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Visualizing Complex Line Integral

[et_pb_section fb_built="1" _builder_version="3.0.47"][et_pb_row _builder_version="3.0.47" background_size="initial" background_position="top_left" background_repeat="repeat"][et_pb_column type="4_4" _builder_version="3.0.106" parallax="off" parallax_method="on"][et_pb_text _builder_version="3.0.106" custom_padding="|20px||20px" box_shadow_style="preset1"] There is nothing complex about complex line integral. It is just vector addition (and taking a limit of that sum). Let's take a concrete example: $$ \oint_{\lambda} \frac {1}{\zeta} d \zeta = 2 \pi i $$ Here, let \(\lambda \) be the unit circle centered at the origin. Then we pick \( \zeta \) from the circumference of the circle. Suppose we work with polar coordinates. Then the coordinate of a typical \( \zeta \) is \( (1, \theta) \).

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