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The Residue Theorem in complex analysis is quite impressive and has profound implications for many applications and integral calculations. The ability to reduce the closed integral of a complex function $f(z)$ (or its Laurent series representation $\sum_{n=-m}^{\infty}a_{n}(z-z_{0})^{n}$) in a domain around, say, an $m$th-order pole $(z-z_{0})^{m}$ to just a constant number ($i2\pi$) multiplied by the coefficient ($a_{-1}$) of the first negative power in Laurent series is a massive computational feat.

But what is more baffling to me is: why, of all other coefficients, did $a_{-1}$ turn out to constantly be the only important one? What is the significance of this specific term (namely, $1/(z-z_{0})$) compared to the others or different orders?

I understand how it was derived mathematically in a step by step fashion and how all terms eventually cancel out except for this term. But I just cannot help to wonder about its deeper significance physically or mathematically, when we look at any arbitrary complex function integral in a model or an application and end up just caring about its $a_{-1}$ coefficient (residue) and nothing else? What is the physical or geometrical or mechanical or "real-world" meaning of the residue that can be more intuitively felt in practice?

Andrews
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user135626
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    In some sense residues contain all the "information" that prevents a loop integral from being zero - if you can intuit why the integrals along closed loops of holomorphic functions are zero this might help. Alternatively, note that any power of $z$ except $z^{-1}$ integrates to another power of the form $z^{-1}$; the fact $\tfrac{d}{dx} \log(x) = 1/x$ creates a problem here. That we get a logarithm as the antiderivative is important, since we know that complex exponentials rotate. Thus the logarithm should involve a rotation somehow, as it does via the argument function... – Brevan Ellefsen Nov 16 '19 at 22:29
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    ... and the winding that happens via the argument function can be used to explain some of this stuff. I hear Needham's Visual Complex Analysis has some stuff on this, though I have not personally read it. I think for me, complex analysis is very "information theoretic"; every point (or more correctly, neighborhood of said point) in the domain of a holomorphic function contains some global information if we can extract it (i.e. analytically continuing to the natural boundary). When we hit a singularity the information gets "wrapped" up there. Not sure if this is helpful, but it is how I think. – Brevan Ellefsen Nov 16 '19 at 22:34
  • take a look here – Masacroso Nov 16 '19 at 23:09

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