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 + (12i)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(spi1)}, 1 <= s <= 1+pi
and combine the two as
Gamma(t) = 2+2i + (12i)t, 0 <= t <= 1
Gamma(t) = e^{i(tpi1)}, 1 <= t <= 1+pi
The reverse contour could be given by
Gamma(t) = e^{i(tpi1)}, 1pi <= t <= 1
Gamma(t) = 2+2i + (12i)(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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 Spring '08
 Grossman

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