Intereting Posts

Prove elements generate a free group
How prove there is no continuous functions $f:\to \mathbb R$, such that $f(x)+f(x^2)=x$.
Why is it called a quadratic if it only contains $x^2$, not $x^4$?
Is the natural map $L^p(X) \otimes L^p(Y) \to L^p(X \times Y)$ injective?
Which rationals can be written as the sum of two rational squares?
How is $ \cos (\alpha / \beta) $ expressed in terms of $\cos \alpha $ and $ \cos \beta $?
Condition on vector-valued function
Surreal numbers without the axiom of infinity
Power series solution of $ f(x+y) = f(x)f(y) $ functional equation
Characterizing injective polynomials
Dominated convergence for sequences with two parameters, i.e. of the form $f_{m,n}$
Relationship between the cardinality of a group and the cardinality of the collection of subgroups
Mapping Irregular Quadrilateral to a Rectangle
proving $ (A \rightarrow B \vee C )\rightarrow((A\rightarrow B) \vee (A\rightarrow C))$
Some equivalent formulations of compactness of a metric space

How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$? Is there a faster-converging method than power series for fractional iteration/functional square roots?

- Calculate the closed form of $\frac{\sqrt{5}}{\sqrt{3}}\cdot \frac{\sqrt{9}}{\sqrt{7}}\cdot \frac{\sqrt{13}}{\sqrt{11}}\cdot …$
- Find all continuous functions from positive reals to positive reals such that $f(x)^2=f(x^2)$
- How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?
- What is the expansion in power series of ${x \over \sin x}$
- Multiplicative Inverse of a Power Series
- Functional Equation (no. of solutions): $f(x+y) + f(x-y) = 2f(x) + 2f(y)$
- Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$
- The value of a limit of a power series: $\lim\limits_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$
- Let $f$ be an analytic isomorphism on the unit disc $D$, find the area of $f(D)$
- Functional Equation $f(mn)=f(m)f(n)$.

Look at this answer:

https://mathoverflow.net/questions/17605/how-to-solve-ffx-cosx/44727#44727

In short, the analytic solution is

$$g^{[1/2]}(x)=\phi(x)=\sum_{m=0}^{\infty} \binom {1/2}m \sum_{k=0}^m\binom mk(-1)^{m-k}g^{[k]}(x)$$

$$g^{[1/2]}(x)=\lim_{n\to\infty}\binom {1/2}n\sum_{k=0}^n\frac{1/2-n}{1/2-k}\binom nk(-1)^{n-k}g^{[k]}(x)$$

$$g^{[1/2]}(x)=\lim_{n\to\infty}\frac{\sum_{k=0}^{n} \frac{(-1)^k g^{[k]}(x)}{(1/2-k)k!(n-k)!}}{\sum_{k=0}^{n} \frac{(-1)^k }{(1/2-k) k!(n-k)!}}$$

Insert here $g(x)=a^x$ The same way you can find not only square iterative root but iterative root of any order.

Unfortunately this does not converge for $g(x)=a^x$ where $a > e^{1/e}$.

Here is a graphic for iterative root of $g(x)=(\sqrt{2})^x$

The question becomes more difficult when speaking about the base $a>e^{1/e}$. But in this case the solution can also be constructed, see this article.

Here’s the proof of a theorem due to Thron (1956), extracted from a article of Laurent Bonavero (available at his webpage).

**Theorem.** There is no entire function $f$ (that is $f:\mathbb C \to \mathbb C$ holomorphic) such that $\exp = f \circ f$.

*Proof.* If such a function $f$ exists, then $f(\mathbb C)= \mathbb C^*$. Indeed, $f(\mathbb C) \supset \exp(\mathbb C) = \mathbb C^*$, but $0$ can’t be in the image of $f$: if $f(x)=0$, then as $x \neq 0$, there exists $y$ such that $x=f(y)$ so that $\exp(y)=0$, absurd.

Therefore $f$ can be lifted by the exponential, $f=\exp \circ g \,$ for $g$ entire. So $\exp = \exp(g \circ f)$, and there must exist a constant $C$ such that $g \circ f(z)=z+C$ for all $z\in \mathbb C$. It follows that $f$ is injective, so $\exp$ should be injective too, which is absurd!

There is a lot of material about this question here and in mathoverflow. There is also a “Tetration forum”, where someone has implemented a version of tetration due to Hellmuth Kneser, see some forum entries there: http://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=8” also in citizendium there is an extensive article of Dmitri Kousznetzov who claims he has a usable interpretation (and implementation) see http://en.citizendium.org/wiki/Tetration

- When is the image of a null set also null?
- What's the probability of rolling at least $k$ on $n$ dice with $s_1,\ldots,s_n$ sides?
- Under what conditions is the product of two invertible diagonalizable matrices diagonalizable?
- Not understanding this proof in Grünbaum-Shephard's Tilings and Patterns
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- how to prove $m^{\phi(n)}+n^{\phi(m)}\equiv 1 \pmod{mn}$ where m and n are relatively prime?
- Proving something is $1$-Lipschitz
- Bounded stopping times and martingales
- Example of Homeomorphism Between Complete and Incomplete Metric Spaces
- Best representation of a polynomial as a linear combination of binomial coefficients
- Real roots plot of the modified bessel function
- Proof of $m \times a + n \times a = (m + n) \times a$ in rings
- Combinatorial proof of $\sum_{k=0}^{n} \binom{n+k-1}{k} = \binom{2n}{n}$
- Is the empty set linearly independent or linearly dependent?
- Indian claims finding new cube root formula