# Iterated integrals wrt one variable

## Main Question or Discussion Point

how are integrals in the form of $$\overbrace{\int \cdots \int \int}^{n \, \mathrm{times}} f(x) \overbrace{\,dx\,dx ... \dx}^{n \, \mathrm{times}}$$ written? ie. if you integrate with respect to x n times, then what is the shorthand notation for that?

EnumaElish
Homework Helper
MathWorld uses D-n f(x) for n repeated integrals; I guess Dn f(x) stands for ∂nf(x)/∂xn.

i was wondering what the superscript number to the d was in the quantum filed theory integrals , like what does $$\int d^4 k$$ mean, and why it isn't written $$\int dk^4$$

Mute
Homework Helper
i was wondering what the superscript number to the d was in the quantum filed theory integrals , like what does $$\int d^4 k$$ mean, and why it isn't written $$\int dk^4$$
It's a short-hand notation that is NOT meant to represent four repeated indefinite integrals. k is a four vector with four components, so the notation means you are to perform the integration over all four components of k, which some specified limits on each integral:

$$\int d^4k = \int_\alpha^\beta dk_x\int_\gamma^\delta dk_y\int_\epsilon^\eta dk_z\int_\lambda^\tau dk_{ct}$$

For generality, I made all the limits different, but I guess usually all would have the same limits of integration.

Also, $dk^4$ would be easily confused with $d(k^4) = 4k^3 dk$.

how are integrals in the form of $$\overbrace{\int \cdots \int \int}^{n \, \mathrm{times}} f(x) \overbrace{\,dx\,dx ... \dx}^{n \, \mathrm{times}}$$ written? ie. if you integrate with respect to x n times, then what is the shorthand notation for that?
It is also standard notation to use Jn for n repeated indefinite integrals (assumed to be single-variable) in the study of integral equations. The form of your result would be (Jnf)(x). This notation is extended in fractional calculus so that n can take on any real value, such that J becomes a unified differintegral operator.

is it standard to write it like this?
(integral sign) stuff (dx n times)

is it standard to write it like this?
(integral sign) stuff (dx n times)
As long as it is not ambiguous, there should be no problem. It is also usually written
$$\idotsint f(x) dx^n$$
where the emphasis is placed on the differential form in the integrand. In other circumstances, the author will also write
$$\int\limits_A f$$
where the emphasis is placed on A, the set being integrated over, and the coordinate-specific differential notation is suppressed. This notation is used in more abstract texts.