It can be written in a parametric form using hyperbolic functions (this explains their name).

Denote y= b·sht , then x2 / a2=1+sh2t =ch2t . Whence x=± a·cht .

Thus, we arrive at the following parametric equations of the hyperbola:

Y= in sht , –< t < . (6)

Rice. one.

The "+" sign in the upper formula (6) corresponds to the right branch of the hyperbola, and the ""– "" sign corresponds to the left branch (see Fig. 1). The vertices of the hyperbola A(– a; 0) and B(a; 0) correspond to the value of the parameter t=0.

For comparison, we can give the parametric equations of an ellipse using trigonometric functions:

X=a cost ,

Y=in sint , 0 t 2p . (7)

3. Obviously, the function y=chx is even and takes only positive values. The function y=shx is odd, because :

The functions y=thx and y=cthx are odd as quotients of an even and an odd function. Note that unlike trigonometric functions, hyperbolic functions are not periodic.

4. Let us study the behavior of the function y= cthx in the neighborhood of the discontinuity point x=0:

Thus the y-axis is the vertical asymptote of the graph of the function y=cthx . Let us define oblique (horizontal) asymptotes:

Therefore, the line y=1 is the right horizontal asymptote of the graph of the function y=cthx . Due to the oddness of this function, its left horizontal asymptote is the straight line y= –1. It is easy to show that these lines are simultaneously asymptotes for the function y=thx. The functions shx and chx have no asymptotes.

2) (chx)"=shx (displayed similarly).

4)

There is also a certain analogy with trigonometric functions. A complete table of derivatives of all hyperbolic functions is given in section IV.

Tangent, cotangent

Definitions of hyperbolic functions, their domains of definitions and values

sh x- hyperbolic sine
, -∞ < x < +∞; -∞ < y < +∞ .
ch x- hyperbolic cosine
, -∞ < x < +∞; 1 ≤ y< +∞ .
th x- hyperbolic tangent
, -∞ < x < +∞; - 1 < y < +1 .
cth x- hyperbolic cotangent
, x ≠ 0; y< -1 или y > +1 .

Graphs of hyperbolic functions

Plot of the hyperbolic sine y = sh x

Plot of the hyperbolic cosine y = ch x

Plot of the hyperbolic tangent y= th x

Plot of the hyperbolic cotangent y = cth x

Formulas with hyperbolic functions

Relationship with trigonometric functions

sin iz = i sh z ; cos iz = ch z
sh iz = i sin z ; ch iz = cos z
tgiz = i th z ; ctg iz = - i cth z
th iz = i tg z ; cth iz = - i ctg z
Here i is an imaginary unit, i 2 = - 1 .

Applying these formulas to trigonometric functions, we obtain formulas relating hyperbolic functions.

Parity

sh(-x) = - sh x; ch(-x) = ch x.
th(-x) = -th x; cth(-x) = - cth x.

Function ch(x)- even. Functions sh(x), th(x), cth(x)- odd.

Difference of squares

ch 2 x - sh 2 x = 1.

Formulas for sum and difference of arguments

sh(x y) = sh x ch y ch x sh y,
ch(x y) = ch x ch y sh x sh y,
,
,

sh 2 x = 2 sh x ch x,
ch 2 x = ch 2 x + sh 2 x = 2 ch 2 x - 1 = 1 + 2 sh 2 x,
.

Formulas for products of hyperbolic sine and cosine

,
,
,

,
,
.

Formulas for the sum and difference of hyperbolic functions

,
,
,
,
.

Relation of hyperbolic sine and cosine with tangent and cotangent

, ,
, .

Derivatives

,

Integrals of sh x, ch x, th x, cth x

,
,
.

Expansions into series

Inverse functions

Areasine

At - ∞< x < ∞ и - ∞ < y < ∞ имеют место формулы:
,
.

Areacosine

At 1 ≤ x< ∞ and 0 ≤ y< ∞ there are formulas:
,
.

The second branch of the areacosine is located at 1 ≤ x< ∞ and - ∞< y ≤ 0 :
.

Areatangent

At - 1 < x < 1 and - ∞< y < ∞ имеют место формулы:
,

Other designations: sinh x, Sh x, cosh x, Ch x, tgh x, tanh x, Th x. Graphs see in fig. one.

Basic ratios:

Geometric G. f. similar to the interpretation of trigonometric functions (Fig. 2). Parametric the equations of a hyperbola allow us to interpret the abscissa and ordinate of a point of an equilateral hyperbola as a hyperbola. cosine and sine; hyperbolic tangent segment AB. The parameter t is equal to twice the area of ​​the sector OAM, where AM- arc of a hyperbola. For a point (at ), the parameter t is negative. Inverse hyperbolic functions are defined by the formulas:

Derivatives and basic integrals of G. f.:

In the entire plane of the complex variable z, the G. f. and can be defined by the series:

thus,

There are extensive tables for G. f. Values ​​G. f. can also be obtained from the tables for e x and e-x.

Lit.: Yanke E., Emde F., Lesh F., Special functions. Formulas, graphs, tables, 2nd ed., Per. from German., M., 1968; Tables of circular and hyperbolic sines and cosines in the measure of angle radiation, M., 1958; tables e x and e-x, M., 1955. V. I. Bityutskov.

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

See what "HYPERBOLIC FUNCTIONS" is in other dictionaries:

Functions defined by the formulas: (hyperbolic sine), (hyperbolic cosine). Sometimes the hyperbolic tangent is also considered: G. f. ... ...

Functions defined by the formulas: (hyperbolic sine), (hyperbolic cosine), (hyperbolic tangent) ... Big Encyclopedic Dictionary

Functions defined by the formulas: shx \u003d (ex e x) / 2 (hynerbolic sine), chx (ex + e k) / 2 (hyperbolic cosine), thx \u003d shx / chx (hyperbolic tangent). Graphs G. f. see in pic...

A family of elementary functions expressed in terms of an exponent and closely related to trigonometric functions. Contents 1 Definition 1.1 Geometric definition ... Wikipedia

Functions defined by the formulas: shx = (ex - e x)/2 (hyperbolic sine), chx = (ex + e x)/2 (hyperbolic cosine), thx = shx/chx (hyperbolic tangent). Graphs of hyperbolic functions, see fig. * * * HYPERBOLIC FUNCTIONS… … encyclopedic Dictionary

Functions. defined by the flags: (hyperbolic sine), (hyperbolic cosine), (insert pictures!!!) Graphs of hyperbolic functions ... Big encyclopedic polytechnic dictionary

By analogy with the trigonometric functions Sinx, cosx, which are known to be determined using the Euler formulas sinx = (exi e xi)/2i, cosx = (exi + e xi)/2 (where e is the base of the Napier logarithms, a i = √[ one]); sometimes brought into consideration ... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron

Functions inverse to hyperbolic functions (See Hyperbolic functions) sh x, ch x, th x; they are expressed by formulas (read: hyperbolic aresine, hyperbolic area cosine, aretangent ... ... Great Soviet Encyclopedia

Functions inverse to hyperbolic. functions; expressed in formulas... Natural science. encyclopedic Dictionary

Inverse hyperbolic functions are defined as the inverses of hyperbolic functions. These functions determine the area of ​​the unit hyperbola sector x2 − y2 = 1 in the same way that inverse trigonometric functions determine the length ... ... Wikipedia

Books

• Hyperbolic functions , Yanpolsky A.R. The book describes the properties of hyperbolic and inverse hyperbolic functions and gives the relationship between them and other elementary functions. Applications of hyperbolic functions to…

Introduction

In mathematics and its applications to natural science and technology, exponential functions are widely used. This, in particular, is explained by the fact that many phenomena studied in natural science are among the so-called processes of organic growth, in which the rates of change of the functions participating in them are proportional to the values ​​of the functions themselves.

If denoted by a function, and by an argument, then the differential law of the process of organic growth can be written in the form where is some constant coefficient of proportionality.

Integration of this equation leads to the general solution in the form of an exponential function

If you set the initial condition at, then you can determine an arbitrary constant and, thus, find a particular solution, which is an integral law of the process under consideration.

The processes of organic growth include, under some simplifying assumptions, such phenomena as, for example, a change in atmospheric pressure depending on the height above the Earth's surface, radioactive decay, cooling or heating of a body in an environment of constant temperature, a unimolecular chemical reaction (for example, the dissolution of a substance in water ), in which the law of mass action takes place (the reaction rate is proportional to the amount of reactant present), the reproduction of microorganisms, and many others.

The increase in the amount of money due to the accrual of compound interest on it (interest on interest) is also a process of organic growth.

These examples could be continued.

Along with individual exponential functions in mathematics and its applications, various combinations of exponential functions are used, among which certain linear and linear-fractional combinations of functions and the so-called hyperbolic functions are of particular importance. There are six of these functions, the following special names and designations have been introduced for them:

(hyperbolic sine),

(hyperbolic cosine),

(hyperbolic tangent),

(hyperbolic cotangent),

(hyperbolic secant),

(hyperbolic secant).

The question arises why exactly such names are given, and here is a hyperbole and the names of functions known from trigonometry: sine, cosine, etc.? It turns out that the relations connecting trigonometric functions with the coordinates of points of a circle of unit radius are similar to the relations connecting hyperbolic functions with the coordinates of points of an equilateral hyperbola with a unit semiaxis. This justifies the name of hyperbolic functions.

Hyperbolic functions

The functions given by formulas are called hyperbolic cosine and hyperbolic sine, respectively.

These functions are defined and continuous on, and is an even function and is an odd function.

Figure 1.1 - Graphs of functions

From the definition of hyperbolic functions it follows that:

By analogy with trigonometric functions, the hyperbolic tangent and cotangent are defined, respectively, by the formulas

A function is defined and continuous on, and a function is defined and continuous on a set with a punctured point; both functions are odd, their graphs are shown in the figures below.

Figure 1.2 - Graph of the function

Figure 1.3 - Graph of the function

It can be shown that the functions and are strictly increasing, while the function is strictly decreasing. Therefore, these functions are reversible. Denote the functions inverse to them, respectively, by.

Consider a function inverse to a function, i.e. function. We express it in terms of elementary ones. Solving the equation with respect to, we get Since, then, from where

Replacing with and with, we find the formula for the inverse function for the hyperbolic sine.

HYPERBOLIC FUNCTIONS- Hyperbolic sine (sh x) and cosine (ch x) are defined by the following equalities:

Hyperbolic tangent and cotangent are defined by analogy with trigonometric tangent and cotangent:

The hyperbolic secant and cosecant are defined similarly:

There are formulas:

The properties of hyperbolic functions are in many respects similar to the properties (see). The equations x=cos t, y=sin t determine the circle x²+y² = 1; the equations x=сh t, y=sh t define the hyperbola x² - y²=1. As trigonometric functions are determined from a circle of unit radius, so hyperbolic functions are determined from an isosceles hyperbola x² - y² = 1. The argument t is the double area of ​​the shaded curvilinear triangle OME (Fig. 48), similarly to the fact that for circular (trigonometric) functions the argument t is numerically equal to twice the area of ​​the curvilinear triangle OKE (Fig. 49):

for circle

for hyperbole

The addition theorems for hyperbolic functions are similar to the addition theorems for trigonometric functions:

These analogies are easily seen if the complex variable r is taken as the argument x. Hyperbolic functions are related to trigonometric functions by the following formulas: sh x \u003d - i sin ix, ch x \u003d cos ix, where i is one of the values ​​of the root √-1. The hyperbolic functions sh x, as well as ch x: can take any large values ​​​​(hence, of course, large units), in contrast to the trigonometric functions sin x, cos x, which for real values ​​cannot be greater than one in absolute value.
Hyperbolic functions play a role in Lobachevsky's geometry (see), are used in the study of the resistance of materials, in electrical engineering and other branches of knowledge. There are also designations of hyperbolic functions in the literature such sinh x; cosh x; tghx.