residues

[Anonymous No.10086924]

greetings /sci/men, i was just thinking about the method of residues. you know even though i've seen the proofs and derivations and everything, this shit is still crazy.

like you get some integral [math]\int_{-\infty}^\infty f
(x)dx[/math] and then you say

>okay so f(x) goes to 0 at +- real infinity
>it also goes to zero at some infinite radius curve around the top half of the complex plane, but not the bottom
>so i can loop back from +infinity to -infinity around the top half of the plane
>now i know my integral is the sum of the simple residues at some poles in the top half of the complex plane
>bam done

shit is fucking fantastic. i know it works and i've seen the proofs, but the fact that it works is still pretty redic

[Anonymous No.10087643]

>>10086924
You get really similar situations arising all the time in elementary electromagnetism. Very cool stuff.

[Anonymous No.10087773]

>>10087643
You talking about Kramers-Kronig relations?

[Anonymous No.10088540]

>>10087773
naw "elementary" electromagnetism.
Applications of maxwell's equations involve relating integrals of fields over certain highly symmetric subspaces to integrals of related fields over lower-dimensional subspaces. In some cases this can look like relating a line integral to a value at a single point (in the simplest case, line integral of magnetic field around a wire = current at a point in a wire). The higher-dimensional cases are also analogous

[Anonymous No.10088552]

Why aren't mathematical synonyms tested for rigor? If they are strong enough we could just develop new languages without anyone noticing the math involved.

e.g. pi is when a counter wants to announce a circle, not break/close a circle.

[Anonymous No.10088953]

>>10088540
Where can I learn the math

[Anonymous No.10088958]

>>10086924
>i was just thinking about the method of residues
Don't do that

[Anonymous No.10089070]

>>10086924
It has to do with complex differentiation being a strong condition that greatly simplifies the properties of functions that are complex differentiable. Cauchy's integral theorem is a concise statement of this consequence. Cauchy's theorem applied to closed loops implies that the integral around any closed loop is zero unless the loop encloses a pole of the function, and the poles yield nonzero contributions to the integrals, which are the residues.

[Anonymous No.10089163]

>>10086924
Reminder that
[eqn]\sum_{n\in\mathbb{Z}}f(n) = \int_\mathbb{C}f(z)\csc(\pi z)[/eqn]

[Anonymous No.10089330]

>>10089163
Is this true for all [math]f[/math]?

[Anonymous No.10090151]

>>10089330
Yes even for [math]f = \operatorname{id}[/math].

[Anonymous No.10091596]

bumx

[Anonymous No.10092471]

>>10088953
get an understanding of what the basic forms of the maxwell equations mean and then attempt to apply them to highly symmetric cases. Most of the time there really is no math. For example, first try to derive expressions for the electric field around a point charge using Gauss's Law, then around a wire of constant charge, then from an infinite sheet of constant charge density. You can then try some slightly more complex cases, just think them up and try them. You can also try magnetism after getting those down.

[Anonymous No.10094075]

blork