Introducing LaTeX Math Typesetting

[saltydog]

I wish to finally evaluate:

$$y(t)=\frac{e^{-\pi/t}}{2 i t^{3/2}}\mathop\lim\limits_{R\to\infty}\left\{\text{Erfi}\left(v-\sqrt{\pi/t}\right)_{\sqrt{Rt}e^{-\pi i/4}}^{\sqrt{Rt}e^{\pi i/4}}\right\}\tag{1}$$

First note:

$$\text{Erfi}(z)=\frac{\text{Erf(iz)}}{i}=\frac{1}{i}\frac{2}{\sqrt{\pi}}\int_0^{iz} e^{-w^2}dw$$

Using this relation, I can re-write the above limit as:

$$\frac{2}{i\sqrt{\pi}}\left\{\mathop\lim\limits_{R\to\infty}\int_0^{i(\sqrt{Rt}e^{\pi i/4})} e^{-w^2}dw-\mathop\lim\limits_{R\to\infty}\int_0^{i(\sqrt{Rt}e^{-\pi i/4})} e^{-w^2}dw\right\}$$

Using Euler's relation, I convert the upper limits to real and imaginary parts and multiply by i:

$$\begin{align*}<br /> i\sqrt{Rt}e^{\pi i/4}&=i\left[\left(\sqrt{Rt/2}-\sqrt{\pi/2}\right)+i\sqrt{Rt/2}\right] \\<br /> &=-\sqrt{Rt/2}+i\left(\sqrt{Rt/2}-\sqrt{\pi/t}\right)<br /> \end{align}$$

Now, as R goes to infinity, this quantity goes to $-\infty+i\infty$.

And the lower limit:

$$\begin{align*}<br /> i\sqrt{Rt}e^{-\pi i/4}&=i\left[\sqrt{Rt/2}+\left(\sqrt{Rt/2}-\sqrt{\pi/2}\right)\right] \\<br /> &=\sqrt{Rt/2}+i\left(\sqrt{Rt/2}-\sqrt{\pi/2}\right)<br /> \end{align}$$

as R goes to infinity, this limit goes to $\infty+i\infty$.

Thus the integrals can be rewritten as:

$$\frac{1}{i}\frac{2}{\sqrt{\pi}}\int_0^{-\infty+i\infty} e^{-w^2}dw-<br /> \frac{1}{i}\frac{2}{\sqrt{\pi}}\int_0^{\infty+i\infty} e^{-w^2}dw$$

I choose to evaluate these line integrals along a parametric path using the following relation from Complex Analysis:

$$\int_C f(z)dz=\int_c udx-vdy+i\int_c vdx+udy$$

Thus the first step is to express the integrand in terms of a real and imaginary part. Relying on Euler's formula:

$$\begin{align*}<br /> e^{-w^2}&=e^{-(x+yi)^2} \\<br /> &=e^{-(x^2-y^2+2xyi)} \\<br /> &=e^{-(x^2-y^2)}Cos(2xy)+ie^{-(x^2-y^2)}(-Sin(2xy))\\<br /> &=u+iv<br /> \end{align}$$

For the first integral, the path goes from the origin to the point $-\infty+i\infty$

Thus if I let:

$$x=-t,\quad dx=-dt$$

$$y=t,\quad dy=dt$$

Making this substitution, note that the exponent term is 1 and we have for the first integral (with t going from 0 to infinity):

$$\begin{align*}<br /> \int_0^{-\infty+i\infty} e^{-w^2}dw&=\left(-\int_0^\infty Cos(-2t^2)dt+\int_0^\infty Sin(-2t^2)dt\right) \\<br /> &+i\left(\int_0^\infty Cos(-2t^2)dt+\int_0^\infty Sin(-2t^2)dt\right) \\<br /> &=\left(-\frac{\sqrt{\pi}}{4}-\frac{\sqrt{\pi}}{4}\right)+i\left(\frac{\sqrt{\pi}}{4}-\frac{\sqrt{\pi}}{4}\right) \\<br /> &=-\frac{\sqrt{\pi}}{2}<br /> \end{align}$$


The same analysis can be made (with the path going from 0 to $\infty+i\infty$ for the second integral with the results that the line integral in that case is:

$$\frac{\sqrt{\pi}}{2}$$

Substituting these values into (1), I obtain:

$$\frac{1}{i}\frac{2}{\sqrt{\pi}}\int_0^{-\infty+i\infty} e^{-w^2}dw=\frac{1}{i}\frac{2}{\sqrt{\pi}}(-\frac{\sqrt{\pi}}{2})=-\frac{1}{i}\frac{i}{i}=i$$

and:

$$\frac{1}{i}\frac{2}{\sqrt{\pi}}\int_0^{\infty+i\infty} e^{-w^2}dw=\frac{1}{i}\frac{2}{\sqrt{\pi}}(\frac{\sqrt{\pi}}{2})=\frac{1}{i}\frac{i}{i}=-i$$

So that the limit is $2i$.


Substituting this into the last expression of the inverse transform, we obtain:

$$y(x)=\frac{e^{-\pi/t}}{2i t^{3/2}}(2i)=\frac{e^{-\pi/t}}{t^{3/2}}$$

 

[opticaltempest]

Testing TeX output from Maple 10

$$\lim _{n\rightarrow \infty } \left(\frac {27}{n^2}\ \sum _{i=1}^{n} i \right )<br />$$

 

[Arust]

Thank you, it's very useful

 

[TinTinn Tabulation]

I'm just $TeX$'ing away on the thin ice of a new Day !
$$\int _{0}^{1}\!{\frac {1}{\sqrt {{x}^{2}+{y}^{2}+{z}^{2}}}}{dx}=-1/2\,\ln \left( {y}^{2}+{z}^{2} \right) +\ln \left( 1+\sqrt {1+{y}^{2}+{z<br /> }^{2}} \right)$$
Oo-er, me ickle preview, it preview-th not !

 

[Ratzinger]

$$a^2$$ I tried to preview my post but the preview only says LaTex graphic is being generated. Reload this page in a moment. But when I reload nothing happens. What can be done?

$$\phi (x) = \frac{1}{{(2\pi )^{3/2} }}\int {d^4 p\delta (p^2 } - m^2 )A(p)e^{ - ip \cdot x}$$

 

[ranger]

Is it possible to show when you cancel out terms i.e show a line through the terms being canceled?

 

[krab]

$$R = \frac{V}{I} = \frac{V}{small no.}$$

I notice a lot of this sort of error. LaTeX math typesetting cannot "know" that "small no." is not the symbols s times m times a times ... If Text appears within math symbols, it should be formatted as such. Then it is in regular type and interword spaces are observed. This is done with mbox, as for the above example:
$$R = \frac{V}{I} = \frac{V}{\mbox{small no.}}$$
And of course similarly for units: it's not
$$64 ft^2$$,
rahter, it is
$$64\mbox{ ft}^2$$

 

[twoflower]

Why can't I insert newline in tex code? I tried \\ but no success...

 

[robphy]

Is it possible to show when you cancel out terms i.e show a line through the terms being canceled?

Here's a crude way
$$a\hspace{-1ex}\slash +b=a\hspace{-1ex}\slash +c$$

I have a fancier way with a macro that requires a little trial and error.

 

[xouper]

For a function $f : \mathbb{N} \mapsto \{0,1\}$$$f(n)=\left\{\begin{array}{cc}0,&\mbox{ if }f(n)=1\\1,&\mbox{ if } f(n)=0\end{array}\right.$$

 

[Doc Al]

Read This: Please delete practice posts :-)

Folks:

Use this thread to practice your Latex (that's what it's for), but unless you've come up with something that hasn't been done before in the last gazillion posts: Please delete your post after you're done!

Of course, if you've come up with something new: Leave it there for others to learn from.

Thanks!

 

[Astronuc]

robphy, I just clicked on kishtik's equation and I can get the LaTeX source. In your post though, it is a png file.

What I have noticed it that if one does not close the LaTeX source code viewing window, subsequent LaTeX code is added to the window (by right clicking on the LaTeX formula), and one has to scroll down to view.

 

[robphy]

actually, it was intended to verify that the IMG tag works here (and possibly in other non-GD areas)

 

[bomba923]

True or False?

$$\boxed{\begin{gathered}<br /> \forall p,q,r \in \mathbb{N}\;{\text{where }}p > q > 0, \hfill \\<br /> \exists \left\{ {a_0 ,a_1 ,a_2 , \ldots ,a_n } \right\} \subset \bigcup\limits_{i = 0}^{p - 1} {\left\{ {\frac{i}<br /> {q}} \right\}} \hfill \\<br /> {\text{such that }}\frac{r}<br /> {q} = \sum\limits_{k = 0}^n {a_k \left( {\frac{p}<br /> {q}} \right)^k } \hfill \\ <br /> \end{gathered}}$$

$$\boxed{\begin{gathered}<br /> \forall p,q \in \mathbb{N}\;{\text{where }}p > q > 0, \hfill \\<br /> \mathbb{N} \subset \bigcup\limits_{k \geqslant 0} {\left\{ {\sum\limits_{n = 0}^\infty {\left( {\left\lfloor {\frac{k}{{p^n }}} \right\rfloor - p\left\lfloor {\frac{k}{{p^{n + 1} }}} \right\rfloor } \right)\left( {\frac{p}{q}} \right)^n } } \right\}} \hfill \\ <br /> \end{gathered}}$$

 

[twoflower]

$$...\ =\ (\mbox{Lagrange})\ \sum_{i=1}^{n} \frac{\partial f}{\partial x_i} \left( \xi^{i} \right) h_i\mbox{ , kde }\xi^{i} \in [a_i, a_i\ +\ h_i]$$

$$\xi^{i} = [a_i, ..., a_{i-1}, a_i\ +\ \vartheta h_i, a_{i+1}\ +\ h_{i+1}, ..., a_n\ +\ h_n]\mbox{, }\vartheta \in (0,1)$$

 

[Cincinnatus]

testing:
$$x^a_i$$

 

[twoflower]

$$\left\{ u_{n}\right\}$$

is bounded in

$$W^{1,N-1}\left( \Omega;\mathbb{R}^{N}\right)$$

,

$$\left\{ \mathrm{adj}\nabla u_{n}\right\} \subset L^{\frac{N}{N-1}}\left( \Omega;\mathbb{R}^{{N}\times{N}}\right)$$

and if

$$u\in BV\left( \Omega;\mathbb{R}^{N}\right)$$


are such that

$$u_{n}\rightarrow u$$

in

$$L^{1}\left( \Omega;\mathbb{R}^{N}\right)$$

and

$$\[ \det\nabla u_{n}\overset{\ast}{\rightharpoonup}\mu \]$$

in the sense of measures, then for

$$\mathcal{L}^{N}$$

a.e.

$$x\in\Omega \[ \det\nabla u\left( x\right) =\frac{d\mu}{d\mathcal{L}^{N}}\left( x\right). \]$$

The result is sharp and counterexamples are provided in the cases where regularity of

$$\left\{ u_{n}\right\}$$

or the type of weak convergence are weakened.

 

[StatusX]

Does anyone know of a program I can use to compile latex code? Any output format is fine, pdf, gif, etc. And simpler is better, I'm no computer programmer.

 

[Psi-String]

$$\oint \vec{E} \cdot d \vec{A} =\frac{q}{\epsilon_0}$$

$$\oint \vec{B} \cdot d \vec{A} =0$$

$$\oint \vec{E} \cdot d \vec{s} = -\frac{d \Phi_B}{dt}$$

$$\oint \vec{B} \cdot d \vec{s} = \mu_0 i + \mu_0 \epsilon_0 \frac{d \Phi_E}{dt}$$

 

[Sartak]

Of course, I have to test too. ;)

$$\frac{d}{dx} f(x) = \lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$$

$$\int^b_a f(x) dx = F(b) - F(a)$$

 

[rea]

sure...

Does anyone know of a program I can use to compile latex code? Any output format is fine, pdf, gif, etc. And simpler is better, I'm no computer programmer.

For Windows systems, there is the option of MikTeX also there exist ghostscript and other similar packages you will need, they are free.


In Linux, I exactly don't know, but normally the distributions contain TeX and LaTeX, also there is a editor that I will like to understand called LyX, LyX is also available IRC for Windows...







Do you know any other good editor for TeX files? for learn the basics?

 

[StatusX]

For Windows systems, there is the option of MikTeX also there exist ghostscript and other similar packages you will need, they are free.

I downloaded MikTex, but it seems to need a lot of other programs to work. I couldn't figure it out. I'm just looking for some program where you can give it a filename and it will compile it into some reasonable output form (preferably pdf). Do you know something like this, or if not, do you know how I can use MikTex to do what I need?

 

[VietDao29]

I dunno, but I think MikTex only provides a Yap viewer, i.e some kind of integrated environment for beautiful formulae (maybe I'm wrong though).
If you want to translate LaTeX code to .pdf files, I'd recommend TeXnicCenter
Just download it, and run it. It can convert LaTeX to PDF, LaTeX to DVI (viewed with yap viewer), or LaTeX to PS.
Hope that's what you are looking for.

 

[StatusX]

I dunno, but I think MikTex only provides a Yap viewer, i.e some kind of integrated environment for beautiful formulae (maybe I'm wrong though).
If you want to translate LaTeX code to .pdf files, I'd recommend TeXnicCenter
Just download it, and run it. It can convert LaTeX to PDF, LaTeX to DVI (viewed with yap viewer), or LaTeX to PS.
Hope that's what you are looking for.

Hey, thanks a lot. That's just what I needed.

 

[rea]

I downloaded MikTex, but it seems to need a lot of other programs to work. I couldn't figure it out. I'm just looking for some program where you can give it a filename and it will compile it into some reasonable output form (preferably pdf). Do you know something like this, or if not, do you know how I can use MikTex to do what I need?

Not all is easy.

You need all the tools, because Windows dosent provide a dvi visor or a ps visor, then you need download ghostscript (that are free also).

I dunno, but I think MikTex only provides a Yap viewer, i.e some kind of integrated environment for beautiful formulae (maybe I'm wrong though)

See the description by instance:

MiKTeX is an up-to-date TeX implementation for the Windows operating system.

TeX is a typesetting system written by Donald E. Knuth, who says that it is "intended for the creation of beautiful books - and especially for books that contain a lot of mathematics".

MiKTeX offers a complete set of utilities, macro packages and fonts, e.g., LaTeX, pdfTeX, ConTeXt, just to name a few.

In fact, I think that TeXnicCenter use MikTeX like the compiler, ie, texnicenter is only the IDE like there are IDEs for control compilers for programming languages in a nice environment, like DEv-cpp use Mingw package, or VS use make, ml and other command line tools.

NOW, with those aclarations, for use miktex, download the minimal installer, about 30Mb, if you need additional packages (aka more things to add), you can use a install center that is provided with miktex and download what you need... and is this one:

Getting updates

Use the update wizard (Start->MiKTeX->MiKTeX Update Wizard) to get the latest package updates.

the ghost script...

GPL Ghostscript 8.50
Posted on Wed, 04 Jan 2006 04:42:00 GMT
http://dojo.miktex.org/blogs/christian_schenk/archive/2006/01/03/247.aspx

that link to: http://sourceforge.net/project/shownotes.php?group_id=1897&release_id=382430

See that is a sf (source forge) page, then the summary of the project is: http://sourceforge.net/projects/ghostscript/ and the home page is http://www.ghostscript.com/

That is all you need:

Core files for TeX (compilation and related things to this process, macros, definitions,...):

1) MikTeX

Viewers:

1) Gostscript and related packages

And an editor for call the above programs:

1) TexniCenter
2) I suguest LyX really (is like WYSIWYG, but not at all, is an WYSIWYM, What You See Is What You Mean, note that because a diferent point of view, switch to it will be extrange XD.For read a more wide introduction in install and other tools: http://www.miktex.org/Links.aspx see that there is listed texnicenter as a editor... also there is listed Lyx, the one I recommend is more nice I think, really you should try it.

 

[StatusX]

Thanks for the info. I think TexnicCenter does what I need, and it's pretty simple.

 

[ksinclair13]

$$e^{\pi i}+1=0$$

 

[Sartak]

$$f(x) = \left\{\begin{array}{cc}x^2 \sin(\frac{1}{x}),& \mbox{ if } x\neq 0\\0, & \mbox{ if } x=0\end{array}\right$$

 

[AKG]

$$\sum _{k=1} ^{K} \left\lfloor\frac{n}{p^k}\right\rfloor < \frac{n}{p-1}$$

$$n\cdot\sum _{k=1} ^{K} \left\lfloor\frac{1}{p^k}\right\rfloor < n\left (\frac{1}{p-1}\right )$$

$$\sum _{k=1} ^{K} \left\lfloor\frac{1}{p^k}\right\rfloor < \frac{1}{p-1}$$

$$\sum _{k=1} ^{K} \left\lfloor\frac{p^K}{p^k}\right\rfloor < \frac{p^K}{p-1}$$

$$p^{K-1} + \dots + 1 < \frac{p^K}{p-1}$$

$$(p^{K-1} + \dots + 1)(p-1) < p^K$$

$$p^K - 1 < p^K$$

 

[twoflower]

$$\xi^{i} = (a_1\ +\ h_1,\ a_2\ +\ h_2,\ ...\ ,\ a_{i-1}\ +\ h_{i-1},\ a_i\ +\ \vartheta h_i,\ a_{i+1},\ a_{i+2}...,\ a_{n})\mbox{, }\vartheta \in (0,1)$$

$$\xi = (a_1 + h_1,\ a_2 + h_2,\ ...\ ,\ a_n + h_n)$$