# Integral definition of factorial

1. Jun 14, 2014

### 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?

2. Jun 14, 2014

### micromass

Staff Emeritus
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

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

3. Jun 14, 2014

### Runei

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

Many thanks Micromass!

4. Jun 14, 2014

### HallsofIvy

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

5. Jun 14, 2014

### 1MileCrash

Multiplication is associative! :P

6. Jun 14, 2014

### micromass

Staff Emeritus
But it's commutative we want! :tongue:

7. Jun 14, 2014

### 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 $\displaystyle\left(\frac 1 r \frac{d}{dr}\right)^nf(r)$.

8. Jun 14, 2014

### Matterwave

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

9. Jun 14, 2014

### 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)

10. Jun 14, 2014

### micromass

Staff Emeritus
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 [Broken]

Last edited by a moderator: May 6, 2017
11. Jun 14, 2014

### 1MileCrash

I fail

12. Jun 15, 2014

### HallsofIvy

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

13. Jun 15, 2014

### pwsnafu

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!