This posting collects together some issues with the equations editor. All of these issues have been raised on the forum in earlier conversations going back to Oct/Nov 2010. (The earlier equations editor was removed with the revamp of Apr/May 2010 and some functionality was restored in Oct/Nov 2010, hence that cutoff.) I posted a bibliography of relevant forum postings yesterday  and won't provide precise citations here; just a pointer to two earlier broad criticisms of the functionality of the equations editor [2-3]. The present posting concerns only editor functionality; there are additional issues with export to print or to pdf and conversion to and from Word or ODT and these issues are left for a separate posting.
There needs to be a way to define new mathematical elements, especially functions and operators that are to be represented in the roman font. Examples of such elements that already exist are the functions sin and cos represented in LaTeX style as \sin and \cos and some other familiar functions. I don’t believe that the GDocs documentation provides a list and in any case it is not reasonable to provide a fixed list suitable for all applications; the system can provide a set of core functions and leave it to the user to define additional ones as needed. Some user may want to refer to the Bessel functions ber and bei. Sure, they are much less common than sin and cos and we don’t expect the GDocs editor to have them predefined, but it should be possible to define these function names and to have them typeset properly as indivisible symbols in the roman font with spacing rules the same as for sin and cos.
In some cases the user may need to override an existing definition and that should be possible too. For example, one may want to write max(x,y) in an equation with “max” in the roman font, but the \max control sequence has a different special treatment. One may try to get (x div y, x mod y) with “div” and “mod” as three-character symbols in roman font, but \div gives the dot-bar-dot symbol for division and \mod is not defined. One may try to get (Re z, Im z) with “Re” and “Im” in roman font, but \Re and \Im give the Fraktur symbols.
There needs to be way to switch to text mode within an equation. This is needed all the time in applied science. For example, in applied science may one have variables T<sub>in</sub> and T<sub>out</sub> or N<sub>girls</sub> and N<sub>boys</sub> in which T and N should appear in the (italic) font used for mathematical variables and the subscripts in, out, boys, girls are ordinary words that should appear in the roman font. Another case where text mode is needed, or the roman font anyway, is for physical units in formulas. It is not an issue for pure mathematics, but is needed all the time in applied science. There are many other applications for switching to text mode, but these examples will suffice.
There need to be ways to toggle the font also for mathematical variables. For example, it is quite common in physical science literature to use bold for vector quantities. Some people may want to follow conventions that use upright bold for vectors and italic sans-serif bold for tensors or other operator symbols. An ability to switch font and to use roman, italic, underline or bold also within equations will cater to many such needs.
There needs to be a way to introduce TeX/LaTeX abbreviations for many more symbols than are presently offered directly within the equations menu. In fact, I don’t know why the GDocs editor shouldn’t just adopt some standard list of TeX/LaTeX symbol codes and include them all rather than the small subset that is offered now. When these are introduced into the standard symbol set for the equation editor then it should be done with sensible rules for spacing left and right, similar to the spacing rules for symbols already in the equations editor. At present, if a symbol such as sqsubseteq is introduced into an equation via the special symbol insert mechanism the spacing is messed up; the user needs to introduce the appropriate space left and right explicitly.
Examples of specific fairly common mathematical symbols that are not available through TeX/LaTeX mode in the equations editor: \cong (we do have \simeq), \oiint (we do have \oint), \hbar; the square order and lattice symbols \sqsubset, \sqsubseteq, \sqsuperset, \sqsuperseteq, \sqcup, \sqcap.
It has been discovered on the forum that LaTeX \dot and \ddot are available (dot and double dot above symbols; fluxion notation for derivatives), although they are not mentioned in the documentation and not available through the point-and-click menu. The bracket notation for binomials is in the Equations menu, but the documentation omits to mention that also \binom is available on the keyboard. Also \vec is available for vector notation, \underline behaves similar to \overline, and \bar gives a fixed-width overline; this is all undocumented.
Several very common TeX/LaTeX mathematical bracket pairs are missing in the GDocs equations editor. Probably the most important missing brackets are the norm symbol (pair of double vertical bars; TeX \norm) and the angular bracket pair. The norm symbol and the angular bracket pair should behave like the round brackets, square brackets, braces and the absolute value symbol that are already in the equations menu: they should adapt in size to whatever is inside. In the class of brackets, \floor and \ceil would be next in line for inclusion.
I emphasize the issue of correct spacing, for which there is a difference between the special characters in the equation bar and the same characters in the math section of the special characters table. Take for example the set inclusion symbol \subseteq. The two instances of the bare symbol differ in an insignificant way on my screen, but the symbol from the equation menu has more whitespace around it and that gives it the correct visual appearance in an equation. Authors of mathematical text should really wish to find their special symbols in the equations menu and not only in the special characters menu, or they should be able to insert their special symbols with correct mathematical layout by using the TeX/LaTeX codes.
Further on the matter of correct spacing, the correct spacing for mathematical symbols gets messed up when one attaches a subscript or superscript to the symbol. For example, compare the equivalents of $a\cosh x$ and $a\cosh^2x$ (cosh vs cosh-squared), or try to attach a subscript “w” to an order symbol, TeX $\leq_w$. The equation $x[leq]y$ looks nice on the screen, with spacing both sides of \leq, but if the subscript is attached to \leq then the spacing disappears. I believe that the simplest reasonably effective general rule is the following: each mathematical symbol (including function names such as \sin and \cos) gets small whitespace left and right, but if a subscript or superscript is attached to the symbol then the whitespace moves outside the subscript or superscript.
There needs to be a convenient way to change the text size within equations in connection with built-up fractions. In the TeX/LaTeX system it is recognized that an in-line equation needs to be treated differently than a displayed equation in this regard; if a built-up fraction must appear in-line then one must use a rather small font, whereas in a displayed equation one can use the standard font also in numerators and denominators. The GDocs editor does not distinguish the two kinds of position (in-line or displayed) and the font for numerator and denominator is unacceptably small in a GDocs displayed equation.
There needs to be a way to produce multiline equations with specified alignment; something such as x=a&+b\\&+c where the \\ denotes a line break and the & is not printed but denotes the position of alignment. It is entirely standard in TeX/LaTeX, but cannot be done using the GDocs equations editor.
There needs to be a way to produce multiline equations with the lines spanned by some left or right brace. Example: the Heaviside function H is defined by (for real variable x) H(x) = 0 (if x.lt.0); 1 (if x.ge.0). One may want to typeset this using two lines spanned by a big left brace after the equals sign. It cannot be done using the GDocs equations editor.
There needs to be a way to typeset matrices or arrays in the GDocs equations editor. This facility is closely linked to the facility to create multiline aligned equations and to create these multiline structures with a spanning brace; probably the underlying code is the same, but the application field is a bit different.