Questions tagged [hyperbolic-functions]

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

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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
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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
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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
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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
121 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
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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
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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
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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
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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
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3 votes
2 answers
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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
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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
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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
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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
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2 votes
3 answers
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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
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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
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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
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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
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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
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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
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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
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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
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31 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
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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
115 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
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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
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3 votes
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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. $\int \sqrt{x^{2}\pm a^{2} }dx= \frac{1}{2}( \sqrt{a^{2}+x^{2}}\cdot x+a^{2}\operatorname{arcsinh}\frac{...
Konan's user avatar
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$\frac{2x}{\sinh(2\tanh x)}<(\cosh x)^2<\frac{2x}{\sinh(2\tanh x)}+x\sinh (2x)$

I am given a question to prove that $$\frac{2x}{\sinh(2\tanh x)}<(\cosh x)^2<\frac{2x}{\sinh(2\tanh x)}+x\sinh (2x)$$ I know people will ask what is your attempt/idea. Honestly I have no idea ...
Mathxx's user avatar
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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
69 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
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Integrating $e^x\sinh x$ gives two different answers [duplicate]

I integrated $e^x\sinh(x)$ by writing out $\sinh x$ as $\frac{e^x-e^{-x}}{2}$ and multiplying to get $\int$$\frac{e^{2x}-e^{0}}{2}dx$ which comes out to be $\frac{e^{2x}}{4}- \frac{x}{2}$ + C however, ...
user1139725's user avatar
11 votes
1 answer
254 views

What is the $\lim_{n\to\infty}\sinh^{\circ n}\sqrt{\dfrac{3}{n}}$?

Let $\sinh^{\circ n}x$ be the $n$-fold iteration of $\sinh x$. I'm interested in the limit $$ \lim_{n\to\infty}\sinh^{\circ n}\sqrt{\dfrac{3}{n}}. $$ This limit exists because the general term is ...
Jianing Song's user avatar
  • 1,779
3 votes
2 answers
228 views

How to minimize the maximum absolute difference between 2 functions?: example $\min_a\{\|\text{erf}(x)-\tanh(\frac2{\sqrt{\pi}}(x+a x^3))\|_\infty\}$

How to minimize the maximum absolute difference between 2 functions?: example $\min_a\{\|\text{erf}(x)-\tanh(\frac2{\sqrt{\pi}}(x+a x^3))\|_\infty\}$ Intro_______________ In this other question I ...
Joako's user avatar
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1 vote
3 answers
243 views

Approximating for the Error function $\text{erf}(x)$ through an Hyperbolic tangent function $\text{tanh}\left(\dfrac{4x}{4-x^2}\right)$

Approximating for the Error function $\text{erf}(x)$ through an Hyperbolic tangent function $\text{tanh}\left(\dfrac{4x}{4-x^2}\right)$ I was plotting some functions and I found that the function $$f(...
Joako's user avatar
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0 answers
13 views

Understanding the connection between the summations of reciprocals of quadratics and hyperbolic trigonometric functions

While messing around with summations of the reciprocals of quadratics from $n=0$ to $\infty$ on wolfie. I made this discovery: $$ \sum_{n=0}^{\infty}\frac{1}{(an)^2+b^2}=\frac{1}{2ab^2}(a+b\coth{\frac{...
uggupuggu's user avatar
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0 answers
95 views

Complex inverse hyperbolic cosine

It is well known that the hyperbolic cosine $\cosh$ induces a bijection from $[0,+\infty)$ to $[1,+\infty)$, with its inverse given by $$ \forall t\in[1,+\infty),\quad\operatorname{arcosh}(t)=\ln(t+\...
Will's user avatar
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0 answers
48 views

Integration by parts, hyperbolic functions

I tried solving this hyperbolic integration using integration by parts, and it doesn’t seem to work. $$ \begin{gather} \int \sinh ^ { 2 } u d u \\ = \sinh u \cosh u - \cosh ^ { 2 } u \end{gather} $$ ...
Chia Fei's user avatar
1 vote
3 answers
162 views

Why inverse hyperbolic cosine is just $\ln(x + \sqrt{x^2 - 1})$ and not $\ln(x - \sqrt{x^2 - 1})$ (source: Wikipedia)?

I am an amateur who loves elementary mathematics. I am not too good by any means, and never considered myself a genius or a math pro etc. Can I please ask a question which I am really curious about? ...
Alexander's user avatar
  • 367
2 votes
0 answers
96 views

Complicated Integral of arccos

Let $R > r > 0$ be constants. I'm trying to work out the following integral: $$ \int_{R-r}^R\arccos\bigg(\frac{\cosh(y)\cosh(r)-\cosh(R)}{\sinh(y)\sinh(r)}\bigg)\sinh(y)dy.$$ So far, I ...
Algebro1000's user avatar
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0 answers
39 views

Area below a curve defined with respect to another curve

Let $a \in (-\infty, -1]$ , $c \in \mathbb R_+$ and $d \in (0,1)$ be constant real numbers. We define the parabola : $$ f(x) := ax^2 + c$$ For all $m \in \mathbb R$, we also define the geometric line $...
Programming Insider's user avatar
7 votes
0 answers
183 views

Evaluating $\int_{-\infty}^\infty \coth x \exp(-a \cosh x + b \sinh x) \, dx$ and $\int_0^b K_0(\sqrt{a^2 - b'^2}) \, db'$

I am trying to evaluate the integral $$I = \int_{-\infty}^{\infty} \underbrace{\coth x}_{f(x)} \exp(-a \cosh x + b \sinh x) \, dx \, ,$$ where $a \in \mathbb{R}^+$, $b \in \mathbb{R}$, and $|b/a| < ...
nullgeodesic's user avatar
5 votes
0 answers
160 views

Definite Integral $\int \tanh(a\cdot \operatorname{atanh}(x)+b)\,\mathrm dx$

I am looking for any type of solution (closed form/recursive integral/special functions/...) or at least a good approximation for this definite integral: $$ \int\limits_{\scriptsize 0}^{\scriptsize y}{...
kamyxx's user avatar
  • 51
-1 votes
1 answer
128 views

What is this constant $\lim_{a\to +\infty}(a-f(u)-\ln(a))=C$?

Problem : Let the function on $x\in(0,1),a=\operatorname{constant}>1$ : $$f(x)=x^{a^{x^{a}}}a^{x^{a^{x}}}-\operatorname{arctanh}(x)$$ Now let : $$f'(u)=0,0.99<u<1$$ Then it seems we have : $$\...
Miss and Mister cassoulet char's user avatar
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0 answers
51 views

Completing the square under a square root

Trying to work through a problem that requires us to complete the square, then use a sinh substitution...but I need to start by remembering back to math from many years ago. $$\int\sqrt{2x^2+3x+4} \ ...
usuallyBadAtMath's user avatar
4 votes
3 answers
207 views

Is there an exact solution to $x \sinh\Big(\frac{1}{x}\Big) = a$?

Is there an exact formula for solutions to the equation $x \sinh\Big(\frac{1}{x}\Big) = a$ where $a,x \in \mathbb{R}^+$? And if not, why? I tried to rearrange to apply Lambert W somewhere to no avail. ...
LeaG's user avatar
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0 answers
34 views

Proving hyperbolic equality

While trying to prove a problem on catenoids intersecting orthogonally with $S^2$ I have to prove that if $a=\frac{1}{T\cosh(T)}$ where $T$ is the only real number such that $T\tanh(T)=1$ and $t$ ...
PunkZebra's user avatar
  • 1,595
4 votes
1 answer
103 views

If $\frac1{\cosh x}+\frac1{\cosh y}+\frac1{\cosh z}=1$, prove that $\sinh x \cdot \sinh y \cdot \sinh z \geq 16 \sqrt{2}$.

Another Olympiad question: Let $x, y, z\in\mathbb{R}^+$ satisfy $\frac{1}{\cosh x}+\frac{1}{\cosh y}+\frac{1}{\cosh z}=1$. Show that $$ \sinh x \cdot \sinh y \cdot \sinh z \geq 16 \sqrt{2} . $$ When ...
user avatar
3 votes
1 answer
129 views

Compact form of solution of $\displaystyle\int_0^{\frac{\pi}{2}}\ln\left(1+\alpha^n\sin(x)^{2n}\right)\mathrm{d}x$

I hope it won't be categorized as a trivial question, I solved this integral and arrived at the following form: $${\int_{0}^{\frac{\pi}{2}}\ln\left(1+\alpha^n\sin(x)^{2n}\right)\mathrm{d}x=\frac{n\pi}{...
Math Attack's user avatar
0 votes
1 answer
93 views

How taut must a stretchable, horizontally-oriented string be in order for a straight line to approximate the string to within a given margin of error? [closed]

My question deals with a string that can stretch due to its own weight. If the string is allowed to stretch then I'd assume there would always be a bit of a bulge due to gravity. The only progress I'...
Simon M's user avatar
  • 657
1 vote
2 answers
113 views

Alternate expression for $\operatorname{atanh}(x)$ for $\lvert x \rvert \to 1$

The inverse hyperbolic tangent function $\operatorname{atanh}$ is defined as: $$\operatorname{atanh}(x) = \frac{1}{2}\log\left(\frac{1 + x}{1 - x}\right)$$ In computers using floating point arithmetic,...
Yimin Rong's user avatar
7 votes
1 answer
382 views

Hyper-closed-forms of series involving hyperbolic functions:$\sum_{n=1}^{\infty}\frac1{n^2\cosh(\pi n)^2},\frac{\tanh(\pi n)}{n^3}$ and the like.

Series like $$ \sum_{n=1}^{\infty} \frac{1}{n^2\cosh(\pi n)^2},\sum_{n=1}^{\infty} \frac{\tanh\left ( \pi n \right ) }{n^3} $$ that used to be recognised as ones with no closed-forms, however, ...
Setness Ramesory's user avatar

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