The complex modulus is a nonnegative real number. Equality therefore requires itself to be real and nonnegative. Its imaginary part gives , and its real part then gives . Conversely every such point satisfies the equality. Hence the complete solution set is
Geometrically this is the closed ray on the real axis starting at .
Rewrite the rational function as . It is undefined at , with the vertical asymptote , and tends to the horizontal asymptote as . At the pole, from the left and from the right.
Its derivative is
vanishing only at . Since and near zero, this is a stationary inflection point, not an extremum. Thus
The graph increases on each side of the vertical asymptote. The left branch lies above , while the right branch lies below it and crosses the origin horizontally.
Figure 1.
The rational-function graph, its two asymptotes, and its stationary inflection at the origin
. Details.
Separate variables near the nonzero initial condition: . Integration gives , and fixes . Therefore
This is the maximal connected interval containing the initial point. Direct differentiation gives ; the pole at prevents continuation as a finite classical solution of a differential equation through that point.
In one period, sine equals at and , and lies below that value between them. Since sine is a periodic function with period , all solutions are:
The endpoints are included because the inequality is non-strict.

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!
We have two killer features:
  1. 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
  2. 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:
    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 5. . 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.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    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