Am I using this integral sign correctly?

Question

I'm not sure how to use this integral symbol--but my guess is that it is used like so

$$\rlap{\sum}\int$$

$$\int_{-\infty}^\infty\frac{\sin x}x\ \mathrm{d}x=\pi$$

$$\sum_{n=-\infty}^\infty\frac{\sin x}x=\pi$$

$$\rlap{\sum_{-\infty}^\infty}\int\ \frac{\sin x}x=\pi$$

I couldn't find anything on the internet except that it was something used in quantum mechanics (now you know why that tag is here). They said that it was used when a summation and integral notation could be used interchangedly. Thank you in advance

Note: I had to use a picture to show the mathematical equations because math stack-exchange doesn't support the notation.

Answer 1

According to reddit, it is used when it is ambiguous whether or not the operation is over a continuous distribution or a discrete one. Recall that integration is just a special kind of summation where we are working with a continuous argument. Thus, one might write $$\text{sumint}_{x=0}^\infty~f(x)$$ for a general definition: it tells the reader the definition works both in the discrete case and the continuous case, and so the reader should know which applies and use the definition accordingly.

What's a practical application of this? Mean value! The mean value for a discrete set is written

$$\left<x\right>=\sum_{x=0}^n x_i p_i$$

where $x_i$ is the $i^{th}$ outcome, $p_i$ is the probability of $x_i$ occuring, and $\left<x\right>$ is the mean outcome. In integral terms

$$\left<x\right>=\int_0^n xp(x)dx$$

So we can then, in general, say

$$\left<x\right>=\text{sumint}_{x=0}^n~xp(x)$$

This conveys all of the information that you need for both the integration and the summation, whichever applies. I have never seen this notation used before, but the above is seemingly how you should use it. I do not use it, but this is definitely a nice way of condensing two definitions into one.

Note, I haven't seen the notation before and online it is rarely referenced. I have omitted the $dx$ from the sumint because I do not think it belongs, but perhaps it does. This answer is to explain how it works, and I urge the community to please edit or comment in corrections to the notation as required.

Answer 2

As just a lowly mathematician, I would use the "Riemann-Stieljes" integral, $\int f(x)d\mu(x)$, where "$\mu(x)$" is a measure function. If $\mu$ is a differentiable function then $\int f(x)d\mu(x)= \int f(x)\mu' dx$ and if $mu$ is a step function, $\mu(x)= 0$ if [$0< x\le 1$, $\mu(x)= 1$ if $1< x\le 2$, etc. then $\int f(x) d\mu= \sum_{n= 0} f(n)$