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Solutions for assignment 4

# Solutions for assignment 4 - Solutions for assignment 4 1...

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Section 4.1 Question 1: See solutions in textbook. Other parameterizations are possible, but you should get something which looks about the same as the answers in the textbook. Question 8: We break up the contour into two curves gamma_1 and gamma_2: gamma_1(t) = -2+2i + (1-2i)t, 0 <= t <= 1 gamma_2(t) = e^{-it}, -pi <= t <= 0 the question asks us to combine these two paramterizations. One way is to shift the second curve (say) from the interval [-\pi,0] to the interval [1, 1+\pi] by the change of variables s = t + pi + i: gamma_2(s) = e^{-i(s-pi-1)}, 1 <= s <= 1+pi and combine the two as Gamma(t) = -2+2i + (1-2i)t, 0 <= t <= 1 Gamma(t) = e^{-i(t-pi-1)}, 1 <= t <= 1+pi The reverse contour could be given by -Gamma(t) = e^{-i(-t-pi-1)}, -1-pi <= t <= -1 -Gamma(t) = -2+2i + (1-2i)(-t), -1 <= t <= 0 Other parameterizations are possible. In practice it is not particularly advantageous to obtain a single paramterization for a complicated contour, as you will most likely need to break it back into components in any event. However occasionally it is useful to have a parameterization by a single function instead of a number of unrelated functions. Question 10: (a) Take gamma(t) = z_1 + t(z_2 - z_1), 0 <= t <= 1; the length formula gives

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Solutions for assignment 4 - Solutions for assignment 4 1...

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