(x) acts O like 22 (ie both ends up) Olike -3 (ie both ends down) like 23 (ie left end down, right end up) Olike -23 -(e. left end up, right end down) O None...
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...
Q3: 5 marks (A) Expand f(z) (2-1)(2-3) in a Laurent series valid for (i) Iz - 11 < 2, and (ii) Iz - 31 < 2. 1.5 marks each part (B) Use Laurent series to find the residue of f(2)= e (x - 2)-2 at its pole z = 2. 2 marks
Complex Analysis:
. (a) Find a single function f(z) which has all of the following properties: f(z) is discontinuous at the origin z = 0, at z = 1, and at all points z with Arg(z) = 7/4, but f(z) is continuous at all other points of C; • f(z) has a simple zero at z = :i; and f(z) has a pole of order 3 at z = n. Justify that your function f(x) has each of the properties...
1 1 + COS Z 8. Define the function f(x) = (2 + 1)2( 23 +1) (a) (6 points) Find all the singularities of f(z) and classify each one as either a removable singulatiry, a pole of order m (and find m), or an essential singularity. (b) (6 points) Let I = 71+72, where 71 and 42 are the directed smooth curves parameterized by -TT TT zi(t) = 2i(1 – 2t), 05t51 z2(t) = 2eit, sts 2' respectively. Compute Sr...
Problem #1. Topics: Z Transform Find the Z transform of: x[n]=-(0.9 )n-2u-n+5] X(Z) Problem #2. Topics: Filter Design, Effective Time Constant Design a causal 2nd order, normalized, stable Peak Filter centered at fo 1000Hz. Use only two conjugate poles and two zeros at the origin. The system is to be sampled at Fs- 8000Hz. The duration of the transient should be as close as possible to teft 7.5 ms. The transient is assumed to end when the largest pole elevated...
Consider the function f(t) = 23 +22-22 Answer all parts: (a) ..(). (a) What is the degree of f(x)? The degree of f() is (b) Which of the following choices describe the end behavior of f(x)? The graph of f(x) acts O like 22 (ie, both ends up) Olike -22 (1.e. both ends down) Olike 23 (i e left end down, right end up) O like-3(e. left end up, right end down) None of the above (C) State the z-intercepts...
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