Complex Roots

One question which has fascinated me in recent years, was the question of what exactly happens, if we start with a complex number as a base, and if we then raise that base to either a rational exponent, or an irrational exponent.

In the following worksheet, I explored that subject for all to see:

http://dirkmittler.homeip.net/Plot_Complex_Points.pdf

(Note : )

The worksheet above was made using a graphical front-end to the open-source Computer Algebra System named ‘Maxima’. That graphical front-end is named ‘wxMaxima’, and adds as features, to be able to convert textual output from Maxima, into typeset equations, as well as to make certain presentations and plots nicer.

Enjoy,

Dirk

 

The failings of low-end consumer software, to typeset Math as (HTML) MathML.

One of the features which HTML5 has, and which many Web-browsers support, is the ability to typeset Mathematical formulae, which is known as ‘MathML’. Actually, MathML is an extension of XML, which also happens to be supported when inserted into HTML.

The “WiKiPedia” uses some such solution, partially because they need their formulae to look as sharp as possible at any resolution, but also, because they’d only have so much capacity, to store many, many image-files. In fact, the WiKiPedia uses a number of lossless techniques, to store sharp images as well as formulae. ( :1 )

But from a personal perspective, I’d appreciate a GUI, which allows me to export MathML. It’s fine to learn the syntax and code the HTML by hand, but in my life, the number of syntax-variations I’d need to invest to learn, would be almost as great, as the total number of software-packages I have installed, since each software-package, potentially uses yet-another syntax.

What I find however, is that if our software is open-source, very little of it will actually export to MathML. It would be very nice if we could get our Linux-based LaTeX engines, to export to this format, in a way that specifically preserves Math well. But what I find is, even though I posses a powerful GUI to help me manage various LaTeX renderings, that GUI being named “Kile”, that GUI relies back on a simple command-line tool named ‘latex2html’. Whatever that command-line outputs, that’s what all of Kile will output, if we tell it to render LaTeX specifically to HTML. ‘latex2html’ in turn, depends on ‘netpbm’, which counts as very old, legacy software.

One reason ‘latex2html’ will fail us, is the fact that in general, its intent is to render LaTeX, but not Math in any specific way. And so, just to posses the .TEX Files, will not guarantee a Linux user, that his resulting HTML will be stellar. ‘latex2html’ will generally output PNG Images, and will embed those images in the HTML File, on the premise that aside from the rasterization, PNG Format is lossless. Further, if the LaTeX code was generated by “wxMaxima”, using its ‘pdfLaTeX’ export format, we end up with incorrectly-aligned syntax, just because that dialect of LaTeX has been optimized by wxMaxima, for use in generating .PDF Files next.

(Updated 05/27/2018 : )

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