New posts in hyperbolic-functions

In deriving the catenary equation, how does integrating $\frac{dy'}{\sqrt{1+(y')^2}}=\frac1a dx$ yield $\sinh^{-1}(y')=\frac{x}{a}$?

calculus trigonometry vector-analysis hyperbolic-functions

Solving the Laplace equation in terms of exponential of hyperbolic trigonometric functions

partial-differential-equations hyperbolic-functions

Proving $~\prod~\frac{\cosh\left(n^2+n+\frac12\right)+i\sinh\left(n+\frac12\right)}{\cosh\left(n^2+n+\frac12\right)-i\sinh\left(n+\frac12\right)}~=~i$

calculus closed-form infinite-product hyperbolic-functions

Geometric meanings of hyperbolic cosine and sine

geometry hyperbolic-functions

Calculate the sum: $\sum_{x=2}^\infty (x^2 \operatorname{arcoth}(x) \operatorname{arccot} (x) -1)$

trigonometry summation hyperbolic-functions

Number of zeros of $f(x)= \frac{1}{2} E\left[ \tanh \left( \frac{x+Z}{2} \right) \right]-\tanh(x)+\frac{x}{2}$ where $Z$ is standard normal

real-analysis probability roots hyperbolic-functions

Compute integral of general form $ \int_0^\infty \left(\frac{x}{\sinh x}\right)^n d x $

integration definite-integrals hyperbolic-functions

Integral of $\ln(x)\operatorname{sech}(x)$

integration definite-integrals improper-integrals gamma-function hyperbolic-functions

How do you prove Osborn's rule?

trigonometry solution-verification hyperbolic-functions

Can the real and imaginary parts of $\dfrac{\sin z}z$ be simplified?

complex-analysis trigonometry roots hyperbolic-functions

Show $\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$ and another

calculus real-analysis sequences-and-series hyperbolic-functions

Showing $\int_0^{\int_0^u{\rm sech}vdv}\sec vdv\equiv u$ and $\int_0^{\int_0^u\sec vdv}{\rm sech} vdv\equiv u$

hyperbolic-functions trigonometric-integrals

An amazing property of the Catenary

ordinary-differential-equations area hyperbolic-functions arc-length

How does one convert an integrand of the form $\frac{x\sinh x-t\sinh t}{\sinh^2x-\sinh^2t}$ into the form $\frac{\ln(x^2-t^2+1)}{\sinh^2x-\sinh^2t}$?

definite-integrals logarithms contour-integration gamma-function hyperbolic-functions

How to prove $\int_{0}^{\infty} \frac{(1-x^2) \, \text{sech}^2\left(\frac{\pi x}{2} \right)}{(1+x^2)^2}\, dx = \frac{\zeta(3)}{\pi}$?

calculus integration definite-integrals riemann-zeta hyperbolic-functions

Geometric definitions of hyperbolic functions

trigonometry hyperbolic-functions

Geometric interpretation of hyperbolic functions

trigonometry hyperbolic-geometry hyperbolic-functions

Prove that $\sinh(\cosh(x)) \geq \cosh(\sinh(x))$

calculus inequality hyperbolic-functions

Integral (Tanh and Normal)

calculus normal-distribution fourier-transform hyperbolic-functions

Closed form for ${\large\int}_0^\infty\frac{x\,\sqrt{e^x-1}}{1-2\cosh x}\,dx$

calculus integration definite-integrals closed-form hyperbolic-functions