- #1
fishturtle1
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Homework Statement
Calculate the following integrals on the given paths. Why does the choice of path change/not change each of the results?
(a) f(z) = exp(z) on
i. the upper half of the unit circle.
ii. the line segment from − 1 to 1.
Homework Equations
∫γf(z) = ∫f(γ(t))γ'(t)dt, with the limits being the limits of the parametrization.
The Attempt at a Solution
i) γ(t) = eit, t ∈ [0, π]
Integral = ∫ez dz = ∫eeitieitdt
u substitution: u = eit, du = ieit
=> Integral =∫eudu, I leave the lower bound at 0 and upper bound at π because I'm going to substitute for u at the end.
Integral = eu]0π
= eeit]0π
= eeiπ - ee0
= eeiπ - e1 = eeiπ - eii)
γ(t) = t, t ∈ [-1, 1]
Integral = ∫e2 (1) dt, with lower bound = -1, upper bound = 1.
= et ]-11
= e1 - e-1
= e - 1/e
So the path does matter because two different paths gave two different answers.
Whats wrong with my answer?