let b = dw, where d = 2. Draw the complex exponential e^jb for w =...
let b = dw, where d = 2. Draw the complex exponential e^jb for w = 0 to 2*pi. describe the shape of the graph and what is the effect of d on the result (also for different values of d).
Homework Answers
The graph is the unit circle. d=2 has no special effect on the nature of the graph, as w varies from 0 to 2pi. For other values of d, it would give the same circle.
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let b = dw, where d = 2. Draw the complex exponential
e^jb for w =...
1) Consider the complex exponential function e* a) Show that the mapping by w = eiz...
1) Consider the complex exponential function e* a) Show that the mapping by w = eiz for fixed 0 is a rotation by 0 b) Show that e* is periodic with imaginary period of 27i, and that e has no other periods (yes there are doubly and multi-periodic functions)
23. Consider the function w(z) = 2-2 (a) Where in the complex z-plane are the poles of w(z)? (b) ...
23. Consider the function w(z) = 2-2 (a) Where in the complex z-plane are the poles of w(z)? (b) Determine the first three terms for the Taylor series expansion of w(z) about 0 (c) Identify the region of convergence for the Taylor series of part (b). (d) Determine the general expression for the n'h coefficient of the Taylor series expansion of part (b) 208 INTRODUCTION TO COMPLEX VARIABLES (e) There is a Laurent series expansion for wC) about-= 0 in...
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to...
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to z2.)
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the...
Question 1. Let S = {a,b}, and consider the language L = {w E E* :...
Question 1. Let S = {a,b}, and consider the language L = {w E E* : w contains at least one b and an even number of a's}. Draw a graph representing a DFA (not NFA) that accepts this language. Question 2. Let L be the language given below. L = {a”62m : n > 0} = {1, abb, aabbbb, aaabbbbbb, ...} Find production rules for a grammar that generates L.
2. Let s() = 200 0 0t be the Complex Exponential Fourier Series representation of the...
2. Let s() = 200 0 0t be the Complex Exponential Fourier Series representation of the periodic signal st). a. (10 points) What is the frequency of 7th harmonic? b. (10 points) Determine the Power Content in tl 93rd harmonic
(1 point) х Suppose w 9 y + where у 2 + sin(2t), and z =...
(1 point) х Suppose w 9 y + where у 2 + sin(2t), and z = z X = e e5t, y 2 + cos(7t). as X. dw A) Use the chain rule to find as a function of x, y, z, and t. Do not rewrite x, y, and z in terms of t, and do not rewrite e e5t d dw 5/y(e^5t)+-x/y^2+1/z(2cos(2t))+(-y/3^2)*(-7sin(7t)) dt Note: You may want to use exp() for the exponential function. Your answer should be...
(Exponential martingales) Suppose O(t,w) = (01(t, w),...,On(t,w)) E R" with Ox(t,w) E VIO, T] for k...
(Exponential martingales) Suppose O(t,w) = (01(t, w),...,On(t,w)) E R" with Ox(t,w) E VIO, T] for k = 1,..., n, where T < 0o. Define 2. = exp{ jQ1, wydBlo) – 4 640,w.do}osist where B(s) ER" and 62 = 0 . 0 (dot product). a) Use Ito's formula to prove that d24 = 2:0(t,w)dB(t). b) Deduce that 24 is a martingale for t <T, provided that Z40x(t,w) € V[O,T] for 1 sk sn.
Let G-(V,E), where V a, b, c,d,e,f.g,h and E- lla, b], [a,c), tb, c),{b,d), [b, g],...
Let G-(V,E), where V a, b, c,d,e,f.g,h and E- lla, b], [a,c), tb, c),{b,d), [b, g], lc, d), se. f, se. g),if,g),if,h], lg, h)). (a) Draw the corresponding graph with no edges crossing. (b) How many paths are there in G from a to h'? (c) How many of these paths have length less than 5? List them.
Explain ur working 4. [6 marks] Using the following graph representation (G(VE,w)): V a, b,c, d,e, fh E -la, b, [a, fl,la,d, (b,ej, [b,d, c,fl,fc,d],Id,el, sd, f) W(a, b) 4, W(a, f)-9, W(a, d)-10 W(b...
Explain ur working
4. [6 marks] Using the following graph representation (G(VE,w)): V a, b,c, d,e, fh E -la, b, [a, fl,la,d, (b,ej, [b,d, c,fl,fc,d],Id,el, sd, f) W(a, b) 4, W(a, f)-9, W(a, d)-10 W(b, e) 12, W (b, d)7, W(c,d) 3 a) [3 marks] Draw the graph including weights. b) [2 + 1-3 marks] Given the following algorithm for finding a minimum spanning tree for a graph: Given a graph (G(V,E)) create a new graph (F) vith nodes (V)...
Can you do part A through B please? 2 Euler's formula relates the complex exponential to...
Can you do part A through B please?
2 Euler's formula relates the complex exponential to trigonometric functions as e" = cos(9) + j sin(9) This problem considers two alternate forms of Euler's formula. (a) Show that we can represent cos(0) in terms of complex exponentials as eje +e-je cos(e) (b) Derive a similar expression to part (a) for sin(e) (c) Use the results of part (a) to hand com pute cos(2). Verify your result with MATLAB. This result conflicts...
1) Consider the complex exponential function e* a) Show that the mapping by w = eiz for fixed 0 is a rotation by 0 b) Show that e* is periodic with imaginary period of 27i, and that e has no other periods (yes there are doubly and multi-periodic functions)
23. Consider the function w(z) = 2-2 (a) Where in the complex z-plane are the poles of w(z)? (b) Determine the first three terms for the Taylor series expansion of w(z) about 0 (c) Identify the region of convergence for the Taylor series of part (b). (d) Determine the general expression for the n'h coefficient of the Taylor series expansion of part (b) 208 INTRODUCTION TO COMPLEX VARIABLES (e) There is a Laurent series expansion for wC) about-= 0 in...
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to z2.)
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the...
Question 1. Let S = {a,b}, and consider the language L = {w E E* : w contains at least one b and an even number of a's}. Draw a graph representing a DFA (not NFA) that accepts this language. Question 2. Let L be the language given below. L = {a”62m : n > 0} = {1, abb, aabbbb, aaabbbbbb, ...} Find production rules for a grammar that generates L.
2. Let s() = 200 0 0t be the Complex Exponential Fourier Series representation of the periodic signal st). a. (10 points) What is the frequency of 7th harmonic? b. (10 points) Determine the Power Content in tl 93rd harmonic
(1 point) х Suppose w 9 y + where у 2 + sin(2t), and z = z X = e e5t, y 2 + cos(7t). as X. dw A) Use the chain rule to find as a function of x, y, z, and t. Do not rewrite x, y, and z in terms of t, and do not rewrite e e5t d dw 5/y(e^5t)+-x/y^2+1/z(2cos(2t))+(-y/3^2)*(-7sin(7t)) dt Note: You may want to use exp() for the exponential function. Your answer should be...
(Exponential martingales) Suppose O(t,w) = (01(t, w),...,On(t,w)) E R" with Ox(t,w) E VIO, T] for k = 1,..., n, where T < 0o. Define 2. = exp{ jQ1, wydBlo) – 4 640,w.do}osist where B(s) ER" and 62 = 0 . 0 (dot product). a) Use Ito's formula to prove that d24 = 2:0(t,w)dB(t). b) Deduce that 24 is a martingale for t <T, provided that Z40x(t,w) € V[O,T] for 1 sk sn.
Let G-(V,E), where V a, b, c,d,e,f.g,h and E- lla, b], [a,c), tb, c),{b,d), [b, g], lc, d), se. f, se. g),if,g),if,h], lg, h)). (a) Draw the corresponding graph with no edges crossing. (b) How many paths are there in G from a to h'? (c) How many of these paths have length less than 5? List them.
Explain ur working
4. [6 marks] Using the following graph representation (G(VE,w)): V a, b,c, d,e, fh E -la, b, [a, fl,la,d, (b,ej, [b,d, c,fl,fc,d],Id,el, sd, f) W(a, b) 4, W(a, f)-9, W(a, d)-10 W(b, e) 12, W (b, d)7, W(c,d) 3 a) [3 marks] Draw the graph including weights. b) [2 + 1-3 marks] Given the following algorithm for finding a minimum spanning tree for a graph: Given a graph (G(V,E)) create a new graph (F) vith nodes (V)...
Can you do part A through B please?
2 Euler's formula relates the complex exponential to trigonometric functions as e" = cos(9) + j sin(9) This problem considers two alternate forms of Euler's formula. (a) Show that we can represent cos(0) in terms of complex exponentials as eje +e-je cos(e) (b) Derive a similar expression to part (a) for sin(e) (c) Use the results of part (a) to hand com pute cos(2). Verify your result with MATLAB. This result conflicts...
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