Contour Integration #12 - Sum of 1/n^2 (The Basel Problem, Part 1/2)
5.5 هزار بار بازدید -
6 سال پیش
-
PART 2:
PART 2: Contour Integration #13: Sum of 1/n^2...
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(Part 2 coming up the day after this video is published!)
In this video, we'll use complex analysis (in particular, the residue theorem) to evaluate the sum of 1/n^2 from n = 1 to infinity. This is known as the Basel problem, and it has numerous solutions, e.g. from Fourier series or other integration tricks. We'll solve the sum (which pops out nicely as a result of the residue theorem) by integrating f(z) = pi*cot(pi*z)/z^2 over the boundary of a square of side length 2N + 1. As N goes to infinity, the poles inside the contour correspond precisely to the integers, which form the basis of our summation.
In this part, we'll calculate the residues of f(z) at each of its poles. In part 2, we'll show that the integral over the square S_N goes to 0 as N goes to infinity, using the estimation lemma.
This technique is central to analytic number theory, and can be applied to other series as well, e.g. the alternating reciprocal-sum of the squares, (-1)^n / n^2, as well as other series, by choosing appropriate functions.
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Video URL: Contour Integration #12 - Sum of 1/n^...
Channel URL: @learnmathsfree
---
SUPPORT ME ON PATREON! Click here: Patreon: crystalmath
---
(Part 2 coming up the day after this video is published!)
In this video, we'll use complex analysis (in particular, the residue theorem) to evaluate the sum of 1/n^2 from n = 1 to infinity. This is known as the Basel problem, and it has numerous solutions, e.g. from Fourier series or other integration tricks. We'll solve the sum (which pops out nicely as a result of the residue theorem) by integrating f(z) = pi*cot(pi*z)/z^2 over the boundary of a square of side length 2N + 1. As N goes to infinity, the poles inside the contour correspond precisely to the integers, which form the basis of our summation.
In this part, we'll calculate the residues of f(z) at each of its poles. In part 2, we'll show that the integral over the square S_N goes to 0 as N goes to infinity, using the estimation lemma.
This technique is central to analytic number theory, and can be applied to other series as well, e.g. the alternating reciprocal-sum of the squares, (-1)^n / n^2, as well as other series, by choosing appropriate functions.
Like us on Facebook: Facebook: learnmathsfree
Follow us on Twitter: Twitter: LearnMathsFree
Google+: https://plus.google.com/u/0/b/1170336...
Video URL: Contour Integration #12 - Sum of 1/n^...
Channel URL: @learnmathsfree
6 سال پیش
در تاریخ 1397/10/30 منتشر شده
است.
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