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Reversing substitution 
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#1
Nov2412, 04:31 PM

P: 242

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)=e^{z} 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 e^{z}=z So, naturally: z=ln(z) From here I saw that a basic substitution was applicable, so the equation can be rewritten: e^{z}=ln(z) Basically, what I have shown is that the function h(z)=e^{z}z has the same zeroes as the function i(z)=e^{z}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
Nov2412, 05:05 PM

Math
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Thanks
PF Gold
P: 39,556

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.



#3
Nov2412, 06:57 PM

P: 242




#4
Nov2412, 11:57 PM

Sci Advisor
P: 834

Reversing substitution
This is related to the Lambert W function.



#5
Nov2512, 12:28 AM

P: 242




#6
Nov2512, 01:56 PM

P: 242

I guess this is kind of unclear. Here is the problem stated more clearly:
Prove that the system of f(z)=e^{z} and g(z)=ln(z) has the same solutions as the system of f(z)=e^{z} and h(z)=z 


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