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

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55
votes
2answers
1k views

Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}\mathrm dx$

Consider the following integral: $$\mathcal{I}=\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}\mathrm dx,$$ where ...
33
votes
2answers
460 views

Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$

Please help me to find a closed form for the infinite product $$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$ where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
19
votes
6answers
236 views

What is the importance of $\sinh(x)$?

I stumbled across $\sinh(x)$. I am only a calculus uno student, but was wondering when this function comes into play, and what is its purpose? Last, does it have world applications, or is it a ...
16
votes
3answers
318 views

Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$

Let $$f(a)=\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx,$$ where $\operatorname{sech}(z)=\frac2{e^z+e^{-z}}$ is the hyperbolic secant. Here are values of $f(a)$ at some ...
14
votes
2answers
318 views

$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$

Is there any closed-form representation for the following integral? $$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx,$$ where $\mathrm{sech}\,x$ is the hyperbolic secant, ...
11
votes
2answers
355 views

Series $\sum\limits_{n=1}^\infty \frac{1}{\cosh(\pi n)}= \frac{1}{2} \left(\frac{\sqrt{\pi}}{\Gamma^2 \left( \frac{3}{4}\right)}-1\right)$

I was playing around with Mathematica and found that $$\sum_{n=1}^\infty\frac1{\cosh(\pi n)} = \frac12\left(\frac{\sqrt{\pi}}{\Gamma \left(\tfrac34\right)^2}-1\right)$$ Does anybody know how to ...
11
votes
1answer
610 views

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

I have calculated the real and imaginary parts of $\dfrac{\sin z}z.$ I've obtained $$\begin{eqnarray} \frac{\sin z}z&=&\frac{\sin(x+iy)}{(x+iy)}\\ &=& ...
8
votes
5answers
639 views

Geometric meanings of hyperbolic cosinus and sinus

In euclidean geometry, $\cos$ and $\sin$ are used for angles in trigonometry. Is there an equivalent for $\cosh$ and $\sinh$ the hyperbolic cosine and sine, and not cosinus and sinus ?
8
votes
3answers
487 views

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

Show that : $$\sum_{n=1}^{\infty}\frac{\cosh(2nx)}{\cosh(4nx)-\cosh(2x)}=\frac1{4\sinh^2(x)}$$ $$\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$$
8
votes
1answer
375 views

Geometric definitions of hyperbolic functions

I've learned in school that all the trigonometric functions can be constructed geometrically in terms of a unit circle: Can the hyperbolic functions be constructed geometrically as well? I know ...
7
votes
5answers
258 views
7
votes
1answer
2k views

How were Hyperbolic functions derived/discovered?

Trig functions are simple ratios, but what does Cosh, Sinh and Tanh compute? How are they related to euler's number anyway?
7
votes
2answers
136 views

Proof $1 -\frac{1}{5} + \frac{1}{9} - \frac{1}{13} + … = \frac{\pi + 2\ln(1+\sqrt2)}{4\sqrt2}$

I'm trying to show that $$1 -\frac{1}{5} + \frac{1}{9} - \frac{1}{13} + \cdots = \frac{\pi + 2\ln(1+\sqrt2)}{4\sqrt2}.$$ I thought of using the power series for $\tanh^{-1}z$ which I found was ...
7
votes
1answer
109 views

Finding taylor expansion for $\tanh(x)$

I am a high school student and am trying to find the taylor expansion of $\tanh(x)$ in terms of a summation form. I have gotten this far, and am aware it might get complicated very quickly. If someone ...
6
votes
1answer
205 views

A conjecture $\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx\stackrel?=\frac\pi2\ln2$

I need to find a closed form for this integral: $$\mathcal{I}=\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx.$$ A numerical integration results in an approximation ...
6
votes
2answers
216 views
+50

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 ...
6
votes
0answers
71 views

Can these two indefinite integrals be evaluated in closed form?

I'm wondering whether any of these two indefinite integrals $$\int \frac{1}{\sqrt{1+\alpha \sinh(x)^{-4/3}}}dx$$ $$\int \frac{\sinh(x)^{-4/3}}{\sqrt{1+\alpha\sinh(x)^{-4/3}}}dx$$ can be evaluated in ...
6
votes
0answers
98 views

Solving a problem of Ramanujan's interest

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} ...
5
votes
5answers
239 views

Evaluating the following integral: $ \int \frac{x^2}{\sqrt{x^2 - 1}} \text{ d}x$

For this indefinite integral, I decided to use the substitution $x = \cosh u$ and I've ended up with a $| \sinh u |$ term in the denominator which I'm unsure about dealing with: $$\int ...
5
votes
5answers
278 views

Prove that $\lim\limits_{x \to 0} \sinh(x)/x =1$.

Prove that $ \lim\limits_{x \to 0} \frac{\sinh x}{x} =1.$ I am having some trouble proving this without derivative. Some help would be much appreciate!
5
votes
3answers
139 views

Find $\int \sinh^{-1}x\hspace{1mm}dx$

Find $\int \sinh^{-1}x\hspace{1mm}dx$ $ $ I am asked to use the following Equation: $$\int \tan^{-1}x\hspace{1mm}dx= x\tan^{-1}x-\ln(\sec(\tan^{-1}x))+C$$ $ $ The confusing part is : What has ...
5
votes
1answer
133 views

Weierstrass $ \tanh \frac{\theta}{2} $ substitution confusion.

I'm already familiar with the trigonometric version of this substitution $ t = \tan \frac{\theta}{2} $ and it's geometrical derivation involving the unit circle found here. However, I'm not sure how ...
5
votes
2answers
117 views

Asymptotics of $\int_{0}^{+\infty}\!\!\frac{dx}{\sinh^2(\epsilon \sqrt{x^2+1}) } $ for $\epsilon$ near $0$

How to find an asymptotic expansion, for $\epsilon$ near $0$, of the following integral $$ I(\epsilon):=\int_{0}^{+\infty}\frac 1{\sinh^2 (\epsilon \sqrt{x^2+1}) } {\rm d}x. $$ As $\epsilon ...
5
votes
1answer
725 views

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
5
votes
1answer
408 views

fourier expansion of $\coth$ and justifying an identity

The problem: Justify the following equalities: $$\cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2}$$ I am trying to figure ...
5
votes
1answer
165 views

Integral with hyperbolic cosine squared

Does anyone can give me a hint how to integrate the following: $$\int_0^\infty{\frac{x^2 {\rm d}x}{\mathrm{cosh}^2(x)}}.$$ The answer is $\frac{\pi^2}{12}$ (taken from the book). I've started with ...
5
votes
0answers
57 views

Inverse Fourier transform using Residues for a ratio of hyperbolic functions.

I'm new and glad to be here. I have a problem relating to an inverse Fourier transform. I have $$g(w)= \frac{\sinh{w(a-b)}}{w \cosh{wa}}$$ and want to find $$G(t)$$. I cannot find this in tables so I ...
5
votes
2answers
378 views

Geometric interpretation of hyperbolic functions

When proving identities like $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ $$\cosh^2(x)=\sinh^2(x)+1$$ algebraically, I am beset by the feeling that there should be a geometrical interpretation that makes them ...
4
votes
4answers
166 views

Evaluating $\int_0^1 x \sinh (x) \ \mathrm{dx}$

I am looking to evaluate $$\int_0^1 x \sinh (x) \ \mathrm{dx}$$
4
votes
4answers
227 views

What is the use of hyperbolic trigonometric functions if they are easily expressible algebraically?

I get that there are uses for $\sin(x)$ and $\cos(x)$ because they are defined with imaginary exponents which aren't as easily worked with but the hyperbolic functions are simply $\frac12(e^x\pm ...
4
votes
1answer
41 views

Why is $\arg(i\cosh x)=\frac{\pi}{2}$?

I was told $\arg(i\cosh (x))=\frac{\pi}{2}$ and $\arg(\cosh (x))=0$ but I can't figure out why. Could someone explain it to me?
4
votes
1answer
72 views

solving $2\cosh2x = 13\cosh x - 12$

I've been asked to solve: $2\cosh2x = 13\cosh x - 12$ I showed earlier in the question that $\cosh2x = 2\cosh^2x -1$ So I can say that: $2(2\cosh^2x -1) = 13\cosh x - 12$ $\therefore 4\cosh^2x ...
3
votes
4answers
405 views

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 ...
3
votes
2answers
826 views

Calculate cosh(x) given sinh(x)

Given the value of sinh(x) for example sinh(x) = 3/2 How can I calculate the value of cosh(x) ?
3
votes
3answers
169 views

Find the limit of $S_n=\sum_{i=1}^n \big\{ \cosh\big(\!\!\frac{1}{\sqrt{n+i}}\!\big) -n\big\}$, as $n\to\infty$?

$S_n=\sum_{i=1}^n\big\{ \cosh\big(\frac{1}{\sqrt{n+i}}\!\big) -n\big\}$ as $n\to\infty$ I stumbled on this question as an reading about Riemannian sums as in $$ \int_a^b f(x)\,dx =\lim_{x\to ...
3
votes
4answers
127 views

Inverse of $(e^x - e^{-x})/2$

What is the inverse of the function $f(x)=\frac{e^x - e^{-x}}2$? I tried replacing $e^x$ by a variable but I still can't get it.
3
votes
1answer
57 views

manipulations with Taylor expansions for log and sinh

How could we derive the equality $$ \frac14 \sum_{m=1}^\infty \frac1m \frac{1}{\sinh^2 \frac{m\alpha}2} = - \sum_{n=1}^\infty n\log (1-q^n)$$ where $q=e^{-\alpha}$ ?
3
votes
2answers
116 views

Why are hyperbolic functions significant?

I'm currently covering Stewart's Early Transcendentals, and there is a whole section dedicated to defining and differentiating hyperbolic functions. The same amount of space is used to cover other ...
3
votes
4answers
94 views

Hyperbolic cosine

I have an A level exam question I'm not too sure how to approach: a) Show $1+\frac{1}{2}x^2>x, \forall x \in \mathbb{R}$ b) Deduce $ \cosh x > x$ c) Find the point P such that it lies on ...
3
votes
1answer
79 views

given $\cosh u = x$ find $\sinh u$

I'm asked to show that:$\newcommand{\arcosh}{\operatorname{arcosh}}$ $\int{x \arcosh x}dx = \frac{1}{4}(2x^2 -1)\arcosh x - \frac{1}{4}x\sqrt{x^2 -1} + C$ If I integrate by parts: let $u = \arcosh ...
3
votes
1answer
37 views

How to find cosh(arcsinh(f(x)))?

With the regular trig functions, if I ever end up with something like $\operatorname{trig}_1(\operatorname{arctrig}_2(f(x))$, where $\text{trig}_1$ and $\text{trig}_2$ are two arbitrary trigonometric ...
3
votes
1answer
127 views

Why do we get two solutions when inverting $y = \sinh x$?

Using the definition $\sinh x = \dfrac{e^x-e^{-x}}{2},\;$ let's say we want to solve $\;y = \sinh x \;$ for $x$. It's not hard to show that $\;\sinh x \;$ is bijective, so this should have exactly ...
3
votes
1answer
47 views

Complicated integral, where $\int\coth(x)dx$ is somehow written in terms of $\int |x|e^{ix}dx$

In Gardiner's Quantum Noise the following integral equality is used (eq 3.3.10, 3.3.14): $$\int_0^{\infty}d\omega ...
3
votes
1answer
181 views

Evaluation of integral involving $ \tanh(ax) $

Is it possible to evaluate in closed form the integral $$ \int_{-\sqrt{x}}^{\sqrt{x}}\frac{r\tanh(ar)}{\sqrt{x-r^{2}}}dr=2\int_{0}^{\sqrt{x}}\frac{r\tanh(ar)}{\sqrt{x-r^{2}}}dr$$ here $a$ is a ...
3
votes
0answers
295 views

Integrating a fractional power of a rational function

I am currently working on a project where I stumbled upon the integral $$ \int \frac{\sinh \left(\frac{R}{2}\right)}{(\coth R - 6R \coth\left(\frac{R}{2}\right) + 9)^{1/4}} \,dR $$ where $R$ is a ...
2
votes
2answers
180 views

The derivative of $\tanh x$

I'm trying to calculate the derivative of $\displaystyle\tanh h = \frac{e^h-e^{-h}}{e^h+e^{-h}}$. Could someone verify if I got it right or not, if I forgot something etc. Here goes my try: ...
2
votes
2answers
65 views

Simplify $\sinh (\log (x))$

$$\sinh (\log (x))=\frac{x^2-1}{2 x}$$ However I do not see how this is done, here is an idea I had but I'm probably way off: $$\sinh \left(\ln \left(\frac{1}{2} ...
2
votes
1answer
174 views

Simplifying $\cosh \mathrm{arcsinh} \ x$

How can I simplify the following: $$\cosh \mathrm{arcsinh} \ x$$ I know that an expression of the form $f(g^{-1}(x))$ where $f$ and $g$ are trigonometric functions can be simplified by constructing a ...
2
votes
1answer
379 views

A hard integral with hyperbolic function

I was self studying integral. I meet a difficult problem here: $$\int_{0}^{\infty }{{{x}^{n}}\frac{\sinh ax}{\cosh bx}}\text{d}x=\frac{\pi }{2b}\cdot \frac{{{\text{d}}^{n}}}{\text{d}{{a}^{n}}}\tan ...
2
votes
1answer
90 views

definite integral involing hyperbolic and trigonometric functions

Trying to prove the following: $$ \int_0^\infty xe^{-c x^2}\sinh(a x)\cos(bx)\,dx = ...