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November 6th, 2017, 02:06 AM   #1
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Simplification of a formula

We consider a fixed parameter $\theta>0$.

For all $t>0$ we note:

$$u(t)=\frac{\sinh\big(\frac{t}{2}\cosh(\theta)\bi g)}{\cosh(\theta)}$$

$$A(t)=\frac{\sqrt{\cosh^2(\theta)u^2(t)+1}-1}{\cosh^2(\theta)}-\Big(\cosh\Big(\frac{t}{2}\Big)-1\Big)$$

$$f(t)=-\ln\Big(1-\frac{2A(t)}{u(t)+\sinh(t)+A(t)}\Big)$$

For all $\theta>0$ the equation $$2\sinh\Big(cosh(\theta)t\Big)\sinh\Big(t\Big)=1$ $ where $t>0$
has a unique solution that we note $t^*>0$.


I want to know if we can simplify $f(t^*)$ or either find a constant $c>0$ such that $f(t^*)\ge c$??

Please help me . Thanks.
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