# Computing the Omega constant

I) Solving a simple-looking equation:

Last Sunday I was trying to compute a complicated integral suggested to me by a friend and was led to consider the problem of solving the following innocent-looking equation:

$x\log(x)=1 \quad (1)$

Little did I know that this little problem would lead me to transcendental functions and the Lambert W function in particular. But, before saying more on those two topics I’ll go through what I tried:

1. Given that $x \in (1,e)$, I tried $x:=e^\alpha$ for $\alpha \in [0,1]$ so that I obtain $\alpha e^\alpha=1$. This resembles the familiar exponential function $f(t)=e^{\alpha t}$.

2. I tried to recover some useful relations and I found the following:

$\forall t \in [0,1]\quad f^n(t)f^{n+1}(t)=\alpha^{2n} \quad (2)$

Using the mean-value theorem:

$\int_{0}^{1} f^n(t)dt=\alpha^{n-1}(\frac{1}{\alpha}-1) \quad (3)$

which isn’t too difficult to show given that

$latex \begin{cases} \exists c_n \in (0,1) \quad f^{n+1}(c_n)=f^n(1)-f^n(0)=\alpha^n f'(c_0) \\ c_0 = \frac{1}{\alpha}\log(\frac{1}{\alpha^2}-\frac{1}{\alpha})\\ \end{cases} &s=1$

After this I still couldn’t figure out how to solve for $\alpha$ analytically. I wondered whether there might be a relation which I failed to discover but at this point it was a bit late so I thought I might want to try a numerical method.

3. Using Newton’s method:

$\large t_{n+1}=t_n-\frac{g(t_n)}{g'(t_{n+1)}}=t_n-\frac{t_n\ln(t_n)+t_n^2}{1+t_n} \quad (4)$

where $g(t)=\log(t e^t)$

I found that $\alpha \approx 0.567143$ and wondered whether it might be related to other numbers I knew. I tried a few calculations which got me nowhere and then I became more interested in whether this problem could be solved without resorting to numerical methods so I asked this on the math stackexchange. From that forum I learned that $\alpha$ is actually known as the Omega constant and it appears as $W(1)$ where $W(x)$ is the Lambert W function, a transcendental function.

II) An equation that can’t be solved analytically:

A transcendental function, $f(x)$ is basically a function that transcends algebra in the sense it can’t be expressed in terms of a finite number of algebraic operations: addition, subtraction, multiplication, division, raising to a power, and taking roots. In particular it can’t satisfy:

$\sum_{k=1}^{n} c_k(x) (f(x))^k = 0 \quad (5)$

where $c_k(x)$ are rational functions(i.e. a ratio of polynomials).

Now, it isn’t too difficult to show that:

$t(x) =e^{x\log(x)}=x^x \quad (6)$

is actually transcendental if we suppose that $t(x)$ actually satisfies (5). If we choose $x \geq 1$, $|t(x)| \geq 1$ we find that

$latex \begin{cases} |c_n(x)| |t(x)|^n \leq b(x) |t(x)|^{n-1} \\ b(x) =\sum_{k=1}^{n} |c_k(x)| \\ \end{cases} \quad (6) &s=1$

It then follows that

$\forall x \in \mathbb{R}, |t(x)| \leq \frac{b(x)}{|c_n(x)|} \quad (7)$

However, in (7) the expression on the left grows exponentially while the expression on the right has at most polynomial growth so $t(x)$ can’t be algebraic and hence can’t be solved analytically.

I’ll write a separate blog post on the Lambert W function as I think this blog post is getting a bit long and that function is worth it’s own post as it has many applications.