CVXI2

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Compute \int_\Gamma \mbox{Re}\ z \ dz along the directed line segment from z = 0\, to z = 1 + 2i\,.

Parametrize the directed line segment by z(t) = t(1 + 2i), 0 \leq t \leq 1. Then

\int_\gamma f(z) \ dz = \int_a^b f(z(t))z^\prime(t) \ dt
\int_\Gamma \mbox{Re}\ z \ dz = \int_0^1 \mbox{Re}(t + 2ti)\cdot (1 + 2i) \ dt
= (1 + 2i)\int_0^1 t \ dt
= \frac{1 + 2i}{2}t^2\left.\right|_0^1
= \frac{1 + 2i}{2}
= \frac{1}{2} + i


Main Page : Complex Variables : Complex Integrals

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