• What am I looking at?

  All subseries of $f$, truncated at the $k^{th}$ degree term, evaluated at some $x \in \mathbb{C}$. Each point on the screen is a different subseries of $f$ evaluated at the chosen point $x$. The image of the evaluations of all subseries is normalized so at to keep it centered on the screen. Taylor expansion of $f$ is about the origin. See below for colors.

• What are subseries?

  Let $A=\sum \alpha_i$, and $B=\sum \beta_j$ be series. Then $B$ is said to be a subseries of $A$ if and only if $\beta_i = \alpha_i$ or $\beta_i = 0$.
  
  This definition is extended to include subseries of analytic functions by considering the subseries of the Taylor series of the function about some point.

• What do the colors represent?

  Subseries which had more points dropped will be bluer, and those with less points dropped will be redder.

• How do I use this?

  Enter your favorite analytic function, (use exp(x) for the exponential function $e^x$. Exponents are entered using **: e.g. x**2 for $x^2$. See the Sympy documentation on parsing for more info on this.) Afterwards, type in a complex number in either polar or cartesian form. If you are on a desktop browser, use the complex number picker on the bottom left to choose a value for $x$. Turn on the "rotation" option to animate $x$ orbiting the origin at its current radius.

• What's this written in?

  The actual construction and evaluation of the subseries occurs in parallel on the GPU and is written in Javascript using WebGL. Symbolic math, in particular computing the Taylor series, is handled by a backend server using Sympy. Math typesetting is done using MathJax.

• Where can I look at the source code?

  On Github.

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