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Can I use inverse of the same function when defining the function itself?

Posted 10 years ago

Can I use inverse of the same function when defining the function itself? For example : b[x_]:=1+Integrate[InverseFunction[b][a],{a,0,x}] b[5] Typing that in I get an error: $RecursionLimit::reclim: Recursion depth of 256 exceeded.

A simpler example: b[x_] := 1 + InverseFunction[b][x] b[5] also gives me the same error.

I understand that it has to do with the fact that a function has its own inverse in its definition, which is not easy to solve (maybe not possible?)

Could you please give me some advice on what to do in the case that I want to solve a problem of this type (my actual problem is more complicated, but I wanted to know on the simpler example).

Are there any other ways to get a solution for this type of problem?

I'm new to Mathematica, so I'm sorry if I'm making some trivial mistakes, all the help would be very appreciated. I'm looking forward to your reply, thank you in advance!

POSTED BY: O. P.
2 Replies

If you're new to programming with mathematica, take a look at the virtual book which will give you some background on how Mathematica is used.

You're looking to solve a functional / Integral equation. So we're not going to use the notation for defining a function, but instead use the notation for a symbolic equality:

b[x] == 1 + InverseFunction[b][x] b[5]

If possible, remove the InverseFunction using some algebra which I hopefully did correctly:

b[(b[x]-1)/b[5]]==x

There's a functional equation you want to solve under the assumption that b has an inverse. That's tough to do. RSolve can manage some functional equations, but not this one. There aren't easy answers for functional equations and I don't know of any methods that'll solve that one.

Taking a look at the integral equation,

b[x]==1+Integrate[InverseFunction[b][a],{a,0,x}] b[5] 

The first thing you do with integral equations is to turn them into differential equations if possible.

b'[x]==b[5] InverseFunction[b][x]

Using the inverse function theorem.

b''[x]==b[5]/b'[b[x]] 

Assuming I didn't mess anything up, we get the differential equation above. This equation isn't easy to solve and DSolve won't be able to do it. If you make some assumptions maybe you can estimate it with a numerical method. Looking online, I don't see any information about solving this kind of equation, but some examples of disproving that a solution exists .

POSTED BY: Sean Clarke

Mathematically, I believe your question yields solving functional equation other than defining a function in terms of the its inverse. For example: the simplest is $f = f^{-1}$ which finds functions are symmetric about line $y=x$ like $line: x+y = 1$, hyperbola $ x y =1$ and so on. I would not expect anything meaningful by $ f \overset{def}{=} f^{-1}$.

POSTED BY: Shenghui Yang
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