Some thoughts on solving transcendental equations

My thinking here is that although transcendental equations don’t have algebraic answers are there still links between them?

That theory is supported by the contour lines in the following 

So I wasn’t going to look into this but sometimes I just get captured by the mathematics.

I have known for a while that most equations can’t be solved. The unsolvability of most equations leads to transcendental numbers. Numbers which aren’t the result of a solvable algebraic equation like 5x+1=7 or 2x^2=3 but can be computed, at least approximately.

These numbers include pi and e.

There isn’t a formula for pi or e, we can define them, we can even define them without the use of geometry. As I have found out in my curious investigations both numbers seem to crop up only when calculus is involved.

e is the solution to the problem of

The functions that produce these two have always seemed arbitrary to me. Why is it the case that when we don’t know the answer to we just say, the answer is the answer?

We seem to do the same with solving for x. We don’t know so we say, the solution function gives you the answer. We just let there be a function such that f(c)=x.

Where does this stop? When do we stop defining functions to be the answer to the problem we are solving rather than actually doing mathematics.

It seems to me that the answer is we try to create the fewest functions possible.

We could define a function such that given the problem such that f(c) = x.

That might be useful but the reason we would want to stick to one such function is that given a little more exploration of it, we can show that the solution to the second problem can be phrased in terms of the first. Here the solution is . (Using our original function).

It turns out that creating a result here is useful in a lot of places.

Even in a post lookup table world. We could just use numeric methods for everything, and we have to when we want an actual result.

One problem which got me wondering on the topic of just going around defining the solutions to the problem was the problem which is quite rampant in (another blog post), xsinx = c.

I have started to look into what happens when you simply create an inverse function such that f(x) sin f(x) = x.

We can start to find out more about this function from the context for example we know that the solution to f(x)=x is 0. We, therefore, know f(0)=0. The forward function here is very easy to calculate for values that we know for sine. We can then work backwards from the result of that creating a map of solutions.

What we need to know is does this mapping of a solution give us any more information?

That I haven't investigated yet.