anonymous
  • anonymous
(iii) Find the most general function that, for all x and y, satisfies the identity t(x) + t(y) = t(z) where z = (x+y)/(1-xy) I can't think of how to do this. I can see some relation to the t=tan((1/2)A) substitution where sin(A) = (2t)/(1-t^2) but don't know how to use it in this context.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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JamesJ
  • JamesJ
tan(a+b) = ( tan a + tan b )/(1 - tan a.tan b) Let t = arctan, x = tan a and y = tan b and you now have an example of this function t. Now, the find the most general form, you're probably going to have to have a look at 2-dimensional Taylor series.
anonymous
  • anonymous
t(x) = arctan(x) ? Now I'm confused because I can't see how this satisfies t(x) + t(y) = t(z) still? :S I think it's the manipulation of these general functions which I am messing up on, it's something I'm unfamiliar with.. Thanks so much for your help James.
anonymous
  • anonymous
Nevermind I understand it now, you have no idea how much this is going to help me James haha.

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anonymous
  • anonymous
2-dimensional Taylor series? Is there any other way? I've only looked at single dimensional taylor series
anonymous
  • anonymous
(as far as I know)
JamesJ
  • JamesJ
It's subtle. I would dread doing the calculations to find the most general function. I think this class of function is good enough.
anonymous
  • anonymous
would you say t(x) = c . arctan(x) would be the general solution? since the constants just cancel out in the equation
JamesJ
  • JamesJ
Yes.
anonymous
  • anonymous
Thanks
JamesJ
  • JamesJ
Or at least that's one family of solutions. Technically, we should prove there are no other such functions. But I hypothesize that this family is sufficient.
anonymous
  • anonymous
Hmm, how would we do that..
JamesJ
  • JamesJ
Multi-dimensional Taylor series. I would ask your lecturer/tutor what they want you to do here.
anonymous
  • anonymous
Ok I will, thanks again.
anonymous
  • anonymous
The trivial solution will work. Let t(s)=0 for all s in R.
anonymous
  • anonymous
Just what I was about to add. z is undefined if xy=-1

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