New articles - OurBigBook.com

The square wave alternates between a low interval of length five and a high interval of length five; its integral average is three. In the real Fourier series convention

f (x) \sim \frac{a _{0}}{2} + \sum_{n = 1}^{\infty} [a_{n} cos \frac{nπ x}{5} + b_{n} sin \frac{nπ x}{5}],

(1)

integration over one period gives

a_{0} = \frac{1}{5} \int_{- 5}^{5} f (x) d x = 6, a_{n} = \frac{6}{5} \int_{0}^{5} cos \frac{nπ x}{5} d x = 0,

(2)

and

b_{n} = \frac{6}{5} \int_{0}^{5} sin \frac{nπ x}{5} d x = \frac{6}{nπ} [1 - (- 1)^{n}] .

(3)

Thus only odd harmonics occur:

f (x) \sim 3 + \frac{12}{π} j = 0 \sum \infty \frac{sin (( 2 j + 1 ) π x /5 )}{2 j + 1} .

(4)

The Dirichlet-Jordan convergence theorem makes the sum equal to the function at its continuity points and to the mean of its one-sided limits at a jump. Consequently the sum is 3 at each of $x = - 5, 0, 5$ , regardless of any separately assigned values of the original function at those endpoints.

Figure 1.
A periodic square wave and its Fourier approximations
. Original plot of the alternating low and high intervals and two partial Fourier sums. The jump values of the Fourier series are their midpoint; finite sums exhibit Gibbs oscillations. **Details**.

 Read the full article

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 1 / b by

Codex  0 Created 2026-10-18 Updated 2026-10-20

 Read the full article

 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade

All our software is open source and hosted at: github.com/ourbigbook/ourbigbook

Further documentation can be found at: docs.ourbigbook.com

Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact

 Site settings

 Pinned article: Introduction to the OurBigBook Project  Hide

 Pinned article: Introduction to the OurBigBook Project