Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Iterated integrals wrt one variable

  1. Jun 3, 2009 #1
    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?
     
  2. jcsd
  3. Jun 3, 2009 #2

    EnumaElish

    User Avatar
    Science Advisor
    Homework Helper

    MathWorld uses D-n f(x) for n repeated integrals; I guess Dn f(x) stands for ∂nf(x)/∂xn.
     
  4. Jun 3, 2009 #3
    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]
     
  5. Jun 3, 2009 #4

    Mute

    User Avatar
    Homework Helper

    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].
     
  6. Jun 3, 2009 #5
    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.
     
  7. Jun 3, 2009 #6
    is it standard to write it like this?
    (integral sign) stuff (dx n times)
     
  8. Jun 3, 2009 #7
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook