0
votes
3answers
66 views

How does $\frac{1}{2}\cosh(2x) -1 = \sinh^2(x)$?

Using hyperbolic trigonometric function identities is there a way to prove the following equation? $$\frac{1}{2} (\cosh(2x)-1) = \sinh^2(x)$$
3
votes
2answers
351 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) ?
1
vote
2answers
78 views

Hyperbolic function identity proof?

On a question i am working thru it says: Obtain the formula:$$ \sinh 2x - \sinh 2y = 2\cosh(x+y)\sinh(x-y) $$and prove that $$\coshθ + \cosh2θ +...+\cosh nθ ...
3
votes
1answer
69 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 ...
1
vote
1answer
51 views

Limit of a Cosh function

Evaluate $$\lim_{t\to\infty} (\cosh x)^{1/x}.$$ I tried to use L'Hopital's but I think I made a mess of the differentiation, and the differentiation doesn't seem like it'll help much.
2
votes
0answers
31 views

Integral Evaluation: Exponential of and Hyperbolic Function

I'm trying to evaluate $$G^{\pm} = \frac{-i}{8\pi^2 X} \partial_X \int_{-\infty}^\infty d\phi e^{i m \left[X \sinh \phi \pm T \cosh \phi \right]}$$ for $T = \pm X$. Where $T, X, m \in \mathbb{R}$ ...
0
votes
2answers
46 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.
0
votes
1answer
37 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 ...
3
votes
4answers
352 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 ...
1
vote
3answers
65 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 $x$ ...
0
votes
1answer
50 views

Rewriting a hyperbolic equation in standard form

$9x^2-4y^2-72x$ = 0 How would that be done? So far, I got up to $\frac{9(x^2-8x)}1-\frac{4(y-0)^2}1=0$
1
vote
2answers
605 views

Hyperbolic Functions

Hey everyone, I need help with questions on hyperbolic functions. I was able to do part (a). I proved for $\sinh(3y)$ by doing this: \begin{align*} \sinh(3y) &= \sinh(2y +y)\\ &= ...
0
votes
1answer
126 views

Integration by parts

Question 2)b) part (ii) is the section that I'm having trouble with: I don't understand the method used in the solutions; how would you deduce the first line or is that something you should know? ...
1
vote
1answer
440 views

Inverse Laplace Transform involving $\cosh$.

While doing an assignment on solving a PDE I stumbled into the following inverse Laplace transform question (involving $\cosh$? I can't believe it). Mathematica gives no solution and I have no idea ...