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kitto6
 Post subject: hyperbolic functions
PostPosted: Sun, 4 Sep 2011 12:07:50 UTC 
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hi i just need some help with manipulating hyperbolic functions!

express: sinh^5x in terms of hyperbolic sines of multiples of x
&
express the following as an algebraic function of x: sinh(arccoshx)

i cancelled this down to
x + sqrt(x^2-1) - (x + sqrt(x^2-1))^-1
but just end up with very messy answers that i cant simplify! :(
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outermeasure
 Post subject: Re: hyperbolic functions
PostPosted: Sun, 4 Sep 2011 13:44:27 UTC 
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kitto6 wrote:
hi i just need some help with manipulating hyperbolic functions!

express: sinh^5x in terms of hyperbolic sines of multiples of x
&
express the following as an algebraic function of x: sinh(arccoshx)

i cancelled this down to
x + sqrt(x^2-1) - (x + sqrt(x^2-1))^-1
but just end up with very messy answers that i cant simplify! :(


(1) Just expand and simplify.
(2) Hint: What is (x+\sqrt{x^2-1})(x-\sqrt{x^2-1})?

_________________
\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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