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I Function as a Solution to Specific Conditions

  1. Nov 7, 2016 #1
    I'm trying to solve some statistical mechanics. This problem appeared.

    ##\frac{f\left(0\right)}{f\left(a\right)}f\left(x+a\right)=f\left(x\right)##

    Any idea as to which function will satisfy this equation?
     
  2. jcsd
  3. Nov 7, 2016 #2

    DrClaude

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    ##f(x) = Ce^{\pm x}##
     
  4. Nov 7, 2016 #3
    Nice. How did you work that out? Is an exponential function the only solution to this problem?
     
  5. Nov 7, 2016 #4

    DrClaude

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    From the observation that ##f(x) \propto f(x+a) / f(a)## requires that an addition in the argument becomes an (inverse) multiplication of the function, and I recognized the exponential. (I actually thought about log first, but then realized I had it backwards :smile:)

    No idea. Apart from the answer I gave and ##f(x) = \mathrm{const.}##, I don't see any.
     
  6. Nov 7, 2016 #5
    That's pretty neat. What branch of mathematics is that under?
     
  7. Nov 7, 2016 #6

    DrClaude

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    I'm not a mathematician. It just comes from years of working with, and getting a feeling for, those kind of mathematical functions.
     
  8. Nov 7, 2016 #7
    No, I was just wondering what topic or branch this question falls under.
     
  9. Nov 7, 2016 #8

    jedishrfu

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  10. Nov 7, 2016 #9

    mfb

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    The general solution is ##f(x) = Ce^{d x}##, C and d arbitrary constants. C and d can be complex if you work with complex numbers.
    It is a typical homework problem to show this.
     
  11. Nov 7, 2016 #10

    Stephen Tashi

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