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Reversing substitution

  1. Nov 24, 2012 #1
    Hi guys... I'm probably missing something pretty basic here but I can't seem to figure this out. I was working on a problem recently: for the complex functions f(z)=ez and g(z)=z, find their intersections. This post is not about the problem, it is about something I noticed while tackling it (incorrectly).

    Anyways, here's what I noticed: If you set these functions equal to each other, you get


    So, naturally:


    From here I saw that a basic substitution was applicable, so the equation can be rewritten:


    Basically, what I have shown is that the function h(z)=ez-z has the same zeroes as the function i(z)=ez-ln(z). Now here's what's troubling me: what if the original problem that I gave you was to find the zeroes of i(z)? Originally, we obtained i(z) from h(z) by using a substitution, but is there some way that we can go in reverse from i(z) to h(z) using a "reverse substitution"? I'm sorry if this is rather unclear. Is there anything fundamental that I am missing?

    Thanks a lot
  2. jcsd
  3. Nov 24, 2012 #2


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    You can do, pretty much any whacky thing you want because everything you do starts from the assumption that z is a real number such that [itex]e^z= z[/itex] and there is NO such number.
  4. Nov 24, 2012 #3
    Note that we are working with complex functions. By the way, I did find an answer to this problem, right now I am just thinking about reversing the substitution
  5. Nov 24, 2012 #4


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  6. Nov 25, 2012 #5
    pwnsnafu, like I said, I already solved the equation using the labert w function. Please read my first post carefully.
  7. Nov 25, 2012 #6
    I guess this is kind of unclear. Here is the problem stated more clearly:

    Prove that the system of f(z)=ez and g(z)=ln(z) has the same solutions as the system of f(z)=ez and h(z)=z
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