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.

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