1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Details about the function n/ln(n)

  1. May 17, 2012 #1
    From the file attached I would like to know the following.
    if Y=n/ln(n), Is there a way of explicitly expressing n in terms of Y.


    Relations I found are:

    There are 2 values of n for every Y. Except at Y=e , the two values converge to n=e.
    If n1 and n2 are the values of n
    then
    n1^n2=n2^n1.

    So is there a way of finding n1, given n2?

    What could be the possible type of functions involved?
    From the graph it is seen that |n1-e| and |n2-e| are related inversely.
    What could be this relation?
     

    Attached Files:

  2. jcsd
  3. May 18, 2012 #2

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    No, there's no way to express n as a function of y using any standard functions. Of course, it's always possible to invent a new one for the purpose.
    Mixtures functions of different types (polynomial, exponential, logarithmic, trigonometric..) are nearly always impossible to invert. E.g. y = x.exp(x), y = sin(x)/x, ...
     
  4. May 20, 2012 #3

    Mute

    User Avatar
    Homework Helper

    There is a special function called the Lambert-W function (aka the Product-Log) which you can use to write n in terms of y. The Lambert-W function is the function w=W(x) such that
    x = w exp(w).

    If you invert your equation so that 1/y = ln(n)/n, and then let n = exp(a), this gives

    [tex]\frac{1}{y} = a e^{-a}[/tex]

    We see that if we multiply both sides by -1 this will be in Lambert-W form, giving a = W(-1/y). Inverting n = exp(a), this gives

    [tex]n = \ln W_k\left(-\frac{1}{y}\right).[/tex]

    Some very important notes: The Lambert-W function has two real-valued branches, corresponding to k = 0 and k = -1. Usually the k=0 branch corresponds to the desired solution. The other branches gives complex values for W, so inverting ln n = W(-1/y) is more complicated if want solutions for these others branches. However, since you seem mostly interested in real values you don't need to worry about this.
     
  5. May 22, 2012 #4
    Thanks 'Mute'. Got my way through.

    tuhinrao
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Details about the function n/ln(n)
  1. Partition Function p(n) (Replies: 20)

  2. Confusion about -x^n (Replies: 7)

  3. P-series ln(n)/n (Replies: 4)

Loading...