# LOLmathbb{C}AT

As a project for my complex analysis class, I had students keep a “complex function field guide,” which consisted of a journal describing important diagnostic features of important elementary complex functions. The complexified versions of standard functions like the exponential or the sine functions have many unexpected properties when the input variable is a complex number, and the purpose of the field guide is to explore those unexpected differences and put them into context when compared with the expected similarities to their real analogs.

One aspect of every field guide is to provide helpful illustrations showing the effect of the function as a mapping on the plane. Most folks opted for the usual demonstrations of “before and after” views of standard shapes like squares or sectors; the more daring opted for full color displays of the plane, such as the mapping by the squaring function w=z2:

One group took a much more interesting approach: they fixed an illustration — in their case, a picture of a grey cat — and then showed the pre-image of that illustration under the mapping, i.e. the set of points in the plane that would, under the given mapping, form the picture. For example, given the same squaring map, they produced the following:

The resulting images are unique and interesting, as is the code that produced them, but the thing that immediately grabbed  my attention was the default image they chose to describe their complex maps: it nothing other than the (in)famous I CAN HAS CHEEZBURGER? LOLcat:

Now, if you haven’t seen a LOLcat before — and if you haven’t, welcome to your first visit to the internet! Enjoy the porn! — it’s a internet meme that is best described by fusing the following xkcd comic below

with the syntax of an illiterate code monkey:

In any event, I felt compelled to take some of the converted pictures of the protoLOLcat and convert them into new   subLOLcats. A first-generation iterative subclass of LOLcat, if you will. They may or may not make you LOL, but they do have cats in them (or at least, objects locally biholomorphic to a cat in them), and if you’re geek enough to find the above comic funny, you’ll hopefully get a kick out of them too. (I’ll leave it as an exercise to the reader to determine what elementary mapping determines the corresponding LOLcat. Solutions due at the midterm.)

In the words of the Defective Yeti, if everyone on the Internet jumped off a bridge, I would probably do that too.

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