Toronto Math Forum
MAT3342020F => MAT334Tests and Quizzes => Quiz 3 => Topic started by: xuanzhong on October 08, 2020, 10:16:37 PM

Compute line integral: $\int_{r} Re(z) \,dz$, where r is line segmemt from 1 to i
$$
f(z) = Re(z)
$$
$$
r(t) = (1t)z_0 + tz_1 = (1t)*1 + ti = 1t+it , 0<= t <= 1
$$
$$
r^{'}(t) = i1
$$
$$
f(r(t)) = Re(r(t)) = 1t
$$
$$
\int_{r} f(z) \,dz = \int_{0}^{1}f(r(t))r^{'}(t) \,dt
$$
$$
=\int_{0}^{1}(1t)(i1) \,dt
$$
$$
=(i1)*(t\tfrac{1}{2} t^2)\Big_0^1
$$
$$
=(i1)(1\tfrac{1}{2}0)
$$
$$
=\tfrac{1}{2}(i1)
$$