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Why can't a^x = x^b be solved analytically?! watch
- 27-10-2015 23:47
- 27-10-2015 23:48
I've seen you around the MAT thread, so I presume you're applying for (maths?) at Oxford.
- 28-10-2015 00:07
(Original post by Callum Scott)
- 28-10-2015 01:02
After reading a little bit about it; how can you still have a useful function that isn't expressed in elementary functions? Is it defined in terms of derivatives/ integrals or something? How can someone put a value into it and get a value out?
The only real difference between things like sin x and the Lambert W function is that you're a lot more familiar with one of them that the other.
You may think "ah, but sin x is defined in terms of triangles and angles", but from the point of view of higher mathematics, that's actually not true (*). Because it turns out that it's really hard to connect our "real world" inttuition about things like angles with precise, rigourous mathematical definitions.
Instead, we usually define sin x to equal and then prove this definition does all the things we want.
Or (less commonly) we might define sin x to be the solution of the equation satisfying y(0) = 0, y'(0) = 1.
Which isn't really any different from defining W(z) to be the value such that .
(In fact, we can diff this to find , then multiply by W(z) to get , then if we replace by z we get , so so and so W(z) satisfies a not terribly scary looking differential equation)).
That said, it would be hypocritical of me if I didn't admit I'm far more comfortable with sin and cos than the W-function. But the existence of such a function certainly doesn't shake my foundations.
(*) I'm sure someone, somewhere has done a rigorous definition of sin/cos based on our intuitive ideas. But I'm also sure it isn't straightforward - there's a reason we normally go for a purely calculus based definition.
- 28-10-2015 08:51
Underlying the questions addressed in this thread are a couple of informal ideas that might be summed up by asking: what problems are amenable to "closed form solutions" in terms of "elementary functions"?
So, what is a "closed form solution" and and what is an "elementary function"? These two have been thought about in one form or another since at least the beginning of the nineteenth century. They are still "active" questions, not least because of the interest in computer algebra systems and the algorithms they use to give "closed form solutions".
For a recent survey on these questions, this is a nice article:
Borwein JM & Crandall RE. "Closed Forms: What They Are and Why We Care", Notices of the American Mathematical Association 60(1): 50–65
- 28-10-2015 10:08