Wikipedia:Reference desk/Archives/Mathematics/2013 May 7

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May 7

Why does Pi appear in Stirling's approximation and Gaussian integrals ?

${\color {blue}e}^{n}\sim {\frac {n^{n}}{n!}}\ {\sqrt {2\ {\color {red}\pi }\ n}}\qquad \iff \qquad \left({\frac {\color {blue}e}{n}}\right)^{n}\cdot \int _{0}^{\infty }{{\color {blue}e}^{-{\sqrt[{n}]{x}}}}\ dx\ \sim \ {\sqrt {2\ {\color {red}\pi }\ n}}$

and

$\int _{-\infty }^{\infty }{\color {blue}e}^{-x^{2}}\,dx={\sqrt {\ \color {red}\pi \ }}\qquad -$ 79.113.219.240 (talk) 17:24, 7 May 2013 (UTC)

See Wallis product about the factorial, the constant is linked to the sin function which is where the pi comes from. The Gaussian integral can be considered as the square root of a 2-dimensional version that can be evaluated fairly easily using radial coordinates the integral turns into an integral over angles from 0 to 2 pi, this is described in Gaussian integral. Dmcq (talk) 22:50, 7 May 2013 (UTC)

Ultimately, the Gaussian integral seems to be related to the Cartesian equation of the circle: x² + y² = R². — 79.113.219.240 (talk) 03:39, 8 May 2013 (UTC)

(ec) Pi and e are linked by Euler's identity and Euler's formula, so when you start doing calculus with e, pi sometimes falls out. Wallis' product can be derived from Euler's formula, and as the article Stirling's approximation notes, the derivation of the approximation goes through a step where Wallis' product can be used to simplify a value. -- 205.175.124.30 (talk) 22:59, 7 May 2013 (UTC)

Here's my problem : Euler's famous formula is an observation... not an explanation. We notice (using the Taylor series) that e^ix just-so-happens to be equal to cos(x) + i sin(x)... But why exactly does this "coincidence" happen in the first place ? What is the reason for its existence ? I just feel like I'm/we're missing something, not seeing the forest for the trees, but I don't know what "it" is... I suspect it's something fundamentally simple and evident... so simple and obvious, in fact, that it's eluding us, by hiding in plain sight... — 79.113.219.240 (talk) 03:39, 8 May 2013 (UTC)

These are interesting questions, but it's fairly unlikely that anyone here will be able to give a definitive answer that will satisfy you. It's not what mathematicians usually think of as mathematics in the narrow sense of the word, though I suppose that might change if anyone started getting actual results. If you like to torture yourself with this sort of thing, see umbral calculus and monstrous moonshine. --Trovatore (talk) 03:52, 8 May 2013 (UTC)

79, I don't know if this will satisfy you, but e^t is a solution of the differential equation f′(t) = f(t), while e^it = cos t + isin t is a solution of f′(t) = if(t). In other words, if you place a particle in the plane at (1,0) at time zero and let it move so that its velocity vector is always equal to its position vector, the particle will move along the real axis with position function (e^t, 0), whereas if its velocity vector is always the position vector rotated by 90 degrees counterclockwise, the particle will move along the unit circle with constant speed 1, so that the position function is given by (cos t, sin t). So both the exponential function and the function cos t + isin t are solutions to similar initial-value problems, which explains their similarity. You just replace "parallel" with "perpendicular". 96.46.198.58 (talk) 04:27, 8 May 2013 (UTC)

Because in a circle, the tangent (graphical representation of the derivative, which is speed) is always perpendicular on the radius (position vector). — 79.113.215.108 (talk) 18:36, 26 May 2013 (UTC)

If you look at Euler's formula#Using the limit definition you'll see the connection using the original definition of e from the limit of compound interest plus the way that imaginary numbers multiply. Dmcq (talk) 08:32, 8 May 2013 (UTC)

Since sin and cos are periodic functions passing countless times back and forth through 0, this allows us to treacherously and mischievously write them as transcendental polynomials (polynomials on infinite degree, with rational coefficients) of the form $\sin(x)=x\prod _{k\in \mathbb {Z} ^{*}}(1-{\tfrac {x}{k\pi }})$ and $\cos(x)=\prod _{k\in \mathbb {Z} }\left(1-{\tfrac {2x}{(2k+1)\pi }}\right)$ ... But e is the "father" of transcendental polynomials, since $e^{x}=\lim _{n\to \infty }\prod _{k\in \mathbb {N} ^{*}}{(1+{\tfrac {x}{n}})}$ ... Hmmm... Yeah, that definitely sheds more light on the topic... Thanks! :-) — 79.113.240.179 (talk) 19:59, 8 May 2013 (UTC)

I was really meaning have a look at the animation, or better still the animation mentioned below at Exponentiation#Imaginary_exponents with base e which makes $e^{i\pi }=-1$ seem rather obvious. Dmcq (talk) 20:30, 8 May 2013 (UTC)

I'm afraid the graphic only illustrates the problem visually, without actually explaining why this is so... This, and the fact that I don't really understand what the intermediary points at each step are supposed to represent... (1 + ^iπ/_N)^k or (1 + ^iπ/_k)^N, where k < N ? They obviously don't represent (1 + ^iπ/_k)^k. — 79.113.240.179 (talk) 20:56, 8 May 2013 (UTC)

The graphic shows the points $\left(1+{{i\pi } \over {N}}\right)^{k}$ for $k=0\dots N$.Each one is the previous multiplied by the factor $\left(1+{{i\pi } \over {N}}\right)$ and as $N$ tends to infinity they become closer and closer to a pure rotations by the small imaginary factor since the movement becomes more an more purely at right angles to the radius. Dmcq (talk) 21:48, 8 May 2013 (UTC)

Like a polygonal nautilus spiral slowly `degenerating` into a perfect semicircle... — 79.113.240.179 (talk) 22:55, 8 May 2013 (UTC)

I would argue as follows. If you define exp(z) as exp(x)[cos(y) + i sin(y)], then this definition makes exp(z) complex differentiable (the Cauchy-Riemann equations are satisfied). According to the theory of complex functions, complex differentiable functions are analytic, therefore exp(z) as defined above is an analytic function. Then for y = 0, exp(z) reduces to the real function exp(x), so exp(z) in the way we defined it, is an analytic continuation of exp(x). Now, from the theory of complex functions, we know that analytic continuations are unique. So, there is no other possible definition of an analytic exp(z) that will reduce to exp(x) on the real line. Count Iblis (talk) 11:37, 8 May 2013 (UTC)

See Exponentiation#Imaginary_exponents with base e for the geometrical explanation why e^ix is actually equal to cos(x) + i sin(x). Bo Jacoby (talk) 12:50, 8 May 2013 (UTC).

For the Gaussian integral (in my point of view), it comes down to the fact that the Gaussian integral computes the square root of the absolute value of the smallest Dirichlet eigenvalue for the one-dimensional Laplacian on [0,1] (via Jacobian theta functions). This is encoded, for instance, in the Poisson summation formula

$\sum _{k\in \mathbb {Z} }e^{-(x+k)^{2}/4t}={\sqrt {\frac {4\pi }{t}}}\sum _{\ell \in \mathbb {Z} }e^{2\pi i\ell x-4\pi ^{2}\ell ^{2}t}.$

Without having worked out the details, I think that Stirling's formula must admit an analogous interpretation using the zeta functional determinant of the one-dimensional Laplacian. Sławomir Biały (talk) 00:22, 9 May 2013 (UTC)

Duh ! :-) — 79.113.240.179 (talk) 00:57, 9 May 2013 (UTC)

Did you ask a serious question, or are you just trolling? Sławomir Biały (talk) 02:49, 9 May 2013 (UTC)

Sorry, I didn't mean to offend... :-( It's just that what you're saying is light-years above my pay-grade... You sort of lost me at hello. :-) The ease and the naturalness with which you speak about such unapproachable topics reminded me of that movie-sequence, based on one of Eliade's novels. — 79.113.240.179 (talk) 03:07, 9 May 2013 (UTC)

Linking Factorials to Geometric Shapes

Newton's binomial ! (God, I can't believe I've been so blind !)

$(a+b)^{m}=\sum _{k=0}^{m}{C_{m}^{k}\cdot a^{k}\cdot b^{m-k}}$

So it should come as no suprise that for a = 1 and b = -ⁿ√x we have:

$\int _{0}^{1}{(1-{\sqrt[{n}]{x}})^{m}}\ dx\ =\ {\frac {m!\ n!}{(m+n)!}}\ =\ {\frac {1}{C_{m+n}^{n}}}\ =\ {\frac {1}{C_{m+n}^{m}}}$

Which, for m = n = ¹/₂ , transforms into the rather famous equation relating to the area of the circle:

$\int _{0}^{1}{\sqrt {1-x^{2}}}\ dx\ =\ {\frac {\pi }{4}}$

From where we deduce that:

${\frac {{\tfrac {1}{2}}!\ {\tfrac {1}{2}}!}{({\tfrac {1}{2}}+{\tfrac {1}{2}})!}}={\frac {\pi }{4}}\ =>\ {\tfrac {1}{2}}!={\frac {\sqrt {\pi }}{2}}$

Now, we also know that Euler's famous constant e is nothing else than the sum of the reciprocals of factorials:

$e=\sum _{n=0}^{\infty }{\frac {1}{n!}}$

So it should come as no surprise when we find out that the generalized expresion of the factorial function involves the exponential one:

$n!=\int _{0}^{\infty }{e^{-{\sqrt[{n}]{x}}}}\ dx$

And there we have -ⁿ√x again ! What a "shocker" !

This last identity, for n = ¹/₂ , becomes the famous Gaussian integral:

${\tfrac {1}{2}}!=\int _{0}^{\infty }{e^{-x^{2}}}\ dx$

Whose value we've already shown to be equal to ${\sqrt {\pi }} \over 2$ .

Now let's write it again, just for efect:

$\int _{0}^{1}{\sqrt[{m}]{1-x^{n}}}\ dx\ =\ \int _{0}^{1}{\sqrt[{n}]{1-x^{m}}}\ dx\ =\ {\frac {{\frac {1}{m}}!\ {\frac {1}{n}}!}{\left({\frac {1}{m}}+{\frac {1}{n}}\right)!}}\ =\ {\frac {\int _{0}^{\infty }{e^{-x^{n}}}\ dx\ \cdot \ \int _{0}^{\infty }{e^{-x^{m}}}\ dx}{\int _{0}^{\infty }{e^{-x^{mn \over m+n}}}\ dx}}$

Which identity helps show that all Gaussian integrals, and -by extension- all values of the Gamma functions are intrinsically linked through Newton's binomial to geometric shapes described in the most general way by Diophantine equations of the form:

$x^{n}+y^{m}=1$

Which, in the special case m = n = 2 describes the unit circle. — 79.113.240.179 (talk) 06:02, 9 May 2013 (UTC)

Resolved