DiGMi

[latexpage]

A Gaussian function (named after one of the greatest mathematician, Carl Friedrich Gauss) is a function of the form: \[ f\left(x\right)=ae^{-\frac{\left(x-b\right)^{2}}{2c^{2}}} \]

We would have like to examine the following integral: \[ I=\intop_{-\infty}^{\infty}e^{-x^{2}}\mathrm{d} x\]

But we are facing a problem, what is the antiderivative function of $e^{-x^{2}}$?

Well, You won’t be able to find one, at least not an elementary one. The integral of a Gaussian is the Gauss error function, but in this special case, there is a cool way to calculate this integral.

At first, Instead of looking at $I$, lets take a look at $I^{2}$. Meaning:
\[ I^{2}=\intop_{-\infty}^{\infty}e^{-x^{2}}\mathrm{d} x\intop_{-\infty}^{\infty}e^{-y^{2}}\mathrm{d} y \]

Note: In the second integral, I’ve changed the variable of integration to y. I can do that, because it’s a completely separated integral.

Now, notice that the following integral is equivalent to the following:\[
\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}e^{-x^{2}-y^{2}}\mathrm{d} x\mathrm{d} y
\]
Well, That’s a bit weird. Now we have a double integral! We’ve made the problem much more complex! Or did we?

Well, yes, now we have a double integral, but there is a good reason for that. Now we can change our coordinate system to the polar coordinate system (Meaning: $x=r\cos\theta, y=r\sin\theta$) and remember to multiply by the Jacobian determinant! In our case, the Jacobian determinant is $r$. So, we got:
\[ \intop_{0}^{2\pi}\intop_{0}^{\infty}e^{-r^{2}}r\mathrm{d}r\mathrm{d}\theta \]
But wait a minute, this looks familiar! Notice that:
\[ =-\frac{1}{2}\intop_{0}^{2\pi}\intop_{0}^{\infty}\frac{\mathrm{d}}{\mathrm{d}r}\left(e^{-r^{2}}\right)\mathrm{d}r\mathrm{d}\theta=-\frac{1}{2}2\pi\intop_{0}^{\infty}\frac{\mathrm{d}}{\mathrm{d}r}\left(e^{-r^{2}}\right)\mathrm{d}r=\] \[ =- \pi \left[e^{-r^{2}}\right]_{0}^{\infty}= -\pi\left[0-1\right]=\pi\]
Therefore:
\[ I^{2}=\pi\Rightarrow I=\intop_{-\infty}^{\infty}e^{-x^{2}}\mathrm{d} x = \sqrt{\pi} \]

If you are using LyX on mac you probably noticed that some symbols are missing while editing an equation.

One of the symbols missing is $latex \Omega$ which is really annoying when you learn probability.
When you try and type \Omega, instead of seeing the symbol you will just see a blank space.

I’ve dived into the code, and found out how the all designer works. For most of the symbols, the translation is quite simple. There are several font files, and a file named symbols (located in /Application/LyX.app/Contents/Resources) which maps every latex code to the relevant font and unicode value.

The problem with $latex \Omega$ and $latex \otimes$ is that their unicode value is 0x00ad. And as it seems, QT4 has bug with characters with that code. LyX source code mentioned this bug in GuiPainter.cpp and it have some code to try and fix it, but for some reason it doesn’t work.

A work around for the $latex \Omega$ is to make a small change in the symbols file, locate the Omega line (with capital O) and change the code from 173 to 937 (a code for different Omega character that is available in the font).

I didn’t find (didn’t really looked for) a way to fix the \otimes, because i don’t really use it. A way to fix it will be using a font editor and changing the font and map the wanted symbols from other unused code. Again, didn’t care enough to do it, If someone want me to just post here. But I believe that LyX developers will soon enough fix it.

When you learn mathematics at the university, and you are a bit of computer’s freak, you will probably want to take lecture notes on the computer.

You are all probably familiar with $latex \LaTeX$ (if not, well… It’s the standard way to write type equations). The best editor for LaTeX is LyX (Well, maybe best is a strong word, The less of all evil is much better).

Recently I’ve bought a new MacBook Air (Mac OS X Lion version 10.7.2. i7 processor), And I’m using it in the university to take some lecture notes. Unfortunately, the LyX release version for mac is a little bit buggy… (for example, when you try to open a file when another file is open it fails most of the time, and if you insist LyX will probably crash. Some characters, such as $latex \Omega$, are not being displayed on editor mode). So I’ve decided to download the source code from their SVN and compile the latest development version.

To make a long story short, I’ve encountered a lot of errors, And I’ve decided to list them here so if you will get them you will be able to solve them much quicker than me.

First error – ICONV_CONST not defined

The first error I’ve encountered was the following:

docstream.cpp: In member function ‘std::codecvt_base::result<unnamed>::iconv_codecvt_facet::do_iconv(void*, const char**, size_t*, char**, size_t*) const’:
docstream.cpp:264: error: expected `>' before ‘ICONV_CONST’
docstream.cpp:264: error: expected `(' before ‘ICONV_CONST’
docstream.cpp:264: error: ‘ICONV_CONST’ was not declared in this scope
docstream.cpp:264: error: expected primary-expression before ‘>’ token

Here the solution is simple. The problem is that the configure doesn’t config the ICONV_CONST properly. The solution to edit the config.h file located in the lyx-devel directory, and add the following line:

#define ICONV_CONST

And now it will work wonderfully.

Second error – Undefined symbols for arh

The second error was much more annoying, and I’ve even destroyed my Mac by mistake while trying to solve it (Had to reinstall it). The solution is a bit more difficult, but no worries.

The error I got, was the following linking error:

Undefined symbols for architecture x86_64:
  "_libiconv_open", referenced from:
      lyx::IconvProcessor::init()     in liblyxsupport.a(unicode.o)
      (anonymous namespace)::iconv_codecvt_facet::iconv_codecvt_facet(std::basic_string<char, std::char_traits<char>, std::allocator<char> > const&, std::_Ios_Openmode, unsigned long)in liblyxsupport.a(docstream.o)
  "_libiconv", referenced from:
      lyx::IconvProcessor::convert(char const*, unsigned long, char*, unsigned long)in liblyxsupport.a(unicode.o)
      (anonymous namespace)::iconv_codecvt_facet::do_out(__mbstate_t&, wchar_t const*, wchar_t const*, wchar_t const*&, char*, char*, char*&) constin liblyxsupport.a(docstream.o)
      (anonymous namespace)::iconv_codecvt_facet::do_in(__mbstate_t&, char const*, char const*, char const*&, wchar_t*, wchar_t*, wchar_t*&) constin liblyxsupport.a(docstream.o)
  "_libiconv_close", referenced from:
      lyx::IconvProcessor::convert(char const*, unsigned long, char*, unsigned long)in liblyxsupport.a(unicode.o)
      lyx::IconvProcessor::Impl::~Impl()in liblyxsupport.a(unicode.o)
      (anonymous namespace)::iconv_codecvt_facet::~iconv_codecvt_facet()in liblyxsupport.a(docstream.o)
      (anonymous namespace)::iconv_codecvt_facet::~iconv_codecvt_facet()in liblyxsupport.a(docstream.o)
ld: symbol(s) not found for architecture x86_64

There seems to be a problem with the built-in libiconv library that comes with the OS. The solution was to get the latest source code from GNU, and installing it.

The way to install it is as follows:

Download the archive file, open a terminal in the folder with the archive and type the following commands:

tar -xzvf libiconv-1.14.tar.gz
cd libiconv-1.14

./configure '--prefix=/usr/local' 'CC=gcc -arch i386 -arch x86_64' 'CXX=g++ -arch i386 -arch x86_64' 'CPP=gcc -E' 'CXXCPP=g++ -E'

make
sudo make install

Note: It is very important NOT to install the lib to “/usr/lib“. but to “/usr/local/lib” (That’s they way I’ve messed up my all OS, and it wouldn’t boot afterwards).

After installing the libiconv, you will need to configure and install LyX  using the following commands (assuming you are in the lyx-devel directory):

LDFLAGS="-L/usr/local/lib" ./configure --with-version-suffix=-2.0 --with-libiconv-prefix=/usr/local --with-x=no --disable-stdlib-debug

make
make install

Note: You will probably encounter again the first error I had, because running the configure recreate the config.h file. Just reedit the config.h file as mentioned in the last section.

And that’s it!

It seems like in the current development version the opening bug is fixed, but the missing symbols are still missing. Hopefully it will be fixed soon (I’m looking into it).