Iterated integrals wrt one variable

  • #1
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Main Question or Discussion Point

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

Answers and Replies

  • #2
EnumaElish
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MathWorld uses D-n f(x) for n repeated integrals; I guess Dn f(x) stands for ∂nf(x)/∂xn.
 
  • #3
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i was wondering what the superscript number to the d was in the quantum filed theory integrals , like what does [tex]\int d^4 k[/tex] mean, and why it isn't written [tex]\int dk^4[/tex]
 
  • #4
Mute
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i was wondering what the superscript number to the d was in the quantum filed theory integrals , like what does [tex]\int d^4 k[/tex] mean, and why it isn't written [tex]\int dk^4[/tex]
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:

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

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

Also, [itex]dk^4[/itex] would be easily confused with [itex]d(k^4) = 4k^3 dk[/itex].
 
  • #5
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how are integrals in the form of [tex] \overbrace{\int \cdots \int \int}^{n \, \mathrm{times}} f(x) \overbrace{\,dx\,dx ... \dx}^{n \, \mathrm{times}}[/tex] 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.
 
  • #6
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is it standard to write it like this?
(integral sign) stuff (dx n times)
 
  • #7
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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
[tex]\idotsint f(x) dx^n[/tex]
where the emphasis is placed on the differential form in the integrand. In other circumstances, the author will also write
[tex]\int\limits_A f[/tex]
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.
 

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