Questions tagged [hyperbolic-functions]
For questions related to hyperbolic functions: $\sinh$, $\cosh$, $\tanh$, and so on.
1,317 questions
0 votes
0 answers
51 views
Fourier transform of the hyperbolic cosine? [closed]
Is there a way to define the Fourier transform of the hyperbolic cosine? $$ \cosh (x/2) $$ My only idea is to expand the exponential functions into taylor series and use the dirac delta derivatives
- 43
3 votes
4 answers
198 views
How to find the locus of points inside a parallelogram such that 'opposite' angles when you connect vertices to the point are supplementary?
After doing problem 1 on the 2011/2012 BMO2 I was encouraged by the book "A Mathematical Olympiad Companion" by Geoff Smith to look at parallelograms, and the case when you have a point ...
- 41
7 votes
2 answers
237 views
Finding a closed-form analytical expression for the improper integral $\int_0^\infty \frac{k^2 \, \mathrm{d}k}{\sinh k + k}$
While working on a fluid mechanics problem involving the rotational drag coefficient in a slit geometry in the Stokes regime, I encountered the following improper integral arising from an inverse ...
- 274
0 votes
1 answer
110 views
Prove $\lim\limits_{R \to \infty} \int_0^{\infty} \frac{Rt + \sinh t}{1 + (Rt + \sinh t)^2}dt = 0$.
This limit has come up when trying to bound the contributions of a certain complex contour integral. Numerical checks shows this limit tends to $0$, albeit very slowly. I think it is easy to show that ...
- 1,529
5 votes
2 answers
210 views
Any other method to compute $\int_0^\pi \tan ^{-1}(1+\cos x) d x$? [duplicate]
The decency of the exact value of $$ \int_0^\pi \tan ^{-1}(1+\cos x) d x=\pi \cot ^{-1} \sqrt{\phi}, \quad \textrm{where $\phi$ is the golden ratio.} $$ attracts me to tackle it. For simplicity, let’...
- 33.2k
10 votes
2 answers
375 views
Evaluation of $\int_{0}^{\infty}\frac{\operatorname{sech}\left(\frac{x}{2}\right)}{(\pi^2+x^2)^2}dx$
I found this integral on FB page : $$I = \int\limits_0^\infty {\frac{1}{{\left( {1 + {x^2}} \right){{\left( {\frac{{{\pi ^2}}}{4} + {{\ln }^2}\left( x \right)} \right)}^2}}}dx} $$ Here is what I ...
- 4,303
6 votes
3 answers
269 views
Is there any other simpler answer/method to the integral $\int _0^{\infty }\frac{x}{1+2\cosh x}dx$?
When I met the integral, $$\int _0^{\infty }\frac{x}{1+2\cosh x}dx,$$ I just wondered whether there is a simpler method to evaluate it and started with an usual substitution: $y=e^{-x}$ and obtained $...
- 33.2k
0 votes
3 answers
132 views
$\cosh (\pi \sin (\pi r))$ representation using trigonometric functions
Investigating the $$\cosh (\pi \sin (\pi r))$$ Looked at the graph and seems it is somehow related to some combination of $sin$ and some unknwon constants $$1.68 \pi (\sin (2 \pi r-1.68)+2.4)-2 \...
- 2,066
0 votes
2 answers
112 views
Integral of linear, Gaussian, modified bessel function of first kind, and hyperbolic cosine functions
Within the context of studying fluids turbulence in wakes, I stumbled upon an integral in the form $$S(\alpha,\beta)=\int\limits_0^\infty x \exp{(-x^2)} I_0(\alpha x) \cosh(\beta x)~dx,$$ where $I_0$ ...
- 91
9 votes
2 answers
536 views
prove that : $\int^{\infty}_0 \frac{\sin\left({\frac{2}{\pi}x^2}\right)}{\sinh^2(2x)} dx =\frac{1}{8}$
Proving : $$\int^{\infty}_0 \frac{\sin\left({\frac{2}{\pi}x^2}\right)}{\sinh^2(2x)} dx =\frac{1}{8}$$ I need help to prove integral without using residue theorem . we have : $$\coth(z) = \sum_{n=-\...
- 3,235
2 votes
2 answers
159 views
Generalisation of $\int \frac{1}{\sqrt{2-\tan (x)}} d x $
The closed post seeks for help to find the elementary primitive function of the integrand. The Wolfram-alpha, by resolving the integrand into partial functions, found a primitive with $i$ $$ \int \...
- 33.2k
1 vote
1 answer
169 views
Evaluation of a Gaussian‑Weighted Hyperbolic Integral
During my investigations into Mordell’s integrals, I came across this interesting and challenging integral. I managed to express it as an infinite series involving the error function, but I’m not ...
1 vote
2 answers
215 views
Is the solution to $ \tanh(\beta K_3) -$ $\tanh(\beta K_1) \tanh(\beta K_2)$ $=$ $0$ unique?
For any parameters $K_1, K_2 > K_3 >0$ there exists a solution $\beta >0$ of the following equation $$ \tanh(\beta K_3) - \tanh(\beta K_1) \tanh(\beta K_2) = 0 $$ This fact is easy to see by ...
- 571
2 votes
1 answer
117 views
Finding a closed form expression for $\sum_{n=2}^{101}\operatorname{arctanh}(n)$
Last evening I was working a problem where a part of it required me to find the the value of the expression $$\tanh\left(\sum_{n=2}^{101}\operatorname{arctanh}(n)\right).$$ At first I just recalled ...
- 1,673
0 votes
0 answers
105 views
Inequality involving $\cosh$.
Let $ x_1, x_2, y_1, y_2 \in \mathbb{R}$, and let $\theta \in (0,1]$. I would like to prove the following inequality: $$ \cosh(x_1)\cosh(y_1)\cosh(x_2)\cosh(y_2) - \cosh(x_1 + x_2)\cosh(y_1 + y_2) \...
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