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

The Exponential of an infinite sum

  1. Oct 24, 2011 #1
    I have a problem that arises in quantum field theory. It involves a problem in combinitorics and about the theory of connected graphs.

    Essentially, I am trying to prove an identity involving an exponential of an infinite series with the ways to decompose an integer into the sum of integers.

    [tex]\exp\left({\sum_{n=1}^\infty a_n}\right) = \sum_{n=0}^\infty \sum_{partitions} \prod_{i=1}^n \frac{a_i^{n_i}}{n_i ! }[/tex]

    Where the partition is over the ways to write n as a sum of integers.


    [tex]\sum_{i=1}^n in_i = n[/tex]

    If you expand out the first few terms of the right hand side, it looks good.

    n=0 (I'm taking it to be 1 just to avoid confusion)


    [tex] a_1 [/tex]


    [tex] a_2 + \frac{1}{2!}a_i^2 [/tex]


    [tex] a_1 + \frac{1}{2!}a_1^2 + a_2 [/tex]


    [tex] a_3 + a_1 a_2 + \frac{1}{3!} a_1^3 [/tex]


    [tex] a_1 + \frac{1}{2!}a_1^2 + \frac{1}{3!}a_1^3+ a_2 + a_1 a_2 + a_3[/tex]


    [tex] a_4 + a_1 a_3 +\frac{1}{2!} a_2^2 + \frac{1}{2!} a_1^2 a_2 + \frac{1}{4!} a_1^4[/tex]


    [tex] a_1 + \frac{1}{2!}a_1^2 + \frac{1}{3!}a_1^3 + \frac{1}{4!}a_1^4 + a_2 + \frac{1}{2!} a_2^2 + a_1 a_2 + \frac{1}{2!}a_1^2 a_2 + a_3 + a_1 a_3 + a_4[/tex]

    As you can see, this seems to be systematically giving us the terms of the exponential.

    Any thoughts on how to prove this generally?
    Last edited: Oct 24, 2011
  2. jcsd
  3. Oct 24, 2011 #2


    User Avatar
    Science Advisor

    Suggestion: this is a math question. Why not move it there?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook