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Questions tagged [hyperbolic-functions]

For questions related to hyperbolic functions: $\sinh$, $\cosh$, $\tanh$, and so on.

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Computing $\int_0^1 \frac{x f(x)}{1+x^2} d x$, where $f(x)$ are the inverse of trigonometric functions.

Recently, I started to investigate the integrals involving the inverse of the six trigonometric functions $f(x)$: $$\int_0^1 \frac{x f(x)}{1+x^2} d x$$ by integration by parts as $$ \begin{aligned} \...
Lai's user avatar
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3 votes
2 answers
84 views

Proving $\sin^2\alpha=\sin^4\phi=\sinh^4\theta$ for real $\alpha$, $\phi$, $\theta$ satisfying $\cosh(\theta+i\phi)=e^{i\alpha}$

If $\cosh(\theta + i \phi) = e^{i \alpha}$ (with $\alpha, \theta, \phi \in \mathbb{R}$), prove that $$\sin^2(\alpha) = \sin^4(\phi) = \sinh^4(\theta)$$ What I've gotten so far: Since: $$\cosh(\theta + ...
BodyDouble's user avatar
0 votes
0 answers
11 views

Is there an exact expression for the full width half maximum of a sech^2 curve convolved on itself?

As some simple math can show, a Gaussian convolved onto itself is also a Gaussian. Importantly, the FWHM of the original gaussian compared to that of its convolved counterpart is different by a factor ...
ChemGuy's user avatar
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2 votes
1 answer
95 views

$ |\sinh(\Im z)| \leq |\cos z| \leq \cosh(\Im z),$ what estimate do we get if $z \in \mathbb{R}$?

Prove that for every $z \in \mathbb{C}$ the following holds: $$ |\sinh(\Im z)| \leq |\cos z| \leq \cosh(\Im z). $$ What estimate do we get if $z \in \mathbb{R}$? Attempt: To prove the given ...
lolip123's user avatar
4 votes
1 answer
159 views

Given the function $f$ with the rule $f(z) = \frac{z \sinh\left(\frac{1}{z^2}\right)}{z^2 + 1}$.

Given the function $f$ with the rule $f(z) = \frac{z \sinh\left(\frac{1}{z^2}\right)}{z^2 + 1}$. (a) Determine and classify the singular points of the function $f$ and calculate the residues at these ...
lolip123's user avatar
1 vote
1 answer
115 views

Two questions about the function $f : \mathbb{C} \to \mathbb{C}$ defined by the expression $f(z) = \sinh z$

Given is the function $f : \mathbb{C} \to \mathbb{C}$ defined by the expression $f(z) = \sinh z$. (a) Prove that for all $z, w \in \mathbb{C}$, the following holds: $\sinh(z + w) = \sinh z \cosh w + \...
GENERAL123's user avatar
6 votes
3 answers
144 views

Integral of principal value of $[\tanh(x+a)-\tanh(x+b)]/x$

How do you solve this integral involving the Cauchy principal value? $$ \mathcal{P} \int_{-\infty}^{\infty} \frac{\tanh(x+a)-\tanh(x+b)}{x} dx \\ = \int_0^\infty \frac{\tanh(x+a)+\tanh(x-a)-\tanh(x+b)-...
Bio's user avatar
  • 1,108
-2 votes
2 answers
99 views

An inequality for arctan in terms of hyperbolic tangent and radicals

Problem : Let $x>0$ then define : $$h\left(x\right)=\arctan\left(x\right)-\tanh\left(x\right)-\frac{1}{7}\left(1-\tanh\left(\frac{1}{x}\right)\right)^{\frac{3}{2}}-\frac{1}{3}\left(1-\tanh\left(\...
Ranger-of-trente-deux-glands's user avatar
0 votes
1 answer
25 views

Prove $Pr[Z \leq 0, -\ln{\tanh{\frac{-Z}{2}}} \leq x] = Pr[Z \geq -\ln{\tanh{\frac{x}{2}}}]$ for any random variable $Z$ and non-negative constant $x$

I am reading an article (Design of capacity-approaching irregular low-density parity-check codes, Richardson et al.) and I stumbled upon the following claim: $Pr[Z \geq 0, -\ln{\tanh{\frac{Z}{2}}} \...
SlowerPhoton's user avatar
6 votes
2 answers
95 views

Optimal length of rope for sliding across a gap

I'm trying to solve a physics problem that I heard ~10 years ago in undergrad that was casually posed to me without a solution in mind; it has been bothering me ever since! Please let me know if this ...
pretzelKn0t's user avatar
1 vote
0 answers
82 views

Understanding the Properties and Applications of My Custom Kernel Function

I am a hobbyist and I have made this kernel that I want to use for varies things. Can you please help me understand the proprieties of what I have created and how I can possibly integrate it correctly ...
Jesse Sanford's user avatar
0 votes
1 answer
61 views

Half plane model, distance between point and hyperbolic line

I'm currently reading chapter 3 of "Hyperbolic Geometry" by James W. Anderson, and try to do exercise 3.19. The crux is to solve the following problem. We are in the half plane model $\...
Traubenzucker's user avatar
3 votes
1 answer
99 views

Help with Integral involving Square-root of Hyperbolic functions (definite)

I'm trying to solve the following integral for a project that I'm doing \begin{equation} I = \int_{x_0}^{x_1} \sqrt{\frac{a^2}{\sinh^2 x}-\left(b - \frac{c}{\tanh x} \right)^2} \, dx \end{equation} ...
MultipleSearchingUnity's user avatar
1 vote
0 answers
30 views

Area of Intersection of Hyperbolic Disks

I am trying to find the area of intersection of two hyperbolic circles using integrals. The problem: Let $D_1$ be the hyperbolic disk of radius $R$ centred at $O$. Consider a point $X \in D_1$ at ...
Algebro1000's user avatar
1 vote
0 answers
87 views

Integral $\int_{-\infty }^{\infty } \frac{x\textrm{tanh}(x)}{\pi^{2}+4\left (x\textrm{cosh}(x)-\textrm{sinh}(x) \right )^{2}} dx$ [closed]

Is there any possibility to evaluate this integral $?$: $$ \int_{-\infty }^{\infty }\frac{x\tanh\left(x\right)}{\pi^{2} + 4\left[x\cosh\left(x\right) - \sinh\left(x\right)\right]^{2}}{\rm d}x $$ I ...
Lorenzo Alvarado's user avatar
0 votes
1 answer
36 views

Troubles with solving a Laplace equation

I'm struggling in solving an exercise about the Laplace equation over the domain $[\frac{\pi}{2}, \pi] \times [0, \pi]$ with the boundary conditions: $f(\frac{\pi}{2},y)=f(\pi,y)=f(x,\pi)=0$ and $f(x,...
Osvaldo Paniccia's user avatar
0 votes
1 answer
39 views

Growth of distance between divergent hyperbolic lines is at least linear in the arc-length parameter

I'm trying to do Chapter 3 Problem 21 in Geometry and Topology (Reid M., Szendroi B.) 2005 ed. The hyperbolic plane ${\cal H}^2=\{(t,x,y):t^2-x^2-y^2=1,t>0\}$ WLOG let $L_1$ be the hyperbolic line ...
hbghlyj's user avatar
  • 3,047
2 votes
0 answers
32 views

cvxpy with a tanh objective

I need to solve an optimisation problem of the form: \begin{align} \textrm{maximize} \quad &\sum_{i=1}^{n} a_i \tanh(c_i x_i/2) \\ \textrm{wrt} \quad &x_i, \quad i \in 1,\dots,n \\ \textrm{...
GingerBreadMan's user avatar
1 vote
1 answer
70 views

If $f(x) = Ae^{x} + Be^{-x}$ and $f(1) = 0$, then $f(x) = C\sinh(x - 1)$

I need to find a function $f(x)$ of the form $$ f(x) = Ae^{x} + Be^{-x} \;\; A,B \in \mathbb{R} $$ with $f(1) = 0$ The professor immediately concluded that $$ f(x) = C \sinh(1-x) \;\;\; C \in \mathbb{...
Tomer's user avatar
  • 436
1 vote
2 answers
85 views

Co- prefix in hyperbolic functions

I was studying calculus when a question came to my mind. As written here, in the context of trigonometric functions, the prefix co- stands for complementary, viz. if f is an angular function (mose ...
Amanda Wealth's user avatar
7 votes
2 answers
294 views

Integral of the product of a Gaussian and a exponential of a hyperbolic function

In a derivation I am working on, I have encountered an integral of the form \begin{equation} \int_{0}^{\infty}e^{-a x^2-b\ \textrm{cosh}(x)}\ dx \end{equation} with $a$ and $b$ real and positive. I am ...
STU's user avatar
  • 117
7 votes
2 answers
258 views

Proving that $0 = \pi - \frac{\pi^5}{5!} - \frac{\pi^7}{7!} + \frac{\pi^{11}}{11!} + \frac{\pi^{13}}{13!} - \frac{\pi^{17}}{17!}-\cdots$

I'm looking for a proof that: $0 = \pi - \frac{\pi^5}{5!} - \frac{\pi^7}{7!} + \frac{\pi^{11}}{11!} + \frac{\pi^{13}}{13!} - \frac{\pi^{17}}{17!}- \frac{\pi^{19}}{19!} + \frac{\pi^{23}}{23!}+\cdots$. ...
Guevara's user avatar
  • 91
0 votes
1 answer
50 views

When is this rational function of exponentials actually rational-valued?

This has come up in my research, and I am sorry if it is obvious. I am looking at the following expression $$ m\frac{\tanh(xm)}{\tanh(x)} = m\frac{e^{2xm}-1}{e^{2xm}+1} \frac{e^{2x}+1}{e^{2x}-1}, $$ ...
Croc2Alpha's user avatar
  • 3,847
0 votes
1 answer
16 views

I need help filling in some in a step from Fomin's calculus of variations

At the bottom of page 20 from Fomin's book on Calculus of Variations, we have: (1) $\frac{x+A}{c}= \ln( \frac{y + (y^2-c^2)^{1/2}}{c})$ Implies that $y = c \cosh(\frac{x+a}{c})$ Can somebody help me ...
PhysicsIsHard's user avatar
3 votes
1 answer
123 views

Solve $\int_{-\infty}^{+\infty}\frac{1}{\cosh x}\ dx$ using residue theory [ANSWERED]

I was trying to solve this exercises which asked to first solve $$I=\lim_{R\to +\infty}\oint_{\Gamma_R}\frac{1}{\cosh z}\ dz $$ where $\Gamma_R=\partial\{z=x+iy\in\mathbb{C}:-R\le x\le R, \ 0\le y\le \...
deomanu01's user avatar
  • 113
1 vote
0 answers
44 views

Continued fraction of Laplace transform

I first learned of the below identity from MathWorld and the works of Ramanujan, but it's completely crazy with polygammas and Laplace transforms of hyperbolic trig. It seems weird that the Laplace ...
Michael Duffy's user avatar
4 votes
3 answers
123 views

How to approach an Hyperbolic Integral that doesn't appear to be solvable in closed form.

I'm interested in tackling the following integral: $$\int_{-\ln (2+\sqrt 5)}^{\ln (2+\sqrt 5)} \sqrt{4+\sinh^2(x)} dx$$ While I've attempted various techniques, it appears challenging to find a closed-...
Mark's user avatar
  • 7,880
0 votes
0 answers
42 views

Geometric proof of $\cosh^2(x)-\sinh^2(x)=1$? [duplicate]

It is easy to show using the analytic continuation of $\sinh^2(x)$ and $\cosh^2(x)$ that the identity$$\cosh^2(x)-\sinh^2(x)=1$$holds. However, what I want to know is, is there any way to prove this ...
CrSb0001's user avatar
  • 2,652
2 votes
0 answers
51 views

Distance Travelled by a Projectile

I wanted to come up with a formula for the total distance travelled by a projectile with some initial velocity $\langle v_x,v_y\rangle$ in $\mathbb R^2$. Its parametrization should be the following: $$...
Leonidas Lanier's user avatar
0 votes
0 answers
51 views

How do I define a function that's first exponential and then logarithmic?

I want to define a continuous function $f(x)$ such that the following properties hold true. $f(0) = 1$ $f(-1) = 1 - 0.5 = 0.5$ $f(1) = 1 + 0.5 = 1.5$ $f(-2) = 1 - 0.5 - 0.25 = 0.25$ $f(2) = 1 + 0.5 + ...
Aadit M Shah's user avatar
0 votes
0 answers
37 views

Fourier transform $\left[\mathrm{csch}(x+i\epsilon-t)\right]^n\left[\mathrm{csch}(x+i\epsilon+t)\right]^m$

In a physics related problem, I am trying to compute the Fourier transform \begin{align} \mathcal{F}\left[\frac{1}{\sinh^{n}\left[\pi T_R\left(x+ i\epsilon-t\right)\right]\sinh^{m}\left[\pi T_L\left(...
hyriusen's user avatar
  • 147
3 votes
2 answers
171 views

Integrals of the form $\int_0^1\text{arctanh}^a(y) y^{2b} dy$

Recently I have been trying to calculate integrals of the form: $$ I(a,b)=\int_0^1\text{arctanh}^a(y) y^{2b} dy$$ for some positive integer-valued $a$ and $b$. The values $a=0$ or 1 have quite trivial ...
Fred Li's user avatar
  • 632
5 votes
1 answer
109 views

Evaluating $\int_{0}^{\infty}{\frac{1}{\cosh^{2k+1}(x)} dx}$

I tried : $$\begin{align}\int_{0}^{\infty}{\frac{1}{\cosh^{2k+1}(x)} dx}&=2^{2k+1}\int_{0}^{\infty}{(e^{x}+e^{-x})^{-(2k+1)}dx}\\&=2^{2k+1}\int_{0}^{\infty}{\frac{1}{u}\left(u+\frac{1}{u}\...
AnthonyML's user avatar
  • 977
0 votes
1 answer
83 views

Prove distance formula in polar coordinates [closed]

I have seen this equality in some youtube video on hyperbolic geometry, but I want to understand the proof of it. dist$((r_1,\theta_1),(r_2,\theta_2)) = \text{arcosh}(\text{cosh}r_1\text{cosh}r_2 - \...
Just do it's user avatar
1 vote
0 answers
133 views

how to evaluate inverse Laplace transform of $e^{- \sqrt{s}} $ using series?

I tried to find an inverse Laplace transform by series as follows $$ f(t)=L^{-1}_s\left(e^{-\sqrt{s}}\right)(t)=L^{-1}_s\left(\sum_{k=0}^{\infty}\frac{(-1)^k}{k!} s^{\frac{k}{2}}\right)(t)$$ and by ...
Faoler's user avatar
  • 1,657
2 votes
3 answers
110 views

Find $x$ such that $\cosh(a + bx) + 1 = cx$

I need to find an analytical solution for $x$ to: $$ \cosh(a + bx) + 1 = cx $$ where a,b and c are real parameters. I have tried to tackle this geometrically, by splitting the problem into finding ...
Gabriele Vecchio's user avatar
-1 votes
2 answers
97 views

Angle-sum identities for $\csc$, $\sec$, $\cot$; $\rm{sech}$, $\sinh$, $\coth$; $\arcsin$, $\rm{arctanh}$, $\rm{arccoth}$? [closed]

I'm sure many of you are aware of the following identities: $$\begin{align} \sin(A \pm B) &= \sin A\cos B \pm \sin B\cos A \\[4pt] \cos(A \pm B) &= \cos A\cos B \mp \sin A\sin B \\[4pt] \tan(A ...
Sam's user avatar
  • 173
1 vote
1 answer
43 views

Using mnemonic triangles for composition of hyperbolic trigonometric functions and their inverses

The composition of circular trigonometric functions, like $\cos(\tan^{-1}(x))$, can be derived drawing a right angle triangle and applying Pythagoras' theorem and the definition of sine and cosine in ...
Jaime Yepes de Paz's user avatar
0 votes
0 answers
14 views

Curves in the plane with hyperbolic secant curvature

I'm searching for a curve in a plane that has a specific curvature, of the form $$ \kappa(s) = A \ \text{sech}(Bs) $$ where $s$ is the arc length parameter, and $A$, $B$ constants. I'm not sure if it ...
Francesco Lorenzi's user avatar
0 votes
0 answers
40 views

Any ideas on how to add complexity?

I am investigating the path of a surfer who, starting at a point on the shore, is trying to paddle to a certain point directly in front of him (perpendicularly to the shore) while a current pushes ...
Mr_Ryder's user avatar
-1 votes
1 answer
43 views

If ψ(ϕ)=ln(secϕ+tanϕ), how do you find an expression for ϕ? [closed]

It should be $2\tan ^{-1}\left( e^{\psi }\right) -\dfrac{\pi }{2}$ but i'm not sure whether that is correct or if the $\dfrac{\pi }{2}$ should be in brackets. It would be helpful if you also found an ...
Ethan's user avatar
  • 1
4 votes
1 answer
161 views

Why do CAS struggle with $\int\frac{dx}{1+\sinh x}$?

$\int\frac{dx}{1+\sinh x}$ is a slightly annoying but still easily solved integral using a weierstrass substitution and PFD. I'm mainly referring to WolframAlpha, but I've seen other computer algebra ...
Nathan29006781's user avatar
0 votes
0 answers
24 views

Integral involving a product of sinh functions in the denominator

The following integrals arise in fluid flow problems. Let $\xi>0$, $-2\pi<\eta_1<0$, $0<\eta_2\le \pi$, and $\eta_1<\eta<\eta_2$. Then evaluate \begin{align*} &\int_0^{\infty} \...
Jog's user avatar
  • 369
0 votes
0 answers
33 views

How to divide a catenary curve into parts of equal length?

I know the basic equation of a catenary is y = a*cosh((x-x0)/a)+b Length of a catenary curve is L = a*sinh((x-x0)/a) where x0 is a symmetry point or vertex or lowest x co-ordinate of a curve. I can ...
vbalaji21's user avatar
0 votes
0 answers
35 views

How to remove a hyperbolic trignometric function from an equation

If I have an equation for example where y and z are known and x is unknown. It is of the form z = sinh(x) +y I want to write this equation of the form x = y + z (some form of this) Inorder to do this, ...
vbalaji21's user avatar
2 votes
1 answer
116 views

What is $\int \frac{-2}{x^2-1}\,dx$

For $x\in\mathbb{R}$, what is the value of $$ \int \frac{-2}{x^2-1}\,dx? $$ Using partial fractions, we get $$ \int \frac{-2}{x^2-1}\,dx= \int \frac{1}{x+1}\,dx+ \int \frac{-1}{x-1}\,dx=\log|x+1|-\...
sam wolfe's user avatar
  • 3,435
0 votes
0 answers
62 views

Why does $\lim\limits_{n\to\infty}\frac{\cosh^{-1}n}{H_n}=1$?

$H_n$ represents the $n\text{th}$ harmonic number. I was messing around with Desmos when I happened to come across this. I typed it into WolframAlpha which confirmed that the limit is equal to $1$ but ...
Dylan Levine's user avatar
  • 1,688
3 votes
1 answer
67 views

The similarity between formula $\int \sqrt{x^{2}+a^{2}} dx$ and $\int \sqrt{a^{2}-x^{2} } dx$

I noticed some similarities between these two formulas when I refer to the basic integral table. $\int \sqrt{x^{2}\pm a^{2} }dx= \frac{1}{2}( \sqrt{a^{2}+x^{2}}\cdot x+a^{2}\operatorname{arcsinh}\frac{...
Konan's user avatar
  • 41
0 votes
3 answers
88 views

Evaluate the following integral involving hyperbolic functions

The following is an integral from MIT integration bee 2023: $$\int_0^{\infty} \frac{\tanh(x)}{x\cosh(2x)} dx.$$ I tried substituting $u=\cosh(x)$, but that just made the integral even more complicated....
User150920's user avatar
1 vote
3 answers
77 views

How do I write asinh in terms of $\log$?

I've found in multiple places (e.g. Wikipedia) that $$ \sinh^{-1} x = \log\left[x + \sqrt{x^2 + 1}\right] $$ Wikipedia says this can be derived via the quadratic formula. Can anyone explain how I ...
byl's user avatar
  • 55

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