Asymptotics of $(\cosh(x+c)-\cosh(c))^{-\frac{1}{2}}$

Question

let $c>0$ be a constant and consider the function $$\frac{1}{\sqrt{\cosh(x+c)-\cosh(c)}}, x>0.$$

I'm wondering how the asymptotic expansion for $x\downarrow 0$ look like!? In case of $c=0$ the asymptotics is quite easy. Unfortunately the proof cannot be applied to this case. How can I get the asymptotics?

Answer 1

Use the Taylor expansion: $$ \cosh(x+c)=\cosh c+(\sinh c)\,x+O(x^2). $$ Then $$ \sqrt{\cosh(x+c)-\cosh c}=\sqrt{(\sinh c)\,x+O(x^2)}=\sqrt{(\sinh c)\,x}+O(x^{3/2}) $$ and $$\frac{1}{\sqrt{\cosh(x+c)-\cosh(c)}}=\frac{1}{\sqrt{(\sinh c)\,x}}+O(x^{1/2}).$$