Articles of hyperbolic functions

Integration Using Hyperbolic Substitution

I’m supposed to prove the following by using the hyperbolic sine double angle identity: $\sinh(2x)=2\sinh x\cosh x$ and archsinhx formula: $\ln(x+\sqrt{x^2+1})$ but can’t seem to figure out the steps. Prove: $\int\sqrt{x^2+a^2}\,dx = \frac{a^2}2 \ln\left(x+\sqrt{x^2+a^2}\right) + \frac{x}2 \sqrt{x^2+a^2} + C$ So far what I have is: $$\int\sqrt{x^2+a^2}dx = a^2\int\cosh^2udu\ \ (\text{substitute }x=\operatorname{asinh}u)$$ $$ = \int(1+\sinh^2u)\ du […]

Identity between $x=y+z$ and $\tan\left(\frac{\theta}{2}\right)=\tan\left(\frac{\nu}{2}\right)\tan\left(\frac{\pi/2-\epsilon}{2}\right) $

I would like to prove that (1) $$\begin{equation} \tan\left(\frac{\theta}{2}\right)=\tan\left(\frac{\nu}{2}\right)\tan\left(\frac{\pi/2-\epsilon}{2}\right) \end{equation}$$ can transformed to (2) $$x=y+z,$$ where (3) \begin{align} x&=&\mathrm{arctanh}\left(cos(\theta)\right)\\y&=&\mathrm{arctanh}\left(cos(\nu)\right)\\z&=&\mathrm{arctanh}\left(\sin\left(\epsilon\right)\right) \end{align} By solving for $\theta$ in 1 and 2, we see that these are indeed equal: For the record, incorrect identity Initially the question was wrongly stated, and the comments below pertain to this: I would […]

How to prove $\tanh ^{-1} (\sin \theta)=\cosh^{-1} (\sec \theta)$

As the question says How to prove $$\tanh ^{-1} (\sin \theta)=\cosh^{-1} (\sec \theta)$$ I have tried to solve it The end result that got for RHS $$=\log \frac{1+\tan\frac{\theta}{2}}{1-\tan \frac{\theta}{2}}$$ I am stuck here Please help

The $n$th derivative of the hyperbolic cotangent

While working on a problem, I came to this: What is the $n$th derivative of the hyperbolic cotangent? For simplicity, let $c=\coth(x)$. $c^{(0)}=c$ $c^{(1)}=-c^2+1$ $c^{(2)}=2c^3-2c$ $c^{(3)}=-6c^4+5c^2-2$ $c^{(4)}=24c^5-34c^3+10c$ Etc. It appears to be representable as a polynomial of $c$. Any ideas on what the coefficients are? Update: It appears the leading coefficient is trivially given by […]

Simplifying hyperbolic compositions like $\sinh (N \operatorname{acosh} a)$

In many occasions, we may meet hyperbolic functions, as well as their combined ones. I want to simplify expressions like $$ \tanh\left( N\left(\textrm{acosh}~ a\right)\right) $$ and $$ \sinh\left( N\left(\textrm{acosh}~ a\right)\right) $$ for any positive integer $N$. My WAY: By the substitution $b=\textrm{acosh}~ a$, then $a=\cosh b$, and we have $$ \tanh\left( N\left(\textrm{acosh}~ a\right)\right) = \tanh\left( […]

Prove that $|\sin z| \geq |\sin x|$ and $|\cos z| \geq |\cos x|$

For any $z=x+iy$, prove the following: $$|\sin z| \geq |\sin x|$$ $$|\cos z| \geq |\cos x|$$ $\epsilon$-$\delta $ proof is not required. I don’t really know how to proceed. I know in order to remove the absolute values I can square both sides and I have tried proving this statement using the hyperbolic forms and […]

Integral $ \int_{0}^{\pi/2} \frac{\pi^{(x^{e})}\sin(x)\tan^{-1}(x)}{\sinh^{-1}\left({1+\cos(x)}\right)} dx$

I need help in evaluating the following integral :- $$ \int_{0}^{\pi/2} \frac{\pi^{\displaystyle (x^{e})}\sin(x)\tan^{-1}(x)}{\sinh^{-1}\left({1+\cos(x)}\right)} dx$$ A brief solution would be very much appreciated.

How do we prove this asymptotic relation $\lim\limits_{k\rightarrow \infty}G(k)=\sqrt{2}$

I made the following observation with Mathematica: Consider the infinite series,for natural number $k$ $$G(k)=\sum_{n=1}^{\infty}\frac{\sinh^{n}(\sqrt{k}\pi)}{\sqrt{(k+1)+\cosh{(2\sqrt{k}\pi n)}}}$$ then $$\lim\limits_{k\rightarrow \infty}G(k)=\sqrt{2}$$ How do we prove this asmptotic relation $G(k)\sim \sqrt{2}$?

Why is $\sum_{k=1}^\infty (-1)^{k+1}\frac{ \text{csch}(\pi k)}{k} = \frac{1}{2}(\ln(64) – \pi)$?

While answering this question I found through Wolfram Alpha that $$\sum_{k=1}^\infty (-1)^{k+1}\frac{ \text{csch}(\pi k)}{k} = \frac{1}{12}(\ln(64) – \pi).$$ Sadly Wolfram Alpha just spits the summation value out without any justification or references. Is there an explanation or reference for this identity?

Reduction of $\tanh(a \tanh^{-1}(x))$

Given $x\in \Re$, $a \in \Re$ where $-1 \le x \le 1$ and $0 \le a \le 4$, is it possible to reduce the following expression: $\tanh(a \tanh^{-1}(x))$ E.g. to some kind of polynomium? I know that if $a$ is an integer then the following holds: $\tanh(0 \tanh^{-1}(x)) = 0$ $\tanh(1 \tanh^{-1}(x)) = x$ $\tanh(2 […]