Questions tagged [hyperbolic-functions]
For questions related to hyperbolic functions: $\sinh$, $\cosh$, $\tanh$, and so on.
988
questions
0
votes
0answers
26 views
Evaluate error of $\tanh{x}$ with inequality [duplicate]
I have to prove the inequality
$\vert \tanh(x)-(x-\frac{x^3}{3})\vert\leq\frac{1}{8}$ for $x\in[-1/4,1/4]$
where $x-\frac{x^3}{3}$ is the 3rd order Taylor expansion at $0$.
I know I have to somehow ...
0
votes
0answers
28 views
integral of Bessel function of first kind with Hyperbolic function
I'd like to solve the following equations:
$$\int_{0}^{\infty} \frac{J_{0}(z)+J_{2}(z)}{z+a_{0}+a_{1}\left(\operatorname{coth}\left(a_{2}z\right)-1\right)} dz$$
where $J_{0}$ and $J_{2}$ are Bessel ...
3
votes
1answer
42 views
$\frac{1}{\cosh{x}}+\log\left ( \frac{\cosh{x}}{1+\cosh{x}} \right )$ for $x \rightarrow \pm \infty$ has a limit
Show from the definition of a limit that $$\frac{1}{\cosh{x}}+\log\left ( \frac{\cosh{x}}{1+\cosh{x}} \right )$$ for $x \rightarrow \pm \infty$ has a limit.
My attempt
This one is really tough for me. ...
0
votes
4answers
81 views
Express $\operatorname{sech}^{-1}(x)$ in terms of logarithms
I'm trying to express the following $\operatorname{sech}^{-1}(x)$ in terms of logarithms, and would warmly appreciate feedback towards my approach. The solution should be :
$$\ln\left(\dfrac{1+\sqrt{(...
1
vote
2answers
71 views
Parameterizing both branches of a hyperbola
Recently I have been studying parametric equations of surfaces and curves, specifically hyperbolic functions. Given by the equations $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \quad\frac{(x-\alpha)^2}{a^2}-\...
1
vote
1answer
68 views
Show $\cosh x$ and $\sinh x$ are continious using $\varepsilon - \delta$ proof
I have to prove that $\sinh x$ and $\cosh x$ are continuous functions. I have to use the hyperbolic addition formula, and the inequalities:
$|\sinh x| \leq 3|x|, \, |x|<\frac{1}{2}$
$|\cosh x -1| ...
0
votes
1answer
22 views
What parameters can be used with the hyperbolic tangent function to enable optimised curve fitting?
If I want to fit a sigmoid curve to data using the logistic function I use something like
$$y = \frac{L}{1 + \exp[-k(x-x_{0})]} + b$$
where $L$, $k$, $x_{0}$ and $b$ are functional parameters that can ...
6
votes
1answer
98 views
Why is $\sin(\tanh x) + \sinh(\tanh x)$ almost exactly $2\tanh x$?
I was trying to come up with some approximations for the solution to the differential equation $y'' + \operatorname{sgn}(y') + y = 0$ and noticed while I was messing around that $\sin(\tanh x) + \sinh(...
0
votes
2answers
59 views
Expansion of $(\sinh(x))^{\frac15}$ around 0 for x > 0
I'm aware of the series expansion of the hyperbolic functions, but how does one expand a fractional power of sinus hyperbolicus, i.e. e.g. $(\sinh(x))^{\frac15}$?
0
votes
1answer
109 views
$\epsilon,\delta$-proof for the limit of $\log \sinh{(x^2)}-x^2$ for $x \rightarrow \infty$
I want to use a $\epsilon,\delta$-proof for the existence and value for the limit of $$\log \sinh{(x^2)}-x^2$$
for $x \rightarrow \infty$.
Now, I know the definition for such proof to be $\forall \...
0
votes
3answers
75 views
Show $\left | \cosh{x}-1 \right |\leq 3 \left | x \right |$ for $\left | x \right |<1/2$
I have to show that $\left | \cosh{x}-1 \right |\leq 3 \left | x \right |$ for $\left | x \right |<1/2$. I can not use derivatives, series of the trigonometric functions etc. I have to use generel ...
-1
votes
0answers
62 views
Hyperbolic sine strictly increasing [duplicate]
Proof without using derivatives (and that $e^x$ is striclty increasing) that $\sinh{x}$ is strictly increasing on $\mathbb{R}$.
I am having troubles with a few things. I know strictly increasing would ...
1
vote
1answer
61 views
Proving $\sinh{x}$ is strictly increasing over all the reals and $\cosh{x}$ is strictly decreasing on $(- \infty , 0]$
How would I be proving $\sinh{x}$ is strictly increasing and $\cosh{x}$ is strictly decreasing on $(- \infty , 0]$
I succeded in showing $\cosh{x}$ is strictly increasing on the interval $[0, \infty)$ ...
0
votes
1answer
20 views
Jacobi $sd(u,m)$ hyperbolic approximation for $m \to 1$
I am following a paper (Fink 1976); the details are a mathematical problem.
We are given a solution (paraphrased from the paper for clarity)
$$y^2 = \alpha m_{1n} sd^2(u,\sqrt{m_n}),$$
where $sd$ is a ...
1
vote
2answers
48 views
Proving $\cosh(\sqrt{1+x^2})$ is not uniformly continuous in $\mathbb{R}$
Given
$f(x) = \cosh(\sqrt{1+x^2})$
I am trying to show that $f(x)$ is not uniformly continuous. Specifically:
$\exists\varepsilon\ \forall\delta\ \exists x,y \in \mathbb{R}: \ |x-y| < \delta \wedge ...
8
votes
0answers
76 views
This (rather long) implicit equation has a short explicit solution, but how can it be found?
I am curious if a method exists for solving for $k$ or $h$ in this implicit equation:
$$\frac{k^2}{h} \mathrm{sech}^2(k) \sqrt{1 + \left(\frac{k}{h} \tanh(k)\right)^2} = \ln\left( \frac{k}{h} \tanh(k) ...
2
votes
0answers
44 views
integration of the product of shifted functions
There is an integral having a following form : $\int f(x) f(ax-b) dx$ . Is there any general way/substitution that makes this kind of integrals easy to solve? Particularly, my functions are shifted ...
0
votes
0answers
69 views
Vector Addition/Translation in Hyperboloid model
I have problems understanding vector addition in Hyperbolic space. In the Poincaré ball model, vector addition/translation is the Möbis addition and defined as:
$$ x \oplus_c y = \frac{(1+2c\langle x,...
0
votes
0answers
41 views
Cosine Transform of Hyperbolic Functions
Does anyone know a closed form of the integral:
$\int^{\infty}_{0}\text{d}x \cos(k x) \frac{\sinh(\frac{3\gamma-\pi}{4} x)}{\sinh(\frac{\gamma}{4} x)\cosh(\frac{2\gamma-\pi}{4} x)},$
where $\frac{\pi}{...
8
votes
2answers
97 views
How do you prove Osborn's rule?
Given a trigonometric identity written in terms of sine and cosine, it is possible to write down the corresponding hyperbolic identity using Osborn's rule:
Replace $\cos$ with $\cosh$
Replace $\sin$ ...
1
vote
2answers
33 views
Evaluate the flux of $F=\langle\sin(xyz), x^2y, z^2e^{x/5}\rangle$ through surface $S$ … $4y^2+z^2=4, \space x\in [-2,2]$
I am trying to find the flux of $\vec F=\langle\sin(xyz), x^2y, z^2e^{x/5}\rangle$ through the surface $S$ where $S$ consists of the elliptical cylinder defined by
$S$ ... $4y^2+z^2=4, \space x\in [-...
0
votes
3answers
47 views
Showing $\lim_{\nu\to\infty}\ln(\coth(\frac\nu2))\to2e^{-\nu}$ and $ \lim_{\nu\to0}\ln(\coth(\frac\nu2))\to-\ln(\frac\nu2)$
I am struggling to derive a couple limits I have come across in a paper I am reading. Both involve the natural log of the hyperbolic cotangent. The paper seems to be saying the two terms trend the ...
1
vote
1answer
38 views
What is wrong with my derivation of the exponential form of $\sinh(x)$
I know the exponential form of $\sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}$. I wish to derive the exponential form of $\sinh(x)=\dfrac{e^x-e^{-x}}{2}$, and I am stuck.
I try to replace $\sin(x)$ with $\sin(i^...
27
votes
6answers
893 views
Unifying the connections between the trigonometric and hyperbolic functions
There are many, many connections between the trigonometric and hyperbolic functions, some of which are listed here. It is probably too optimistic to expect that a single insight could explain all of ...
3
votes
2answers
81 views
What is a slower decay rate than hyperbolic?
Hyperbolic decay
$$f_\alpha(x)=\frac{1}{\alpha x + 1}
$$
is slower than exponential decay
$$f(x) = e^{-x}$$
where $\alpha > 0$ is a scaling factor. The larger that $\alpha$ is, the steeper the ...
-1
votes
2answers
126 views
How to simplify $cosh(arccoth(x))$? [closed]
I was given the problem to simplify $\cosh(\text {arccoth} (x))$ for $|x| > 1$, and I was just wondering how I would do that.
1
vote
1answer
139 views
Fourier transform of hyperbolic function $\frac{d}{dx}\log({\sinh{x}})$
I am trying to calculate the FT of $\frac{d}{dx}\log({\sinh{x}})$.
I did not calculate it directly but I used the result of its derivative.
This is a tentative: I found the following result in the ...
-2
votes
1answer
35 views
What do sinh and cosh have to do with exp? [closed]
My friend told me that $\sinh$ and $\cosh$ result from an exponential function, but I can't figure out why
-2
votes
1answer
66 views
Show that $ 2 \sinh(z)=\exp(z)-\exp(-z)$ [closed]
$ 2 \sinh(z)=\exp(z)-\exp(-z)$;
$ 2 \cosh(z)=\exp(z)+\exp(-z)$
where $z \in \mathbb{C} $
$$\sin(z) := \sum_{k=0}^{\infty}\frac{(-1)^k}{2k+1!} z^{2k+1}$$
$$\cos(z) := \sum_{k=0}^{\infty}\frac{(-1)^k}{...
0
votes
1answer
40 views
Indefinite integral of $\int \frac 1 x \operatorname{arsech} \frac x a \, \mathrm d x$
Spiegel's "Mathematical Handbook of Formulas and Tables" (Schaum, 1968), item $14.668$ gives the indefinite integral of the area hyperbolic secant (that is, the "inverse" ...
0
votes
2answers
16 views
Range of Real Inverse Hyperbolic Cosine — can it be negative?
There are several results in Spiegel's "Mathematical Handbook of Formulas and Tables" (Schaum, 1968) concerning $\cosh^{-1}$ which are presented in the following format:
$$\cosh^{-1} x = \pm ...
1
vote
5answers
71 views
Maclaurin expansion for sech$(x)$
I am a bit unsure where I have gone wrong in working this out.
Sech$(x)=2/(e^x+e^{-x}).$
Maclaurin expansions:
$e^x = 1+ x + x^2/2+ x^3/6 + x^4/24;\; e^{-x} = 1- x + x^2/2 - x^3/6 - x^4/24;$
so sech$...
2
votes
1answer
36 views
When proving hyperbolic identities why do we add one or two
Example:
$\cosh^2x+\sinh^2x=\cosh2x$
Proof:
$$\frac{1}{2}(e^x+e^{-x})^2+\frac{1}{2}(e^x-e^{-x})^2$$
Where does this two's come from?
$$\frac{1}{4}(e^{2x}+e^{-2x}+2)+\frac{1}{4}(e^{2x}+e^{-2x}-2)$$
0
votes
3answers
42 views
Hyperbolic functions simplifying
How do you simplify
$$\cosh(\sinh^{-1}(x))$$
to become
$$(1+x^2)^{1/2}$$
I have managed to get $(1+\sinh^2(\sinh^{-1}(x))^{1/2}$ but haven't been able to progress from there.
0
votes
0answers
20 views
If $f,g \in PSL(2,\mathbb{C})$ then $\langle f,g \rangle$ is not discrete
Let $H^3 = \mathbb{C} \times \mathbb{R}^+$, the $Isom(H^3) = PSL(2,\mathbb{C})$
Now.
I try to show that if $f,h \in PSL(2,\mathbb{C})$ with a exactly one common fixed point then the group $\langle f,h\...
2
votes
1answer
32 views
How to deduce that $\arctan\left(\frac{a}{i}\right)=-i\cdot \text{ arctanh}(a)$
I was doing some calculations in wolframalpha and I found the following equality:
$$\arctan\left(\frac{a}{i}\right)=-i\cdot \text{ arctanh}(a)$$
This is the first time I've seen this equality. How is ...
2
votes
1answer
46 views
Verifying the hyperbolic law of cosines and law of sines..
I am trying to understand hyperbolic triangles. These two laws are supposedly universal in the hyperbolic plane.
However, whenever I've tried to verify them using applets which let me construct ...
0
votes
0answers
25 views
Existence of two constants related to the prime counting function
Well it's the continuation of Conjecture : A lower bound for the prime counting function . .
Rearranging some terms and adding a fixed exponent we work with the expression :
$$S(n)=\sum_{k=1}^{n}\frac{...
1
vote
1answer
73 views
Good approximations for $\int_{-\infty}^0 \bigg(\tanh(ax+b)\tanh(cx+d) - 1\bigg)~\mathrm{d}x$
I have an integral
$$I(a,b,c,d) = \int_{-\infty}^0 \bigg(\tanh(ax+b)\tanh(cx+d) - 1\bigg)~\mathrm{d}x$$
which I need to approximate analytically (to use it in further steps within a machine learning ...
0
votes
0answers
22 views
Find the Volume V of the solid obtained by rotating the region bounded by the given curves about the y-axis: zy=8, z=0, y=1, y=2
I am aware that I am finding the volume of a hyperbola about the y-axis. I will probably do this using the shell method: $V=2\pi \int_a ^b x(f(x))\,dx$.
Any help would be appreciated
4
votes
1answer
49 views
Conjecture : A lower bound for the prime counting function .
Well it take my a little bit of time to find it but now I think that my conjecture is ready .
Conjecture :
Let $n\geq 100$ and then define the sum :
$$S(n)=\sum_{k=1}^{n}\frac{1}{\operatorname{...
1
vote
1answer
56 views
Integrating $\text{sech}(x)$ using a hyperbolic substitution method
I have been tasked to find $\int{\text{sech}(x)dx}$ using both hyperbolic and trig substitutions, for the trig substitution method I did the following.
$$I=\int{\frac{2e^x}{e^{2x}+1}dx} $$ $$\text{Let}...
1
vote
1answer
52 views
Prove the following and use it to evaluate the integral: [duplicate]
I want to prove that:$$\int_{-\infty}^\infty f(x)dx=\int_{-\infty}^\infty f\left(x-\frac1x\right)dx$$
And use the result of this proof to evaluate:$$\int_{-\infty}^\infty\frac{x^2}{x^4+1}dx$$
0
votes
0answers
32 views
Show that the hyperbolic sine function is bijective
I am supposed to show that sinh : R -> R is bijective and I'm a bit lost. I suppose the part of showing that it's injective isn't so bad but I'm having some problems with showing that it's ...
2
votes
1answer
45 views
Find $\int\cosh^\frac{1}{2}u~du$ and/or $\int\frac{1}{\cosh^\frac{1}{2}\theta}~d\theta$
I am trying to find a function of the form $y=f(x)$ such that the volume of the solid generated by the function between any two points around the $x$ axis is numerically equal to its length bewteen ...
1
vote
0answers
42 views
Prove $\tanh^{-1}\left(\sqrt{\tanh(x)}\right)-\tan^{-1}\left(\sqrt{\tanh(x)}\right)\geq 0$ without using derivative
Claim :
Let $0\leq x$ then prove (without using derivative) that :
$$\tanh^{-1}\left(\sqrt{\tanh(x)}\right)-\tan^{-1}\left(\sqrt{\tanh(x)}\right)\geq 0$$
Trick :
We have using integral :
$$\tanh^{-1}\...
1
vote
1answer
29 views
Prove $\coth\;2v = \frac{x^2 + y^2 + 1}{2y}$
Q:
Given that $x + jy =\tan (u + jv)$, prove that
$$\coth\;2v = \frac{x^2 + y^2 + 1}{2y}$$
I would like to ask this question, how can we prove it?
I had tried to expand the equation $x + jy =\tan (u + ...
2
votes
1answer
53 views
Proving that $|\sin(z)|^2 = \sin(x)^2+\sinh(y)^2$
Goal: Prove $|\sin(z)|^2 = \sin(x)^2+\sinh(y)^2$
I have
\begin{align*}
\sin(x+iy)
&=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\
&=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\
|\sin(z)|^2
&=\sin(x)^2 \cosh(x)^...
0
votes
0answers
12 views
Symmetry of piecewise defined function
Let
$$
f(x) =
\begin{cases}
2-\cosh(x), & \text{for } |x| \leq d \\
\alpha e^{-|x|}, & \text{for } |x| > d
\end{cases}
$$
with a positive constant $d$ and parameter $\alpha$.
How can I ...
3
votes
0answers
124 views
$L\{f(x)\}^{-1}=\int^x_a \frac{dx}{f(x)}$
let $a<b\in \overline {\mathbb R}$ such that $\lim_{x\to a}f(x)=0$, Let $f:(a,b) \to \mathbb R$ be continuous and positive on $(a,b)$
$$L\{f(x)\}^{-1}=\int^x_a \frac{dx}{f(x)}$$
Where $f(x)^{-1}$ ...
