Thursday, October 29, 2015
[U_20151029ITCX] Roger Cotes’ Integral of the Reciprocal of x^n – 1
The book VisualComplex Analysis by Tristan Needham recounts the story of RogerCotes who considered the above problem. Without ostensibly using complex numbers, Cotes discovered a geometrical principle that helped to factorise the denominator xn – 1, and hence decompose the above integral.
In this article, I am going to “cheat” by using complex numbers to split up the denominator. The fact that the denominator splits completely into a product of simple linear factors makes it easy to decompose the integrand into partial fractions. Once this is done, I can single out the one or two fractions with purely real linear denominators. Then I can pair up the conjugate fractions to get fractions with real quadratic denominators. In other words, I apply a divide-and-conquer strategy, splitting up a big problem into smaller problems (Heuristics!). Then I collect all the partial answers together to form my final answer.
Note that I have only used real integration, not complex integration. Complex numbers are used only to derive the various algebraic fractions.
H02. Use a diagram / model (mentally: imagine roots of unity in a circle)
H04. Look for pattern(s)
H10. Simplify the problem
H11. Solve part of the problemH13* Use Equation / write a Mathematical Sentence
* University / college level calculus
* high school students very strong and interested in calculus and complex numbers* anybody who loves a challenging calculus problem and complex numbers