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

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1answer
41 views

Laplace's equation-separation of variables

I am looking at the $2$-D Laplace's equation $$\nabla^2u=u_{xx}+u_{yy}=0$$ $$u(x,0)=f(x), x \in (0,a)$$ $$u(x,b)=0, x \in (0, a)$$ $$u(0,y)=u(a,y)=0, y \in (0,b)$$ The solution is in the form ...
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2answers
29 views

Complex Numbers and Hyperbolic Functions

How would you evaluate: $\mathfrak{R}\left[(1+i)\sin\left(\dfrac{(2+i)\pi}{4}\right)\right]$? I know that $\cos x = \dfrac{e^{ix}+e^{-ix}}{2}$ and $\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}$. I have also ...
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1answer
27 views

Disappearing negative signs when evaluating a sinh^-1 integral

$$\int_{-2}^{6} \frac{1}{\sqrt{1+(-x)^2}} \, dx$$ When performing this integral on paper, I get $$\sinh^{-1}(6) - \sinh^{-1}(-2) $$ But when I type it on wolframalpha, I get the unintuitive answer ...
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1answer
36 views

Inverse trig and trigh in integration?

I have just done part (iii) of this question and can get the right answer but am a bit confused why do we take arcosh i.e. just the principle value of cosh and not the other value. I presume this is ...
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3answers
61 views

Separate Into Real and Imaginary Parts

Separate the following trigonometric function into Real and Imaginary Parts $$\tan^{-1}e^{i\theta} $$ or $$\tan^{-1}(\cos\theta+i\sin\theta)$$ I Have made till here Assuming $x+iy$ is the final ...
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1answer
22 views

Find the value of hyperbolic $\tanh x$ function from the equation

If $\sinh x-\cosh x=5$, find $\tanh x$ I have done till the following steps but dont know how to proceed further from solving this equation in Euler's form ...
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1answer
622 views

Taylor Series of Hyperbolic Cotangent Coth(x)

Expanding about 0 gets me a divergence on the first term, and the wikipedia article says nothing about how to derive it other than taylor series. It makes me think I'm supposed to use Laurent Series, ...
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2answers
16 views

Does the definite integral of (1 - tanh t) from 0 to x diverge as x goes to infinity?

Decidedly in the category of things I used to know how to prove but have forgotten: Does $$ \int_0^x (1 - \tanh t) \,dt $$ converge or diverge as $x \to \infty$? (I know that the indefinite ...
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1answer
25 views

hyperbolic inequality

Calculating some contour integral, I have to prove that $\int^{R+i}_{R}\frac{cosh(az)}{cosh(\pi z)}dz$ goes to zero if R goes to infinity. And we know that $\left|a\right|<\pi$. I want to use the ...
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2answers
40 views

Is this a typo, or am I missing something?

I have a handout for my precalc II class. It says $\sinh(-x) = -\sin(x)$ It should be $\sinh(-x) = -\sinh(x)$ right? I don't see how a negative input could make a hyperbolic function circular.
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1answer
29 views

Solving the Laplace equation in terms of exponential of hyperbolic trigonometric functions

I'm solving the Laplace equation $U_{xx}+U_{yy}=0$ subject to BC's: \begin{align} U(0,y) &= 0 \\ U(a,y) &= 0 \\ U(x,0) &= 0 \\ U(x,b) & = \left\{x \text{ for } x \in ...
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2answers
19 views

Calculate ch(0.2) to the nearest 0.01

Help me calculate ch(0.2) to the nearest 0.01. I tried to rewrite ch as a series but I still don't know how to evaluate it and what to do with factorial Help me please. it's very important
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1answer
287 views

Integration Using Hyperbolic Substitution

I'm supposed to prove the following by using the hyperbolic sine double angle identity: $\sinh(2x)=2\sinh x\cosh x$ and archsinhx formula: $\ln(x+\sqrt{x^2+1})$ but can't seem to figure out the steps. ...
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1answer
232 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 ...
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2answers
76 views

I cannot find the following integral in an integral table.

In the appendix A of this paper there is an integral that the author says can be solved using any good integral table. However I cannot seem to find it on any integral table (ex: gradshteyn and ...
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0answers
42 views

Poles (and order of) of $(\cosh(1/(z-\pi)))^2$ and Residue at $z=\pi$

Hi I need help finding the poles and the order of the poles of the following function: $$\left(\cosh\frac1{z-\pi}\right)^2$$ and the residue at $z=\pi$. I have tried a number of different methods ...
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1answer
63 views

How do I solve this integral with hyperbolic functions?

I was studying mechanics when I f ound a problem that lead to an integral that I can't solve. Basically the problem asked to find the period of oscillation function of the energy $E$ of a particle ...
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1answer
38 views

Problem with Taylor (asymptotic) expansion of hyperbolic functions at infinity

(Note: I chose a general title, because I believe this discussion will be applicable to all other hyperbolic functions having an asymptote at infinity, but I will specifically be focusing on ...
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5answers
378 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 ?
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1answer
85 views

Maximum value of tanh Z

$$ f(z) = \tanh (z) = \dfrac{e^z - e^{-z}}{e^z + e^{-z}} $$ Find the point $z$ with $|z| \leq1$ where $|f(z)|$ attain its maximum. I figured out that the maximum is probably at the edge (concluded ...
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2answers
22 views

Write the following as an algebraic expression of x; sinh(lnx)

So I have a questions asking for $sinh(log_ex)$ sinh(x) = $\frac{e^x-e^{-x}}{2}$ So $sinh(log_x)$ = $\frac{e^{lnx}-e^{-lnx}}{2}$ $\frac{e^{lnx}-e^{lnx^{-1}}}{2}$ = $\frac{x - x^{-1}}{2}$ ...
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1answer
56 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 = ...
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0answers
42 views

How to find the value of this sum?

The sum below numerically (to 13th digit at least) is the same as $\ln 2$. So there should be a way to prove it analytically, but I haven't succeeded. Any suggestions? ...
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3answers
147 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 ...
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4answers
266 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 ...
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1answer
75 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 ...
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1answer
42 views

Finding a hyperbola's equation based off given asymptotes

I need help finding the equation of a hyperbola that opens vertically with asymptotes $y=2x+11$ and $y=-2x-1$. I also need help finding the equation of a different hyperbola that also opens upwards ...
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1answer
44 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 ...
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1answer
42 views

find $\int{artanh(x)} dx$

I've been asked to find $\int{artanh(x)} dx$ The first thing I did was I said: let $y = artanh(x)$ $\therefore \dfrac{dy}{dx} = \dfrac{1}{1-x^2}$ $\therefore dx = (1-x^2)dy$ Also, from $y = ...
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1answer
133 views

Need help with a Fourier Transform Question

I need an way to solve this Fourier transform problem. $$ f(t)= \begin{cases} \cosh(t) & \text{ For } |t|<1\\ 0 & \text{ For }|t|>1 \end{cases} $$ The given answer for the ...
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2answers
44 views

Expressing $\mathrm{B}(\sinh(x), \cosh(x))$ in terms of elementary functions

Is it possible to express: $\mathrm{B}(\sinh(x), \cosh(x))$ (where $\mathrm{B}$ is the beta function) In closed form, in terms of elementary functions?
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1answer
34 views

Find a linear combination of $u_n$'s satisfying $u(x,1) = \sin(2\pi x) -\sin(3\pi x)$

I have the following problem: $$u_n(x,y) = \sin(n\pi x)\sinh(n\pi y), \;\;\;n = 1, 2, 3, ...$$ Find a linear combination of the $u_n$'s that satisfies: $$u(x,1) = \sin(2\pi x) -\sin(3\pi x)$$ Any ...
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26 views

Questions about hyperbolas and integration

I have a couple of questions regarding hyperbolas and their integrals. If it's too much, don't feel like you have to answer all 3 questions. My first question: The integral of a function like 1/x^2 ...
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12 views

Solving for hyperquadratic?

I'm working on defining a decision plane for two different classes but am running into trouble. For those familiar with stats I'm working with two classes that have arbitrary distribution, mean and ...
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3answers
49 views

Showing that $\sinh(\mathrm{e}^z)$ is entire

I am attempting to show that $\sinh(\mathrm{e}^z)$, where $z$ is a complex number, is entire. The instructions of the problem tell me to write the real component of this function as a function of $x$ ...
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1answer
60 views

How to find the angle subtended to the origin by the unit hyperbola through the point (1,0)?

I'm trying to find the angle subtended by the unit hyperbola through the point (1,0). I think that I should be integrating something, but I'm not sure how to set it up. I've been trying to think of ...
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3answers
77 views

Derivatives of $\sinh x$ and $\cosh x$

Can someone give me an intuitive explanation about the derivatives of $\sinh x$ and $\cosh x$? Something similar to: Intuitive understanding of the derivatives of $\sin x$ and $\cos x$ Thanks!
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4answers
95 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.
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0answers
21 views

How are the sine functions along with the hyperbolic functions visualized with imaginary rotations?

Since we know that: cos(t)=cosh(it) and isin(t)=sinh(it) I've been thinking about this, and obviously this is referring to how if you move at a right angle from a circle on a conic section, you end ...
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0answers
48 views

Why are hyperbolic trigonometric functions avoided in (my) high school and early post-secondary school?

I remember seeing hyperbolic trigonometric functions (sinh, cosh, tanh, etc.) in my precalculus textbook back in high school and see them today in my calculus textbook. However, I have not had a ...
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1answer
171 views

Find area of a curvilinear triangle that includes hyperbolic functions

We were given this question in class and I tried to compute it and it looks to be pretty crazy. Can anyone take a look and let me know if I did it correctly? I would really appreciate it. ...
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0answers
385 views

Solving system of equations with hyperbolic functions

Do you maybe know how to solve a system of equations with hyperbolic functions? Imagine the problem of the form: $$ x=\textrm{sech}(x^2+y^2) \\ y=1-\textrm{sech}^2 (x+y) $$ Any ideas how to solve it ...
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2answers
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Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx$

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

Nonlinear Differential Equation

During my study of calculus of variations I came across a nonlinear DE. I know that its solution is hyperbolic cosine with some constants yet do not know how to proceed since the function of interest ...
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2answers
48 views

Differentiate $y=\cosh^{3} 4x$.

Differentiate $y=\cosh^{3} 4x$. $$\frac{dy}{dx} = 3 \cosh^{2} (4x) \sinh (4x)\cdot 4$$ These are the parts that I don't quite understand: \begin{align*} \frac {dy}{dx} &=12 \cosh^{2} ...
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$\lim\limits_{n\to\infty}\operatorname{arsinh}(1 + \operatorname{arsinh}(2 + \operatorname{arsinh}(3 + \dots\operatorname{arsinh}(n+\dots)\dots)))=?$

Does the limit $$\lim\limits_{n\to\infty}\operatorname{arsinh}(1 + \operatorname{arsinh}(2 + \operatorname{arsinh}(3 + \operatorname{arsinh}(4+\dots\operatorname{arsinh}(n+\dots)\dots))))$$ exist ...
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0answers
65 views

Closed form for $\int^1_0 \mathrm{arccsch}\left( \frac{x^2-x-1}{x^2+x+1}\right)\;\mathrm dx$

Find a closed form for $$\int^1_0 \mathrm{arccsch}\left( \frac{x^2-x-1}{x^2+x+1}\right)\;\mathrm dx$$ What I have tried Expanding the $\mathrm{arccsch}$ into its logarithmic form, however I ...
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1answer
36 views

Finding the derivative of $\;\operatorname{arccoth}(\sin x)$

I have tried to solve it but I don't why it's wrong. I need to take the derivative of $\;\operatorname{arccoth}(\sin x)$: By using chain rule, I get: $$\dfrac 1{1 - \sin^2 x}\cdot \cos x = \dfrac ...
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1answer
67 views

When I choose arctanh or arccoth?

I want to know When I choose $\operatorname{arctanh}$ or $\operatorname{arccoth}$ ? E.x $$\int_0^3 \frac{1}{49-4x^2} dx$$ It will be $(1/14) \operatorname{arctanh} (2x/7)$ or $(1/14) ...
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2answers
79 views

How to integrate this $\frac{1}{(1-e^{2x})^{1/2}}$?

Please how to integrate this $$\frac{1}{(1-e^{2x})^{1/2}}$$ I have tried $u= e^x$ But I think that is wrong So can anyone help me ?