I just read this question about finding the solution to the equation $\ln(x) = \sin(x)$. All the answers focus on using a numerical method to approximate the solution. This is interesting in its own regard, but I am interested in why this is the case. Several questions immediately jump out at me:
- Is it really the case that there is no analytic expression that represents the solution? Are there general techniques one can use to prove this is the case for similar equations?
- Is there a connection between these types of solutions and transcendental numbers? Is $x$ a transcendental number?
- Are there additional complex solutions? Are they more easily expressed? I am not very familiar with complex analysis.
For the purposes of this question, I will follow the wikipedia definition of an Analytic Expression. This allows the following constructions in the expression: Constant, Variable, Elementary arithmetic operation, Factorial, Integer exponent, N-th root, Rational exponent, Irrational exponent, Logarithm, Trigonometric function, Inverse trigonometric function, Hyperbolic function, Inverse hyperbolic function, Gamma function Bessel function, Special function, Continued fraction, Infinite series
If it is more interesting, what about just a 'closed-form', which allows everything up to the Gamma function in the above list.
EDIT: I've attempted to use PARI to calculate the continued fraction without much luck. I used the command "x = exp(sin(x))", which returns a truncated power series. I can't really find anything in the documentation for how to get this as a continued fraction; simply doing "contfrac(x)" just returns the power series again inside of some square brackets.
Even if I could get it to display a continued fraction, it seems that PARI is resorting to a numerical method to calculate an approximate value and then displaying this approximate value in different representations. I believe this means the continued fraction series in not simply a truncated version of the "real" (probably) infinite continued fraction, but the entire finite sequence representation of the approximate value. If this is the case, then I cannot look at the sequence to search for a pattern...
Thus question 4. Is there a numerical method for generating the continued fraction?