In the complex line integral, we integrate along a path or curve. In the case of a closed curve how is the integral value 0. And how does singularity points affects the line integral along a closed curve?
How does line integration work in complex analysis?
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complex-analysis
complex-numbers
complex-integration
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0The value is only guaranteed to be $0$ if the function is holomorphic / analytic on the region enclosed by the curve. Other functions can give whatever result you want. – 2017-02-07
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0As an example, take the function $f(z) = \operatorname{Re}(z)$. The integral along a closed curve in this case is equal to the area of the region enclosed by the curve. – 2017-02-07
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0Does the presence of singularity points inside the closed curve affect it(cauchy integral theorem) – 2017-02-08
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0It might. Integrating $\frac 1z$ once counterclockwise around the origin gives an answer of $2\pi i$. $\frac1{z^2}$, on the other hand, still gives you zero. – 2017-02-08
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0Sorry i can't get u – 2017-02-08