Re: How do you solve Int( x to 2*x +1 exp(g(u))du ) = Pi ?

From: Gaetano Barbaro (ingegno07_at_libero.it)
Date: 09/29/04 

Date: Wed, 29 Sep 2004 17:00:04 +0000 (UTC)

ALAIN Verghote wrote:
>Dear friends,
>
>I will be pleased to know how you handle such an equation.
>Here limits are x and 2*x +1 ,integration with u .
>g is R->R ,continuous. Pi =3.14...
>
>Thanks for ideas...Alain.

Ciao Alain.

I think that your equation should be handled as follows.
Let us suppose that F(x) is a function such that its first derivative
                       dF(x) = exp(g(x))
Therefore, your equation to be solved is:
                        F(2x+1)=F(x)+PI
The most general solution I have found for this functional equation
is:
            F(x)=PI*ln(x+1)/ln2 + T(ln(x+1)/ln2) + constant
wherein T(.) is a periodic function with period 1.
Thus the function g(x) is
                        g(x) = ln(dF(x))

I'm looking for a book about the solution of functional equations, in
particular the equations of the type:
                        H(F(x))=F(G(x))
wherein F(x) is the function to be calculated and H(.) and G(.) are
two known functions. Such a book should clearly illustrate which kind
of functional equations are still unsolved and which techniques of
solutions have been devised so far.
Could you suggest any book on this topic?

Gaetano

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