Pfthb

I have created the following macros in LaTeX:

deftwomate(#1,#2,#3,#4)beginpmatrix#1&#2\#3&#4endpmatrix
defpwr(#1)^#1
deftwoclmn(#1,#2)beginpmatrix#1\#2endpmatrix

Where twomate is a 2x2 matrix, twoclmn creates a 2x1 column vector, and pwr is a command to raise the argument to the power.
However, if I try to do the following:

$$twomate(3epwr(-3t),epwr(2t),-epwr(-3t),-2epwr(2t) )$$

I get an error. However, if I replace the last entry with something that is not a def command:

$$twomate(3epwr(-3t),epwr(2t),-epwr(-3t),5 )$$

The document complies successfully. In addition, the following does not compile successfully:

$$twoclmn(frac625epwr(2t)-frac15tepwr(2t),-frac225epwr(2t) + frac25 t epwr(2t))$$

But the following does:

$$twoclmn(frac625e^2t-frac15te^2t,-frac225e^2t + frac25 t e^2t)$$

So the problem seems to be the nested def commands. How may I nest def commands, without the compiler complaining?

You can use xparse for this job, because the r argument type takes care of nesting.

documentclassarticle
usepackageamsmath
usepackagexparse

NewDocumentCommandtwomate>SplitArgument3,r()%
 maketwomate#1%

NewDocumentCommandmaketwomatemmmm%
 beginpmatrix#1&#2\#3&#4endpmatrix%

NewDocumentCommandpwrr()^#1

begindocument

[
twomate(3epwr(-3t),epwr(2t),-epwr(-3t),-2epwr(2t) )
]

enddocument

However, using () as delimiters doesn't seem a good idea. The following is as clear and behaves better with syntax coloring of front ends.

documentclassarticle
usepackageamsmath
usepackagexparse

NewDocumentCommandtwomate>SplitArgument3,m%
 maketwomate#1%

NewDocumentCommandmaketwomatemmmm%
 beginpmatrix#1&#2\#3&#4endpmatrix%

NewDocumentCommandpwrm^#1

begindocument

[
twomate3epwr-3t,epwr2t,-epwr-3t,-2epwr2t
]

enddocument

Your def contains a very specific sequence defined as the parameter text:

% 1 2 3 4 5
deftwomate(<1>,<2>,<3>,<4>) ... 
% ^ ^ ^ ^ ^
% │ │ │ │ │
% │ └ comma ┘ │
% └─── bracket ───┘

This parameter text is matched exactly in order (almost like a first-come-first-served style) to extract the four arguments <1>, <2>, <3> and <4>. Here's how the elements are grabbed for twomate with the above notation:

% 1 2 3 4 5
twomate(3epwr(-3t),epwr(2t),-epwr(-3t),-2epwr(2t))
% ^ ^ ^ ^ ^
% │ │ │ │ │
% │ └────── comma ────────┘ │
% └───────────────── bracket ──────────────────┘

It should be clear that the last bracket ) isn't properly captured for pwr. The way around it is to hide pwr(.) from twomate:

documentclassarticle

usepackageamsmath

deftwomate(#1,#2,#3,#4)beginpmatrix#1&#2\#3&#4endpmatrix
defpwr(#1)^#1

begindocument

[
 twomate(3epwr(-3t),epwr(2t),-epwr(-3t),-2epwr(2t))
]

enddocument

This solves the problem only temporarily. If you nest elements, you'll run into similar problems because of the parameter text pattern matching. In general, it is safer to group arguments using ....

TeX's delimited parameters parsing doesn't take nesting into account properly. So if you call twomate(3epwr(-3t), ...), the final ) for twomate isn't found at the end of that line but at the end of pwr(-3t). The improperly formed call of pwr then causes trouble.

To hide nested calls of your commands, put them into ... groups (though this probably defeats the purpose):

documentclassarticle
usepackageamsmath

deftwomate(#1,#2,#3,#4)beginpmatrix#1&#2\#3&#4endpmatrix
defpwr(#1)^#1
deftwoclmn(#1,#2)beginpmatrix#1\#2endpmatrix

begindocument
[ twomate(3epwr(-3t),epwr(2t),-epwr(-3t),-2epwr(2t)) ]
enddocument

By the way, don't use $$ ... $$ for display math environments but LaTeX's or amsmath's variants like [ ... ].

Here is how to define macros expecting comma separated arguments enclosed in parentheses. The code converts any standard LaTeX command (macro with non-delimited parameters) into one expecting (..., ..., ...) input syntax. When used, the number of arguments must be the one expected by the macro, there is no error-checking.

Nesting works, as we will test with this kind of input:

foo(Bar(3,5,7),Bar(9,foo(a,Bar(W,X,Y),c,d),13),Bar(15,17,19),Bar(21,23,25))

Attention that there is no special handling of spaces in the input, and also some brace-stripping may occur in certain circumstances.

As this all works purely by expansion, expandable commands remain expandable commands.

_{Important: for a fuller implementation of the analogy with "functional notation", the macro foo(a, b, c) should first expand completely the a, b, c before doing whatever it wants to do (if only for reasons of efficiency). But this would limit from the start such constructs to the realm of expandable macros. And furthermore, only the new expanded primitive allows (easy) such complete expansion. Else, one can make do restricting to only "first-fully" expandable macros using some romannumeral-`Q governed expansion (with a problem if macro expands to something starting with a space token.) Anyway, here I did not add the automatic expansion of a, b, c, before applying the foo macro, I stayed more on the side of usual macro expansion in TeX for things such as fooabc.}

documentclassarticle
usepackageamsmath

% copy over some utility code from package xintexpr
catcode`_ 11

makeatletter
letxint_c_monem@ne
letxint_c_z@
letxint_c_i@ne
longdefxint_bye #1xint_bye %
makeatother
% THE XINT_isbalanced_... MACROS COPIED OVER FROM XINTEXPR CODE MADE
% HERE ALL long
% % endmacrocode
% subsubsectioncshXINT_isbalanced_a for cshnolabelXINT_expr_onliteral_seq_a
% lverb|Expands to xint_c_mone in case a closing ) had no opening ( matching
% it, to @ne if opening ( had no closing ) matching it, to z@ if expression
% was balanced.|
% beginmacrocode
% use as XINT_isbalanced_a relax #1(xint_bye)xint_bye
longdefXINT_isbalanced_a #1(XINT_isbalanced_b #1)xint_bye %
longdefXINT_isbalanced_b #1)#2%
 xint_bye #2XINT_isbalanced_cxint_byeXINT_isbalanced_error %
% endmacrocode
% lverb|if #2 is not xint_bye, a ) was found, but there was no (. Hence error -> -1|
% beginmacrocode
longdefXINT_isbalanced_error #1)xint_bye xint_c_mone%
% endmacrocode
% lverb|#2 was xint_bye, was there a ) in original #1?|
% beginmacrocode
longdefXINT_isbalanced_cxint_byeXINT_isbalanced_error #1%
 xint_bye #1XINT_isbalanced_yesxint_byeXINT_isbalanced_d #1%
% endmacrocode
% lverb|#1 is xint_bye, there was never ( nor ) in original #1, hence OK.|
% beginmacrocode
longdefXINT_isbalanced_yesxint_byeXINT_isbalanced_dxint_bye )xint_bye xint_c_ %
% endmacrocode
% lverb|#1 is not xint_bye, there was indeed a ( in original #1. We check if
% we see a ). If we do, we then loop until no ( nor ) is to be found.|
% beginmacrocode
longdefXINT_isbalanced_d #1)#2%
 xint_bye #2XINT_isbalanced_noxint_byeXINT_isbalanced_a #1#2%
% endmacrocode
% lverb|#2 was xint_bye, we did not find a closing ) in original #1. Error.|
% beginmacrocode
longdefXINT_isbalanced_noxint_bye #1xint_byexint_bye xint_c_i %

% NEW EXPANDABLE UTILITY

longdefapplyfunction #1#2)%
%
 ifcaseXINT_isbalanced_a relax #1#2)(xint_bye)xint_bye
 expandafterapplyfunction_b
 orexpandafterapplyfunction_again
 elseexpandafterwe_are_doomed
 fi #1#2%
%
longdefapplyfunction_again #1applyfunction #1)%
longdefmy_bbye #1my_bbye %
longdefapplyfunction_b #1applyfunction_c #1,my_bbye,%
longdefapplyfunction_c #1(applyfunction_d #1%
% we will take care of brace removal another day
longdefapplyfunction_d #1#2#3#4,%
%
 ifcaseXINT_isbalanced_a relax #3#4(xint_bye)xint_bye
 expandafterapplyfunction_e
 orexpandafterapplyfunction_d_again
 elseexpandafterwe_are_doomed
 fi #1#2#3#4%
%
longdefapplyfunction_d_again #1#2#3#4%
%
 applyfunction_d #1#2#3#4,%
%
longdefapplyfunction_e #1#2#3#4%
%
 my_bbye#4applyfunction_finishmy_bbye
 applyfunction_g #1#2#3#4%
%
longdefapplyfunction_g #1#2#3applyfunction_d #1#2#3%
longdefapplyfunction_finishmy_bbyeapplyfunction_g #1#2#3#1#2%

catcode`_ 8

%% USAGE
% to define a "function" to be used as foo(a,b,c,...),
% we need a foomacro which uses standard non-delimited parameters,
% thus e.g. defined via newcommand, and then one only needs
% newcommandfooapplyfunctionfoomacro
% to get a foo to be used with parentheses.

newcommandtwomateapplyfunctiontwomatemacro
newcommandpwrapplyfunctionpwrmacro
newcommandtwoclmnapplyfunctiontwoclmnmacro

% define here the **non-delimited** auxiliary macros
newcommandtwomatemacro[4]beginpmatrix#1&#2\#3&#4endpmatrix
newcommandpwrmacro[1]^#1
newcommandtwoclmnmacro[2]beginpmatrix#1\#2endpmatrix

%% TESTING NESTING
newcommandfooapplyfunctionfoomacro
newcommandfoomacro[4]left[#1+#2+#3+#4right]
letBarrelax
newcommandBarapplyfunctionBarmacro
newcommandBarmacro[3]left(#1*#2*#3right)

delimiterfactor1001

begindocument

[
 twomate(3epwr(-3t),epwr(2t),-epwr(-3t),-2epwr(2t))
]
[
 twoclmn(frac625epwr(2t)-frac15tepwr(2t),-frac225epwr(2t)
 + frac25 t epwr(2t))
]
[foo(Bar(3,5,7),Bar(9,foo(a,Bar(W,X,Y),c,d),13),Bar(15,17,19),Bar(21,23,25))]
enddocument