Articles of hyperbolic functions

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 […]

If $f$ is holomorphic and $\,f'' = f$, then $f(z) = A \cosh z + B \sinh z$

Suppose $f$ is holomorphic in a disk centered at the origin and $f$ satisfies the differential equation $$f” = f.$$ Show that $f$ is of the form $$f(z)=A \sinh z + B \cosh z,$$ for suitable constants $A,B\in \mathbb C$. Of course if $f$ were a function of one real variable, usual techniques to solve […]

Integral of the ratio of two exponential sums

I am trying to find a lower bound on the following integral \begin{align*} \int_{y=-\infty}^{y=\infty} \frac{ (\sum_{n=[-N..N]/\{0\}}n e^{-\frac{(y-cn)^2}{2}})^2} {\sum_{n=[-N..N]/\{0\}} e^{-\frac{(y-cn)^2}{2}}}dy \end{align*} I have been trying to find the upper and the lower bounds on the summations in here. I have also look at the Jacobian-Theta functions but I wasn’t able to find theta function that would […]

An alternating series identity with a hidden hyperbolic tangent

How does one use the inverse Mellin transform to prove that the following identity holds? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi – 5\log(2))$$ The identity follows from MO189199 and the penultimate identity on this page (for $x=\frac{1}{2}$). Scratch-work: I computed the Mellin transform of $$f(x) = \frac{(-1)^x}{x(e^{x\pi} + 1)}$$ and re-wrote the function in terms of […]