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

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Derivatives of hyperbolic functions and Osborne's rule.

I am slightly confused when it comes to Osborne's rule when you take derivatives of hyperbolic functions. For example. The derivative of cotx is -cosec^2x, so there is a product of sines. So should ...
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48 views

Difference between hyperbolic sector, hyperbolic angle and hyperbolic argument

I've been working with hyperbolic functions and am completely confused by the Wikipedia definitions of hyperbolic sectors and angles. Are they the same thing? Based on my trial calculations, they seem ...
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34 views

Hyperbolic functions calculator values differing from the graph

I was exploring hyperbolic functions and noticed something weird while comparing the analytical definition (e^x+e^-x)/2 with the geometrical definition using the hyperbola x^2 - y^2 = 1. For the angle ...
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59 views

Geometric interpretation of hyperbolic functions and the hyperbolic angle/argument

I've been reading up on hyperbolic functions and was wondering if there was a geometric definition for the hyperbolic angle and hyperbolic function. In particular I was reading this: Definition of ...
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3answers
69 views

Integrating $\frac{1}{(x^2+b)^{3/2}}$?

How to integrate $$\int\frac{1}{(x^2+b)^{3/2}}dx$$ using the hyperbolic sine substitution ?
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3answers
40 views

How to integrate $I = \int_{-a/2}^{a/2}\frac{1}{\sqrt{x^2 + b}}dx$

$$I = \int_{-a/2}^{a/2}\frac{1}{\sqrt{x^2 + b}}dx$$ I tried to integrate by parts but failed. I think I'm supposed to change variables using a hyperbolic sine, but I don't know this method.
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20 views

Exponential curve with hyperbolic sine behavior on the tails

I have a dataset that I've fitted an exponential curve to that looks like a great fit at midrange values of the domain but is not such a good fit at low and high end domain values. Instead, at these ...
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2answers
76 views

Evaluating: $ \int\sqrt{\tanh(\ln(\sqrt{x}))} dx$ ; $ \int \ln\left(\sqrt{\tanh(\ln(\sqrt{x}))}\right) dx$

I don't have much experience with hyperbolic trig funnctions... So I don't know how to start solving this. How do I evaluate the following integrals? Any advice, hint or well-thought solution will be ...
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2answers
182 views

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

I was able to calculate $$\int_0^\infty\frac{\sqrt{e^x-1}}{1-2\cosh x}\,dx=-\frac\pi{\sqrt3}.$$ It turns out the integrand even has an elementary antiderivative (see here). Now I'm interested in a ...
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21 views

Rearrangingg equations using hyperbolic transcendental functions [duplicate]

I have tried and tried but cannot for the life of me see how one equation follows onto the other... can anybody help?? $$\Omega(\theta)=-b.\coth(\operatorname{arsinh}(\exp a\theta . \sinh(c_0)))$$ ...
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41 views

Osborne's rule for hyperbolic functions?

I am confused as to why you only change the sign for powers of sine that are 4n+2. As I understand, $sin(i\theta)=isinh(\theta)$ $sin^2(i\theta)=-sinh^2(\theta)$ $sin^3(i\theta)=-isinh^3(\theta)$ ...
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0answers
25 views

Rearranging equations using hyperbolic transcendental functions [duplicate]

I have tried and tried but cannot for the life of me see how one equation follows onto the other... can anybody help?? $$\Omega(\theta)=-b.\coth(\operatorname{arsinh}(\exp a\theta . \sinh(c_0)))$$ ...
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2answers
61 views

Computing Residue for a General, Multiple-Poled function?

I'm trying to compute the residue of the following function at $a$. I'm having a little trouble seeing which poles are relevant: Compute $\,Res_f(a)$ for the following function: $$f(z) = ...
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1answer
24 views

Differentiating hyperbolic functions.

$\DeclareMathOperator{\sech}{sech}$Can anyhow me how to differentiate the following? I already tried using the product rule, but I can't quiet seem to succeed. $\sech^{2} x$. $2\bigl(\cosh(2x) - ...
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1answer
39 views

Hyperbolic Trig Functions - Identities

I don't understand how the 3rd step (the 4 divisions) happens? Can someone explain how they arrived at that.
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3answers
56 views

Solving an equation with hyperbolic functions

I'm trying to prove that for a given $s,t\in\mathbb{R}$ there exists $w\in\mathbb{R}$ such that $\cosh(t)e^{i(s+w)}+\sinh(t)e^{i(s-w)}\in\mathbb{R}$. How to solve this?
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2answers
235 views

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 ...
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2answers
44 views

Proving $\ln \cosh x\leq \frac{x^2}{2}$ for $x\in\mathbb{R}$

It seems to me that $\ln \cosh x\leq \frac{x^2}{2}$ for $x\in\mathbb{R}$, as suggested by graphing the difference between both functions as well as the fact that the Taylor series expansion of ...
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1answer
38 views

What is the $n$th derivative of $\coth(x)$?

I would like to know the $n$th derivative of the Hyperbolic Cotangent, i. e., $\frac{\partial^n}{\partial x^n} \coth( x )$. So far, I have only found an expression for the $n$th derivative of the ...
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3answers
98 views

A level Integration: $\int\frac{x^3}{\sqrt{x^2-1}}dx$

Using the substitution $x=\cosh (t)$ or otherwise, find $$\int\frac{x^3}{\sqrt{x^2-1}}dx$$ The correct answer is apparently $$\frac{1}{3}\sqrt{x^2-1}(x^2+2)$$ I seem to have gone very wrong ...
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1answer
31 views

Solve for a hyperbolic Laplace Transform by expressing as exponents and shiftig on s-axis (5.3-21)

I cannot get past a certain point on this problem as shall be shown. I need guidance in order to complete the problem. The exercise as stated in the text: Represent the hyperbolic function in terms ...
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3answers
43 views

How can I calculte the probability of $X$ with a Generlized Hyperbolic Distribution?

I would like to know how to calculate the probability of $X$ when I have fitted a Generalized Hyperbolic Distribution to my data set. The depth of my knowledge is basic t-tests and z-tests. I am ...
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1answer
31 views

Stuck on an integration question…

$$\int x^{-\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)dx$$ The answer I should get is $$2x^{\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)-4(x+4)^{\frac{1}{2}}$$ but I keep going wrong. Can someone show me how to ...
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0answers
30 views

Complex Mapping of $\mathrm{cosh}(w)=z$

Mapping in complex analysis has not been very easy for me unfortunately. I am having difficult trying to find the mapping between the z and w plane. I attempted to simply write that ...
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1answer
19 views

Generalizing 1D function to higher dimensions

I have a function in 1D given by $f(x) = \tanh(x-x_1) + \tanh(x-x_2)$. I want to generalize this to two dimensions, such that it describes a circle. The function $f(x,y)$ has to have a form such that ...
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1answer
37 views

A level Integration question.

1a) Prove that $$e^x\operatorname{sech} x\equiv\frac{2e^{2x}}{e^{2x}+1}$$ b) Find $$\frac{d}{dx}[\arcsin(\tanh x)]$$ Simplify your answer as far as possible. c) Hence, or otherwise, solve $$\int ...
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2answers
304 views

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 ...
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2answers
38 views

Simplifying a hyperbolic trigonometric expression [closed]

How can I rewrite $$\tanh(x)\left(-\frac12 \mathrm{sech}^4(x) + \frac12 \mathrm{sech}^2(x)\right)$$ as $$\frac12 \mathrm{sech}^2(x)\tanh^3(x)?$$
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26 views

Hyperbolic trigonometric inequality

Is the following hyperbolic trigonometric inequality correct and if so, is there a simple derivation? $$\tanh A- \tanh B \geq (A-B)(\operatorname{sech}{^2}(A)), \qquad \forall A\ge B\ge 0.$$
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1answer
27 views

Trouble finding the equation of the inverse of the hyperbolic tangent

I'm trying to find the equation of the inverse of the hyperbolic tangent as follows: Take an $x \in ]-1,1[$ and define $y:=\text{arctanh}(x)$ so that $\tanh y=x$. This means that ...
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1answer
98 views

Solve $x \tanh(x) = constant$

Does the following equation admit a real solution: $x\cdot \tanh(x) = C$ with $C$ a constant. While I was not able to find a specific answer with symbolic calculations, this solutions seems to ...
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2answers
66 views

Integral question showing the primitive functions differ only by a constant?

$$\int \frac{dx}{\sqrt{x^2-6x+13}}$$ $$\int \frac{dx}{\sqrt{\left(x-3\right)^2+4}}$$ It can be solved by Method 1 Let $$x-3=2\tan u$$ $$dx=2\sec^2 u\,du$$ Therefore, using the trigonometric ...
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2answers
56 views

hyperbolic tangent vs tangent

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side (in a right triangle). Is there a similar definition for the hyperbolic tangent? The reason ...
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2answers
171 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 ...
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1answer
61 views

Hard integral of root function and hyperbolic function

I need to calculate this integral: $$\int^A_B\frac{\sqrt{x-B}}{\cosh^2x}dx$$ Is there any way to do this?
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6answers
298 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 ...
7
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1answer
180 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 ...
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1answer
40 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 ...
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4answers
217 views

Taylor series extension of tanh

I know how to find the taylor expansion of both sinh and cosh, but how would you find the taylor expansion of tanh. It seems you can't just divide both the taylor series of sinh and cosh so how would ...
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1answer
40 views

Hyperbolic Intuition.

I am working with hyperbolic functions and was wondering where they actually came from. I am under the understanding that Ricotta (and I think Johann Heinrich Lambert also did work in this area), did ...
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1answer
35 views

Evaluating $\lim_{x\to\infty}\left(\frac{\sinh x}{x}\right)^{{1}/{x^2}}$

$$\lim_{x\to\infty}\left(\frac{\sinh x}{x}\right)^{\dfrac{1}{x^2}}$$
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How to find the inverse function of $f(x)=\frac{\sinh(\ln(\cosh x))}{\sinh x}$

How to find the inverse function of $f(x)=\frac{\sinh(\ln(\cosh x))}{\sinh x}$? I' ve tried the following: $y=\frac{\sinh(\ln(\cosh x))}{\sinh x}$ . Now I should express $x$ in terms of $y$. Then: ...
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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 ...
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1answer
46 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?
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1answer
49 views

Solve ${y}' = \cosh^{-1}\left ( x \right ) + \mathrm {Si}(x)$

I am wondering how to find an explicit, closed-form solution for the following first-order differential equation: $${y}' = \cosh^{-1}\left ( x \right ) + \mathrm {Si}(x)$$ Where $\mathrm {Si}(x)$ ...
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2answers
138 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 ...
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27 views

inverse hyperbolic function of a complex argument

It is not too difficult to prove that $f(z)=\cosh z$ is a bijection from $$\def\C{{\Bbb C}}D=\{\,z=x+iy\in\C\mid 0<y<\pi\,\}$$ to $$R=\{\,w=u+iv\in\C\mid v\ne0\,\}\cup\{\,w\in\C\mid ...
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60 views

Help with a hyperbolic trig problem

$$\tanh n=\operatorname{csch}n$$ Solve so that $n=\ln(x\pm x^{1/2})$ $%replace "x^{1/2}" with "\sqrt{x}" if you want. - editor$ I need some advice with this problem; I answered a similar one ...
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1answer
42 views

How to solve transcendental hyperbolic equation

How can I solve the functional relation $$ e^{-af'(x)}\cosh( f(x) ) = bx $$ for $f(x)$? It would suffice to solve for $x>0$, $a>0$ and $b>0$.
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1answer
36 views

Recognising that $\sum_{n=0}^\infty \frac{a^2-b^2(2n+1)^2}{(a^2+b^2(2n+1)^2)^2}=-\frac{\pi^2\mathrm{sech}^2\left(\frac{a\pi}{2b}\right)}{8b^2}$

So I know from Mathematica that: $$\sum_{n=0}^\infty \frac{a^2-b^2(2n+1)^2}{(a^2+b^2(2n+1)^2)^2}=-\frac{\pi^2\mathrm{sech}^2\left(\frac{a\pi}{2b}\right)}{8b^2}$$ I am wondering how someone could ...