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!

About Recursive Formula

  1. Oct 3, 2012 #1


    User Avatar

    Hey guys,

    I would like to get some general form of recursive formula

    let f(s) = (a+bs)/(c+ds)

    given this I would like to get nth composite function of f

    i.e the general form of f^n(s)= fofofofofo........of(s) (nth composite)

    I can conjecture that the form of f^n(s) would be the same form with f(s) like

    f^n(s) = (α+βs)/(γ+δs)

    and α,β,γ,δ are would be expressed as function of a,b,c,d and n

    So, I would like to get the general from of α,β,γ,δ with the function of a,b,c,d and n.

  2. jcsd
  3. Oct 3, 2012 #2


    User Avatar
    Homework Helper

    Your function is called a Mobius transformation. By looking up references about this function you may be able to find some information about the composition of functions that you want to find. A brief skim of the wikipedia article didn't yield a discussion of composition of the function, but I did see a section about fixed points of the transformation, which is related to composing the function infinitely many times.
  4. Oct 3, 2012 #3


    User Avatar
    Science Advisor
    Gold Member


    Your conjecture is correct. The solution ends up consisting of matrix multiplies. For example, if
    z_1 = \frac{a z + b}{c z + d}
    z_2 = \frac{a_1 z_1 + b_1}{c_1 z_1 + d_1}
    then if we want to write
    z_2 = \frac{a_2 z + b_2}{c_2 z + d_2}
    we have
    \left( \begin{array}[cc] \\ a_2 & b_2 \\ c_2 & d_2 \end{array} \right)
    \left( \begin{array}[cc] \\ a_1 & b_1 \\ c_1 & d_1 \end{array} \right)
    \left( \begin{array}[cc] \\ a & b \\ c & d \end{array} \right) .

    You should work through the algebra yourself to verify that I didn't make a mistake. Composition of more than two of these leads to simply more matrices to multiply. In your case, [itex]a_1=a, b_1=b,[/itex] etc. so you would simply have an nth power of a matrix. If your matrix can be diagonalized then this nth power is relatively simple.

    By the way, this functional form is often called a Mobius (or bilinear) transformation. If you have ever learned about conformal mapping, you may recall that it maps circles/lines in the complex plane to other circles/lines.

    Last edited: Oct 3, 2012
  5. Oct 4, 2012 #4


    User Avatar

    Thanks buddy! It is really helpful.
    I will try to get the specific form!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook