Originally Posted by khh
That would make sense if I started out with the intention "I'm gonna test if these are equal". But I didn't. I started out with the intention "I'm gonna show this ragamuffin I'm right".
(I really should have known better, though. We've argued math before and I have yet to win)
Hah, oh, I see. I thought the graph was quite an elaborate method!
I don't believe there's any way of mathematically proving two functions are the same other than to just show the conversion from one form to the other in small, recognizable steps. Of course, proving them not to be equal can be much simpler. But if you find something interesting on the topic, please post it
Hm well, I just had one of those weird kind of moments were I realised I'd never tried to do this before, and had been using some shaky assumptions.
I thought disproving would be easy too, but then I realised this ends up involving, as a random example, showing that arctan(1/2) is not equal to e^2 or something; and I've been subliminally trained to treat calculators as abhorrent, so it's a lot harder than it seems.
The shaky assumptions I realised I had were that certain functions are simply fundamentally different from others, and so equivalences between them, or functions that involve them, can be dismissed; this only works if you know all of the important relationships between functions (which I'm sure I do by now), and are hence able to see when there is incompatibility. For instance, I 'know' that for real x, there are no identities linking trig, exponential, or algebraic functions. That's why I could discount your relationship.
However, for something like
f(x) = cos(x)
g(x) = 1/2*( exp(ix) + exp(ix) )
for complex x, it's not clear at all that the functions are identical (they are), and if you hadn't learn Euler's formula you'd probably think they were totally different, and discount this relationship on the same basis as me. It's never actually been proven to me that I can't turn a trig into an exponential or whatever, it's just that it'd be so fundamental that I would definitely know it.
Anyway, I think the proper, general basis for this kind of thing is Taylor series, which at a higher level are actually usually said to define exp, trig, etc. (rather than being derived). If you determine the Taylor series of two given functions and compare coefficients, that'll determine it for you, as Taylor series are unique.
Well, if it gives you pleasure, your little mistake opened up a whole can of worms for me and I've spent quite a long time Googling stuff now. The above is a basic answer, but then there's the problem of the fact that
 The problem of checking coefficients may well be no easier than checking values of the function in the first place, as you may get two different looking formulae for the coefficients. If your functions are the same, no algorithm will ever be able to tell you, because it'll be checking forever.
 You may also have to check infinite intervals of the function because some Taylor series only work for finite distances from a point, and I've just discovered that there are even functions where the Taylor series only converges for the point about which it is taken, along with a host of other analysis concepts I've never seen before...
Right, bed now.


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