What is Hyperbolic functions: Definition and 81 Discussions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, just as the derivatives of sin(t) and cos(t) are cos(t) and –sin(t), the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t).
Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.
The basic hyperbolic functions are:
hyperbolic sine "sinh" (),
hyperbolic cosine "cosh" (),from which are derived:
hyperbolic tangent "tanh" (),
hyperbolic cosecant "csch" or "cosech" ()
hyperbolic secant "sech" (),
hyperbolic cotangent "coth" (),corresponding to the derived trigonometric functions.
The inverse hyperbolic functions are:
area hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh")
area hyperbolic cosine "arcosh" (also denoted "cosh−1", "acosh" or sometimes "arccosh")
and so on.
The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.
In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane.
By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument.Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. (sinus/cosinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names, but altered the abbreviations to those used today. The abbreviations sh, ch, th, cth are also currently used, depending on personal preference.

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  1. chwala

    Solve the given trigonometry equation

    I was able to solve with a rather longer way; there could be a more straightforward approach; My steps are along these lines; ##\sinh^{-1} x = 2 \ln (2+ \sqrt{3})## ##\sinh^{-1} x = \ln (7+ 4\sqrt{3})## ##x = \sinh[ \ln (7+ 4\sqrt{3})]## ##x = \dfrac {e^{\ln (7+ 4 \sqrt{3})} - e^{-[\ln 7+ 4...
  2. chwala

    Solve the given trigonometry problem

    My question is on the highlighted part (circled in red); Why is it wrong to pre-multiply each term by ##e^x##? to realize , ##5e^{2x} -2-9e^x=0## as opposed to factorising by ##e^{-x} ## ? The other steps to required solution ##x=\ln 2## is quite clear and straightforward to me.
  3. chwala

    Prove the given hyperbolic trigonometry equation

    I have, Using ##\ cosh 2x = 2 \cosh^2 x - 1## ##\cosh x = 2 \cosh^2\dfrac{x}{2} -1## Therefore, ##\cosh x -1 = 2 \cosh^2\dfrac{x}{2} -1 - 1## ##\cosh x -1 = 2 \cosh^2\dfrac{x}{2} -2## ##=2\left[ \cosh^2 \dfrac{x}{2}...
  4. chwala

    Solve the given trigonometry equation

    In my approach i have the following lines ##\ln (x + \sqrt{x^2+1}) = 2\ln (2+\sqrt 3)## ##\ln (x + \sqrt{x^2+1} = \ln (2+\sqrt 3)^2## ##⇒x+ \sqrt{x^2+1} =(2+\sqrt 3)^2## ##\sqrt{x^2+1}=-x +7+4\sqrt{3}## ##x^2+1 = x^2-14x-8\sqrt 3 x + 56\sqrt 3 +97## ##1 = -14x-8\sqrt 3 x + 56\sqrt 3 +97##...
  5. Safinaz

    How to write this expression in terms of a Hyperbolic function?

    The eqution can be written as: ## Eq= e^{t( -h + \sqrt{ x} )} + e^{t( -h -\sqrt{ x} )} ## Can this be written in terms of Cosh x ?
  6. Vectronix

    Analysis What’s a good book on transcendental functions?

    I found one ages ago about the hyperbolic functions, but it hadn’t been translated to English from German yet. Anyone know of a good book on hyperbolic functions and other transcendental functions besides the circular functions (trigonometric)?
  7. chwala

    Write the given hyperbolic function as simply as possible

    My take; ##2\cosh x = e^x +e^{-x}## I noted that i could multiply both sides by ##e^x## i.e ##e^x⋅2\cosh x = e^x(e^x +e^{-x})## ##e^x⋅2\cosh x = e^{2x}+1## thus, ##\dfrac{e^x}{1+e^{2x}}=\dfrac{\cosh x + \sinh x}{e^x⋅2\cosh x}## ##= \dfrac{\cosh x +...
  8. chwala

    Prove the hyperbolic function corresponding to the given trigonometric function

    ##8 \sin^4u = 3-4\cos 2u+\cos 4u## ##8 \sinh^4u = 3-4(1+2\sinh^2 u)+ \cosh ( 2u+2u)## ##8 \sin^4u = 3-4-8\sinh^2 u+ \cosh 2u \cosh 2u + \sinh 2u \sinh 2u## ##8 \sinh^4u = 3-4+1-8\sinh^2 u+ 4\sinh^2u +4\sinh^4 u + 4\sinh^2 u + 4\sinh^4 u## ##8 \sinh^4u = -8\sinh^2 u+ 8\sinh^2u +8\sinh^4 u##...
  9. chwala

    Find the roots of the given hyperbolic equation

    This is a textbook question and i have no solution. My attempt: We know that ##\cosh x = \dfrac{e^x + e^{-x}}{2}## and ##\cosh u = \dfrac{{x^2 + 1}}{2x}## it therefore follows that; ##e^{2u} = x^2## ##⇒u = \dfrac {2\ln...
  10. chwala

    Comparing Hyperbolic and Cartesian Trig Properties

    I came across this question; i noted that the hyperbolic trigonometry properties are somewhat similar to what i may call cartesian trigonometry properties... My approach on this; ##\tanh x = \sinh y## ...just follows from ##y=\sin^{-1}(\tan x)## ##\tan x = \sin y## Therefore...
  11. Samama Fahim

    I Deriving Lorentz Transformations: Hyperbolic Functions

    While deriving Lorentz transformation equations, my professor assumes the following: As ##\beta \rightarrow 1,## $$-c^2t^2 + x^2 = k$$ approaches 0. That is, ##-c^2t^2 + x^2 = 0.## But the equation of the hyperbola is preserved in all inertial frames of reference. Why would ##-c^2t^2 + x^2##...
  12. S

    MHB Fourier Series involving Hyperbolic Functions

    Hello everyone first time here. don't know if it's the correct group... Am having some issues wiz my maths homework that going to count as a final assessment. Really Really need help. The function (f), with a period of 2π is : f(x) = cosh(x-2π) if x [π;3π].. I had to do a graph as the first...
  13. Leo Liu

    I How did mathematicians discover the expressions of hyperbolic functions?

    The hyperbolic function ##\cosh t \text{ and } \sinh t## respectively represent the x and y coordinate of the parametric equation of the parabola ##x^2-y^2=1##. The exponential expressions of these hyperbolic functions are $$ \begin{cases} \sinh x = {e^x-e^{-x}} /2; \: x \in \mathbb R, \: f(x)...
  14. B

    A Can this difficult Gaussian integral be done analytically?

    Here is a tough integral that I'm not quite sure how to do. It's the Gaussian average: $$ I = \int_{-\infty}^{\infty}dx\, \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\sqrt{1+a^2 \sinh^2(b x)} $$ for ##0 < a < 1## and ##b > 0##. Obviously the integral can be done for ##a = 0## (or ##b=0##) and for...
  15. universal2013

    I Exploring Maxwell Equations & Hyperbolic Functions

    I am trying to understand why maxwell equations are correct in any reference frames? While i started to understand of his laws of physics a bit i could not imagine why he uses hyperbolic functions such as coshw instead of spherical ones in position and time relation between moving frames...
  16. R

    Modulus of a complex number with hyperbolic functions

    Homework Statement For the expression $$r = \frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha L)-i\delta\sinh(\alpha L)} \tag{1}$$ Where ##\alpha=\sqrt{\kappa^{2}-\delta^{2}}##, I want to show that: $$\left|r\right|^{2} = \left|\frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha...
  17. T

    B Problem solving with hyperbolic functions

    Mod note: Because his caps-lock key is stuck, it's OK for this post to be in all caps. FIRSTLY, MY LAPTOP'S CAPS LOCK IS BEHAVING REALLY WEIRD AND I HAVE NO CONTROL ON IT WHATSOEVER. SO SORRY FOR POSTING IN ALL CAPS/ALL SMALL LETTERS I HAVE RECENTLY LEARNED HYPERBOLIC FUNCTIONS. HOWEVER, I AM...
  18. J

    Analytic Integration of Function Containing the Exponential of an Exponential

    Homework Statement Can this function be integrated analytically? ##f=\exp \left(-\frac{e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32 \sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right),## where ##a##, ##b## and ##L## are some real positive...
  19. K

    Tangent to Hyperbolic functions graph

    Homework Statement Show that the tangent to ##x^2-y^2=1## at points ##x_1=\cosh (u)## and ##y_1=\sinh(u)## cuts the x-axis at ##{\rm sech(u)}## and the y-axis at ##{\rm -csch(u)}##. Homework Equations Hyperbolic sine: ##\sinh (u)=\frac{1}{2}(e^u-e^{-u})## Hyperbolic...
  20. K

    I An identity of hyperbolic functions

    Prove: ##(\cosh(x)+\sinh(x))^n=\cosh(nx)+\sinh(nx)## Newton's binomial: ##(a+b)^n=C^0_n a^n+C^1_n a^{n-1}b+...+C^n_n b^n## and: ##(a-b)^n~\rightarrow~(-1)^kC^k_n## I ignore the coefficients. $$(\cosh(x)+\sinh(x))^n=\cosh^n(x)+\cosh^{n-1}\sinh(x)+...+\sinh^n(x)$$...
  21. H

    I Graphs of inverse trigonometric vs inverse hyperbolic functions

    I noticed the graphs of ##y=\cos^{-1}x## and ##y=\cosh^{-1}x## are similar in the sense that the real part of one is the imaginary part of the other. This is true except when ##x<-1## where the imaginary part of ##y=\cos^{-1}x## is negative but the real part of ##y=\cosh^{-1}x## is positive. I...
  22. G

    Limit of arccosh x - ln x as x -> infinity

    Homework Statement find the limit of arccoshx - ln x as x -> infinity Homework Equations ##arccosh x = \ln (x +\sqrt[]{x^2-1} )## The Attempt at a Solution ## \lim_{x \to \infty }(\ln (x + \sqrt{x^2-1} ) - \ln (x)) = \lim_{x \to \infty} \ln (\frac{x+\sqrt{x^2-1}}{x}) \ln (1 + \lim_{x \to...
  23. j3dwards

    Splitting function into odd and even parts

    Homework Statement Split the function f(x) = ex + πe−x into odd and even parts, and express your result in terms of cosh x and sinh x. Homework Equations f(x) = 0.5[f(x) + f(-x)] +0.5[f(x) - f(-x)] The Attempt at a Solution So i know that: ex = 0.5[ex - e-x] + 0.5[ex + e-x] = sinh(x) +...
  24. E

    What is the solution for the attached equation?

    Good afternoon, i was just wondering if this equation is possibly solvable where I(z) is a function of z. The equation is: I(z)=cosh(1/2 ∫I(z)dz) I know it looks stupid but is it possible? How would you approach this problem? Thank you.
  25. N

    Proof using hyperbolic trig functions and complex variables

    1. Given, x + yi = tan^-1 ((exp(a + bi)). Prove that tan(2x) = -cos(b) / sinh(a)Homework Equations I have derived. tan(x + yi) = i*tan(x)*tanh(y) / 1 - i*tan(x)*tanh(y) tan(2x) = 2tanx / 1 - tan^2 (x) Exp(a+bi) = exp(a) *(cos(b) + i*sin(b))[/B]3. My attempt: By...
  26. E

    Hyperbolic partial differential equation

    What is the general solution of the following hyperbolic partial differential equation: The head (h) at a specified distance (x) is a sort of a damping function in the form: Where, a, b, c and d are constants. And the derivatives are with respect to t (time) and x (distance). Thanks in advance.
  27. P

    Inverse hyperbolic functions (logarithmic form)

    To express the ##\cosh^{-1}## function as a logarithm, we start by defining the variables ##x## and ##y## as follows: $$y = \cosh^{-1}{x}$$ $$x = \cosh{y}$$ Where ##y ∈ [0, \infty)## and ##x ∈ [1, \infty)##. Using the definition of the hyperbolic cosine function, rearranging, and multiplying...
  28. Kyuutoryuu

    Are hyperbolic functions used in Calculus 3?

    More than just a few problems that happen to pop up in the textbook, I mean.
  29. J

    Geometry with hyperbolic functions

    Is known that in every rectangle triangle the following relationships are true: But, how use geometrically the function sinh, cosh, and tanh?
  30. O

    C) Am I Solving this Hyperbolic Functions Homework Correctly?

    Homework Statement Attached is the problem Homework Equations My question is am i going about it the right way for question C). I have done A and B and am sure they are correct. The Attempt at a Solution Attached
  31. samjohnny

    Matrices with hyperbolic functions

    Homework Statement I thought it would be better to attach it. Homework Equations The Attempt at a Solution So for the first part I've found that A^2=the Identity matrix, but from there I don't have much of an idea on how to substitute that into the equation for M and end up with...
  32. S

    Locus and hyperbolic functions

    show that the locus of the point \left(\dfrac{a(cosh\theta + 1)}{2cosh\theta},\dfrac{b(cosh\theta - 1)}{2sinh\theta}\right) has equation x(4y^2 + b^2) = ab^2 working: http://gyazo.com/4c96af128d0293bce7f18029c2f54b0d where have I gone wrong :(
  33. aNxello

    MHB Fourier Series Involving Hyperbolic Functions

    [SOLVED] Fourier Series Involving Hyperbolic Functions Hello everyone! Sorry if this isn't the appropriate board, but I couldn't think of which board would be more appropriate. I was running through some problems I have to do as practice for a test and I got stuck on one I'm 99% sure they'll...
  34. L

    Understanding Hyperbolic Functions

    Will someone help me to understand sinh, cosh, and tanh. I know they have some relevance to hyperbolas and trigonometric identities. Thank you.
  35. Calculuser

    Unraveling the Mystery of Hyperbolic Functions: A Mathematical Proof

    I've searched and thought on it for a long time but I couldn't find any mathematical proof or something else about the formula of hyperbolic functions. sinh=\frac{e^{x}-e^{-x}}{2},cosh=\frac{e^{x}+e^{-x}}{2} How do I get these formulas mathematically??
  36. R

    Problem with hyperbolic functions demostrations

    Homework Statement Prove that cosh (\frac{x}{2}) = \sqrt{\frac{cosh(x)+1}{2}} Homework Equations cosh(x) = \frac{e^{x}+e^{-x}}{2} The Attempt at a Solution \frac{\sqrt{e^{x}}+\sqrt{e^{-x}}}{2} \ast \frac{\sqrt{e^{x}}-\sqrt{e^{-x}}}{\sqrt{e^{x}}-\sqrt{e^{-x}}} \rightarrow...
  37. P

    Integrating hyperbolic functions

    Hi, I am trying to integrate (tanh(x)+coth(x))/((cosh(x))^2) I am substituting u=tanh(x), du=dx/((cosh(x))^2) and end up with 1/2(tanh(x))^2 + ln |tanh(x)| + C which is incorrect. What am I doing wrong??
  38. T

    Finding Indefinite Integral of a combination of hyperbolic functions

    Homework Statement Compute the following: \int \frac{cosh(x)}{cosh^2(x) - 1}\,dx Homework Equations \int cosh(x)\,dx = sinh(x) + C The Attempt at a Solution I had no clue where to start, so I went to WolfRamAlpha, and it used substitution but it made u = tanh(\frac{x}{2})...
  39. T

    Proving an identity involving hyperbolic functions

    Homework Statement Prove sin(x-iy) = sin(x) cosh(y) - i cos(x) sinh(y) Homework Equations The Attempt at a Solution I tried to prove it by developing sinh into it's exponential form, but I get stuck. sinh(x-iy) = [ ei(x-iy) - e-i(x-iy) ] /2i = [ eixey - e-ix e-y ] /2i...
  40. N

    Solving Complex Hyperbolic functions

    Homework Statement I am a little confused on the steps to take to solve these kinds of functions. Solve: cosh z = 2i The Attempt at a Solution We were given identities for sinh z = 0 and cosh z = 0 and also other identities like cosh(z) = cos (iz) So cos (iz) = 2i cos...
  41. DryRun

    Integral involving hyperbolic functions

    Homework Statement Find \int \frac{x}{\sqrt{2x^2-2x+1}}\,dx The attempt at a solution First, i complete the square for the quadratic expression: 2x^2-2x+1=2((x-\frac{1}{2})^2+\frac{1}{4}) \int \frac{x}{\sqrt{2x^2-2x+1}}\,dx=\int \frac{x}{\sqrt 2 \sqrt{(x-\frac{1}{2})^2+\frac{1}{4}}}\,dx...
  42. C

    MHB Help with Hyperbolic Functions

    Bany's question from Yahoo Questions: CB
  43. B

    Why Isn't the Answer to the Hyperbolic Function e^10x?

    Homework Statement why is the answer not e^10x ? If you ignore the e's it should be 5x - 5x + 5x - - 5x, or 5x - 5x + 5x + 5x, which is 10x
  44. P

    Manipulating hyperbolic functions

    Homework Statement Express the function cosh(6x) in terms of powers of cosh(x) Homework Equations The Attempt at a Solution Okay the problem booklet also asks me to do the opposite. Express cosh(x)^6 as mutiples of cosh(x). I can do that fine, I just simply write it out as [1/2(e^x + e^-x)]^6...
  45. O

    Rearrangeing Inverse Hyperbolic functions

    Hi, My brain is not working today. So can someone please tell me what I am doing wrong. (^2 = squared) coshy^2 - sinhy^2 = 1, how do I rearrange this for coshy^2 I keep getting: coshy^2 = 1 + Sinhy^2 The book that I'm looking at has it this way: coshy^2 = Sinhy^2 + 1 Thanks Obs
  46. N

    Why are hyperbolic functions defined in terms of exponentials?

    Where do the definitions of hyperbolic functions in terms of exponentials come from ?
  47. A

    Integration of hyperbolic functions

    Homework Statement \int cosh(2x)sinh^{2}(2x)dx Homework Equations Not sure The Attempt at a Solution This was an example problem in the book and was curious how they got to the following answer: \int cosh(2x)sinh^{2}(2x)dx = \frac{1}{2}\int sinh^{2}(2x)2cosh(2x) dx =...
  48. S

    Volume of the Solid involving Hyperbolic functions

    Homework Statement The area bounded by y=2 coshx, the x-axis, the y-axis, and the line x=4 is revolved about the x-axis. Find the volume of the solid generated. Homework Equations I sliced the area along the axis of revolution. That is the strip is dx. So the equation necessary is...
  49. I

    Understanding Hyperbolic Functions

    Hyperbolic functions :((( Homework Statement Question is: and Homework Equations Now from what i can recall the formula for sin(A+B) = sinAcosB+sinbcosA so same goes for hyperbolic function i suppose? sinh(2x+x) = sinh2xcoshx+cosh2xsinx (2sinhxcoshx)coshx +...
  50. B

    Differentiating Hyperbolic Functions

    1. Differentiate cosh(x) using first principles 2. cosh(x) = (e^x+e^-x)/2 From previous exercises, I know the answer will be sinh(x)= (e^x-e^-x)/2 but I cannot get to the answer. I seem to be left with the equation: lim h ---> 0 (e^2x*e^2h +1-e^h*2e^x +e^h)/(2h*e^x*e^h) But when...
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