# Composition of a function with itself

#### Andre' Quanta

Is possible to study the composition of a function f with itself when the number of compositions goes to infinity? I am interessed in the functions that can be written as a complex exponential of the function itself. Where i can study this kind of things?

#### micromass

Is possible to study the composition of a function f with itself when the number of compositions goes to infinity?
This is studied in dynamical systems and chaos theory.

I am interessed in the functions that can be written as a complex exponential of the function itself.
Like functions such that $f(x) = e^{if(x)}$ for each $x$?

Gold Member

#### Andre' Quanta

Thank you for the answer.
I am interesed in a function that can be written in this way f(x)=c*exp{i*k*x*f(x)} where c and k are costants and f is a function fom R to R and x is the variabile.
Does exist a similar function? If the answer is positive, does it have special propriety?

#### disregardthat

Science Advisor
Thank you for the answer.
I am interesed in a function that can be written in this way f(x)=c*exp{i*k*x*f(x)} where c and k are costants and f is a function fom R to R and x is the variabile.
Does exist a similar function? If the answer is positive, does it have special propriety?
If the function is differentiable with continuous derivative, then it must be constant, as you can see by differentiating both sides.

#### Andre' Quanta

Why it is a costant? If i take the derivative of that expression i get a differential equation for f(x), easy to resolve only respect to x: so it doesn' t seems to me that the solution is a costant

#### pasmith

Homework Helper
Why it is a costant? If i take the derivative of that expression i get a differential equation for f(x), easy to resolve only respect to x: so it doesn' t seems to me that the solution is a costant
From your earlier post:

Thank you for the answer.
I am interesed in a function that can be written in this way f(x)=c*exp{i*k*x*f(x)} where c and k are costants and f is a function fom R to R and x is the variabile.
Does exist a similar function? If the answer is positive, does it have special propriety?
If $f : \mathbb{R} \to \mathbb{R}$ and $k$ is real, then taking the modulus of both sides yields $|f(x)| = |c|$. If $f$ is to be continuous then either $f(x) = |c|$ or $f(x) = -|c|$, ie. $f$ is constant.

If instead $f: \mathbb{C} \to \mathbb{C}$ then you might be able to find a non-constant $f$ which satisfies that equation.

#### Andre' Quanta

From your earlier post:

If $f : \mathbb{R} \to \mathbb{R}$ and $k$ is real, then taking the modulus of both sides yields $|f(x)| = |c|$. If $f$ is to be continuous then either $f(x) = |c|$ or $f(x) = -|c|$, ie. $f$ is constant.

If instead $f: \mathbb{C} \to \mathbb{C}$ then you might be able to find a non-constant $f$ which satisfies that equation.[/QUOTE
Thanks, you are right :)
Now i am really interessed in find those complex functions that satisfies that condition

#### disregardthat

Science Advisor
If f(x) is differentiable, even as a complex function, yields
$f'(x) = ikf'(x)e^{ikf(x)} \Rightarrow f'(x)(1-ike^{ikf(x)}) = 0$
So we either have $f'(x) = 0$, or $1 = ike^{ikf(x)} \Rightarrow log(ik)+ ikf(x) = 0$
If it has a continuous derivative, then this implies that f is constant.

If you're dealing with analytical functions for example, then your function is infinitely differentiable hence has a continuous derivative, leaving only constant functions.

#### Andre' Quanta

If f(x) is differentiable, even as a complex function, yields
$f'(x) = ikf'(x)e^{ikf(x)} \Rightarrow f'(x)(1-ike^{ikf(x)}) = 0$
So we either have $f'(x) = 0$, or $1 = ike^{ikf(x)} \Rightarrow log(ik)+ ikf(x) = 0$
If it has a continuous derivative, then this implies that f is constant.

If you're dealing with analytical functions for example, then your function is infinitely differentiable hence has a continuous derivative, leaving only constant functions.
That is not my function: mine is f(x)=exp{ik x f(x)} you have forgotten the x

#### wabbit

Gold Member
Your equation can be written $x=\frac{1}{ikf(x)}\ln\frac{f(x)}{c}$ so you are trying to find an inverse to the function $g(y)=\frac{1}{iky}\ln\frac{y}{c}$, which will not exist globally but is possible for well chosen domains.

I wonder if you might be able to relate f to Lambert's W function, the equation is somewhat similar?

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#### disregardthat

Science Advisor
That is not my function: mine is f(x)=exp{ik x f(x)} you have forgotten the x
Oops, I was looking at the post above yours. Wabbit is right, however. The Lambert's W function is defined as the inverse of $y = xe^x$, so that $W(x)e^{W(x)} = x$. Since $y = xe^x$ is neither injective nor surjective, W is really "doubled valued", or we have to choose a branch of W. Now, the formula

$f(x) = ce^{ikf(x)}$ really asks for solutions to $y = ce^{iky}$. Rearranging, we get $-ikye^{-iky} = -ikc$. Substituting $z = -iky$, we get $ze^{z} = -ikc$. Thus $z = W(-ikc)$. I.e. $f(x) = y = \frac{iz}{k} = \frac{iW(-ikc)}{k}$. While the Lambert W function is double valued, any continuous solution for f(x) must be constant. The constant value of f(x) may be chosen to be any value of $\frac{iW(-ikc)}{k}$ (if it exists). I am unsure if such a complex value always exists.

#### wabbit

Gold Member
OP's equation however is $f(x)=c e^{ikxf(x)}$, not $f(x)=c e^{ikf(x)}$, so the possible link with Lambert's W function isn't as direct as that, and the solution is not constant if complex valued functions are allowed.

#### disregardthat

Science Advisor
OP's equation however is $f(x)=c e^{ikxf(x)}$, not $f(x)=c e^{ikf(x)}$, so the possible link with Lambert's W function isn't as direct as that, and the solution is not constant if complex valued functions are allowed.
I can't believe I made that mistake twice.

#### disregardthat

Science Advisor
Returning to the correct type of function: $f(x) = ce^{ikxf(x)}$. Like above, we are asking for a solution to $y = ce^{ikxy}$. Rearranging, we get
$ye^{-ikxy} = c$. So $-ikxye^{-ikxy} = -ikcx$. Substituting $z = -ikxy$ we get $ze^z = -ikcx$. Thus $z = W(-ikcx)$. Plugging back yields $-ikxy = W(-ikcx)$, so $y = \frac{W(-ikcx)}{-ikx}$.

Hence $f(x) = \frac{W(-ikcx)}{-ikx}$. Thus any branching of $W$ will yield this unique solution for a continuous $f$. Here of course we assume that both x and k are non-zero. It is likely possible to extend the domain of this function to include x = 0, by putting f(0) = 0, possibly depending on the branch chosen for W. In any case, it will depend on the behavior of f(x) as x approaches 0.

#### wabbit

Gold Member
Indeed ! I missed that, the link with W is much closer than I was suspecting.

$f(0)=c$ for any solution however, from the equation itself.

And to wrap it up, if $k$ is real then the only real-valued continuous solution is the constant $f(x)=c$ since $\forall x, |f(x)|=|c|$ and $f(0)=c$

#### disregardthat

Science Advisor
$f(0)=c$ for any solution however, from the equation itself.
Right, so depending on the behavior of W around x = 0, we may possibly continuously extend the domain to x = 0 by putting f(0) = c.

#### disregardthat

Science Advisor
Just to add to this, We have from the equation $W(z)e^{W(z)} = z$ that $W'(z)e^{W(z)}+W'(z)W(z)e^{W(z)} = 1$, so that $W'(z) = \frac{1}{z+e^{W(z)}}$. Since $W(0) = 0$, we may apply l'hopital to the expression

$\lim_{x \to 0} \frac{W(-ikcx)}{-ikx} = \lim_{z \to 0} \frac{W(cz)}{z} = \lim_{z \to 0} \frac{cW'(cz)}{1} = \lim_{z \to 0} \frac{c}{z+e^{W(z)}} = c.$

So we may extend the definition of f(x) to x = 0 by putting f(0) = c.

#### Andre' Quanta

To sum up: my function is only a costant everywhere?

#### disregardthat

Science Advisor
To sum up: my function is only a costant everywhere?
Not constant, it will have the form $f(x) = \frac{W(-ikcx)}{-ikx}$ for any choice of branching of the lambert W-function. You may also extend it to x = 0 by putting f(0) = c. It is only constant if c = 0, or k = 0.

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