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Mathman23
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Homework Statement
Solve [tex]I = \int_{\gamma} f(z) dz[/tex] where [tex]\gamma(t) = e^{i \cdot t}[/tex] and [tex]0 \leq t \leq \pi[/tex]
Homework Equations
Do I use integration by substitution??
The Attempt at a Solution
If I treat this as a line-integral I get:
[tex]I = \int_{a}^{b} f(\gamma(t)) \cdot \gamma'(t) dt = \int_{0}^{\pi} cos(e^{i \cdot t}) \cdot (i \cdot e^{i \cdot t}) dt[/tex]
then If I choose
[tex]u = e^{it}[/tex] and [tex]dt = i \cdot e^{i \cdot u} du[/tex]
where [tex]t=0, u=e^{i \cdot 0} = 1[/tex] and [tex]t=\pi, u=e^{i \cdot 0} = -1[/tex]
[tex]\int_{0}^{\pi} cos(e^{i \cdot t}) \cdot (i \cdot e^{i \cdot t}) dt = \int_{1}^{-1} (i \cdot (cos(u) \cdot u) du = (i \cdot (cos(u) + u \cdot sin(u))) \cdot i \cdot e^{i \cdot u} ]_{1}^{-1} = 2 sin(1) \cdot ((cos(1) + sin(1)) \cdot i[/tex]
Am I on the right path here??
Best regards
Fred
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