Integral definition of factorial

[Runei]

I'm watching V. Balakrishnan's video lectures on Classical Physics, and right now he's going through statistical mechanics.

In that regards he's talking about Stirlings formula, and at one point, he wrote an integral definition of the factorial like the following

$n! = \int_{0}^{\infty}dx\hspace{0.1cm}e^{-x}\hspace{0.1cm}x^n\hspace{0.1cm},\hspace{2cm} \text{where}\hspace{1cm} n={1,2,3 ...}$

Why is he writing the integral in that way? With the dx first and the exponentials afterwards?

I thought the definition was

$n! = \int_{0}^{\infty}e^{-x}x^ndx$

Can anybody explain this?

Many thanks in advance :)

 

[micromass]

It's the same thing. But physicists tend to use the notation of writing $dx$ first for some reason. It's just a notational issue. For all intents and purposes, we have

$$\int f(x)dx = \int dx f(x)$$

 

[Runei]

Ah, I was wondering that that might be the case :)

Many thanks Micromass!

 

[HallsofIvy]

We can't really expect physicists to write mathematics correctly, can we?

 

[1MileCrash]

Multiplication is associative! :P

 

[micromass]

Multiplication is associative! :P

But it's commutative we want!

 

[AlephZero]

If you think of $\displaystyle\int_0^\infty \cdots \,dx$ as an operator, writing $\displaystyle\int_0^\infty\!\! dx\,f(x)$ is just as sensible as something like $\displaystyle\frac{d^n}{dx^n}f(x)$ - or even $\displaystyle\left(\frac 1 r \frac{d}{dr}\right)^nf(r)$.

 

[Matterwave]

Does anyone have any historical perspective on why some people prefer $\int f(x)dx$ and some people prefer $\int dxf(x)$?

 

[Runei]

While I perfectly understand the idea of the notation, i think its confusing in the way where the following two ways are different:

$\int(dx x) = \int x dx = 1/2 x^2$

And

$\int(dx) x = x^2$

(Mind the constants not shows)

 

[micromass]

$\int(dx) x = x^2$

OK, but this is bad notation. The $x$ in the $dx$ is the dummy variable. The other $x$ is an actual variable. You shouldn't use the same name for them.
And indefinite integrals are tricky things so I'm not sure if the thing above is well-defined. See https://www.physicsforums.com/blog.php?b=4566

 

[1MileCrash]

But it's commutative we want!

I fail

 

[HallsofIvy]

While I perfectly understand the idea of the notation, i think its confusing in the way where the following two ways are different:

$\int(dx x) = \int x dx = 1/2 x^2$

And

$\int(dx) x = x^2$

(Mind the constants not shows)

No this is wrong. The second form should be $\left(\int dx\right)x= x^2$.

 

[pwsnafu]

It's the same thing. But physicists tend to use the notation of writing $dx$ first for some reason. It's just a notational issue. For all intents and purposes, we have

$$\int f(x)dx = \int dx f(x)$$

We can't really expect physicists to write mathematics correctly, can we?

It's worth pointing out that mathematicians do this as well, specifically in papers dealing with n-fold integration. Cauchy's formula for repeated integration is very often written
$\int_{a}^{x} dt_n \int_{a}^{t_n} dt_{n-1} \ldots \int_{a}^{t_2} f(t_1) \, dt_1 = \frac{1}{(n-1)!}\int_a^x (x-t)^{n-1}\,f(t)\,dt$
Observe this uses both notations mixed!