Understanding Stirling's Formula: The Bent Equals Sign Explained

[PageWizard]

Hello everybody,

My name is Daniel. I have a simple question. I'm just doing some research for my theory and I found myself just in the topic of stirling's formula. It is denoted with an equals sign that is bent in like the top bar of the equals sign in bent down, and the bottom bar of the equals sign is bent up. What is this symbol? (like it looks like = but, is modified). I apologize if this is a simple answer just my book doesn't give the written definition of the formula. Thank you for your time :).

 

[HallsofIvy]

In the case of Stirling's formula it means "approximately equal". Stirlings formula gives an approximation of n!, the approximation becoming closer to n! as n gets larger.

 

[Gib Z]

Perhaps the sign you meant was "~" ?

That sign denotes asymptotic equally, which roughly, as Halls has already said, means that Stirlings formula approximates n!, which the error reducing as n becomes larger.

Formally, if \lim_{n\to \infty} \frac{A(n)}{B(n)} = 1, then we state A(n) ~ B(n).

From this we can see that the error does not necessarily reduce as n grows large, but that the ratio of the error becomes small with respect to the size of the function itself.

Eg (x+1) ~ x, but with a constant error of 1.

 

[D H]

n!\approx\sqrt{2\pi n} \left( \frac n e \right)^n

The \approx means "approximately equal to". In other words, Sterling's formula (better: Sterling's approximation) does not yield n factorial. It does yield something that is fairly close to n factorial.

 

[PageWizard]

no it's not approximately equal to. It is an equal sign with the top bar bent downwards, and the bottom bar bent upwards. It could in meaning be approximately or precisely equal to but it looks like if you cut a empty half elipse in half, put the bottom half on the top where the topbar of the equal sign was, and the other on the bottom bar.

I can't exactly find the symbol unfortunately.

 

[gel]

There's a list of mathematical symbols in Latex http://www.csulb.edu/~fnewberg/Research/latexsymbols2.pdf" . Sounds like you're describing \succ or \prec to me.

 

[tiny-tim]

Welcome to PF!

Hi Daniel! Welcome to PF!

There's an non-pdf list of symbols at http://www.physics.udel.edu/~dubois/lshort2e/node61.html#SECTION008100000000000000000

Can you find it there?

 

[PageWizard]

\succ\prec it looks like that except the gap between the top and bottom pieces is a bit more and they are together not separated. now what is this symbol the one that is highlighted as \asymp, cause that is the symbol it uses? and thank you :).

 

[gel]

You mean x \asymp y? Not sure, but it could mean x is http://en.wikipedia.org/wiki/Asymptotically_equal_to" y. That is, x/y -> 1 in the limit.
I'm more familar with x\sim y for this though.

 

[PageWizard]

i'll get the quote here:

"This is a startling fact that Stirling discovered:

n! (that symbol here) sqrt(2*pi)*e^(-n)*n^(n+(1/2))

To my mind, this is one of the quintessential discoveries of 18th century mathematics. A formula like this give us some idea of the astonishing transformation of mathematics"​

does that make sense

 

[gel]

Yes, that makes sense. See D.H.'s post above or http://en.wikipedia.org/wiki/Stirling%27s_formula" . They use \approx and \sim near the bottom of the page.

 

[PageWizard]

thank you :).