EntranceHyperbolic functions formulas. Reference data on hyperbolic functions - properties, graphs, formulas
It can be written in parametric form using hyperbolic functions (this explains their name).
Let us denote y= b·sht, then x2 / a2=1+sh2t =ch2t. Where x=± a·cht .
Thus we arrive at the following parametric hyperbola equations:
У= in ·sht , –< t < . (6)
Rice. 1.
The "+" sign in the upper formula (6) corresponds to the right branch of the hyperbola, and the ""– "" sign to the left one (see Fig. 1). The vertices of the hyperbola A(– a; 0) and B(a; 0) correspond to the parameter value t=0.
For comparison, we can give parametric equations of an ellipse using trigonometric functions:
X=a·cost ,
Y=в·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 the even and odd functions. Note that, unlike trigonometric functions, hyperbolic functions are not periodic.
4.
Let us study the behavior of the function y= cthx in the vicinity of the discontinuity point x=0: 
Thus, the Oy axis is the vertical asymptote of the graph of the function y=cthx. Let us define oblique (horizontal) asymptotes:
Therefore, the straight line y=1 is right horizontal asymptote 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 (shown similarly).
4) 
There is also a certain analogy with trigonometric functions. 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
Hyperbolic sine graph y = sh x
Graph of hyperbolic cosine y = ch x
Graph of hyperbolic tangent y = th x
Graph of hyperbolic cotangent y = cth x
Formulas with hyperbolic functions
Relation to trigonometric functions
sin iz = i sh z ; cos iz = ch z
sh iz = i sin z; ch iz = cos z
tg iz = i th z ; cot iz = - i cth z
th iz = i tg z ; cth iz = - i ctg z
Here i is the 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 the 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 the products of hyperbolic sine and cosine
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,
,
,
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Formulas for the sum and difference of hyperbolic functions
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,
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Relation of hyperbolic sine and cosine with tangent and cotangent
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Derivatives
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Integrals of sh x, ch x, th x, cth x
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Series expansions
Inverse functions
Areasinus
At - ∞< x < ∞
и - ∞ < y < ∞
имеют место формулы:
,
.
Areacosine
At 1 ≤ x< ∞
And 0 ≤ y< ∞
the following formulas apply:
,
.
The second branch of the areacosine is located at 1 ≤ x< ∞
and - ∞< y ≤ 0
:
.
Areatangent
At - 1
< x < 1
and - ∞< y < ∞
имеют место формулы:
,
Introduction
In mathematics and its applications to 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 organic growth processes, in which the rates of change of the functions involved in them are proportional to the values of the functions themselves.
If we denote it through a function and through an argument, then the differential law of the organic growth process can be written in the form where some constant coefficient proportionality.
Integrating this equation leads to a 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 that represents the integral law of the process under consideration.
Organic growth processes include, under certain simplifying assumptions, such phenomena as, for example, change atmospheric pressure depending on the height above the Earth's surface, radioactive decay, cooling or heating of the body in environment constant temperature, 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 available amount of the reactant), the proliferation of microorganisms and many others.
The increase in a sum of money due to the accrual of compound interest (interest on interest) is also a process of organic growth.
These examples could be continued.
Along with individual exponential functions, various combinations are used in mathematics and its applications. exponential functions, among which some linear and fractional-linear 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 these names are given, and here is a hyperbola 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 on a circle of unit radius are similar to the relations connecting hyperbolic functions with the coordinates of points on an equilateral hyperbola with a unit semi-axis. This justifies the name hyperbolic functions.
Hyperbolic functions
The functions given by the formulas are called hyperbolic cosine and hyperbolic sine, respectively.
These functions are defined and continuous on, and - even function, a is an odd function.
Figure 1.1 - Function graphs
From the definition of hyperbolic functions it follows that:
By analogy with trigonometric functions, hyperbolic tangent and cotangent are determined respectively by the formulas
The function is defined and continuous on, and the function is defined and continuous on the set with a punctured point; both functions are odd, their graphs are presented in the figures below.
Figure 1.2 - Function graph
Figure 1.3 - Function graph
It can be shown that the functions and are strictly increasing, and the function is strictly decreasing. Therefore, these functions are invertible. Let us denote the functions inverse to them by respectively.
Let's consider the function inverse to the function, i.e. function. Let's express it through elementary ones. Solving the equation relatively, 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 (сh x) are defined by the following equalities:
Hyperbolic tangent and cotangent are defined by analogy with trigonometric tangent and cotangent:
Hyperbolic secant and cosecant are defined similarly:
The following formulas apply:
The properties of hyperbolic functions are in many ways similar to those of (see). The equations x=cos t, y=sin t define the circle x²+y² = 1; the equations x=сh t, y=sh t define the hyperbola x² - y²=1. Just 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 how for circular (trigonometric) functions the argument t is numerically equal to the double area of the curvilinear triangle OKE (Fig. 49):
for a circle
for hyperbola
Addition theorems for hyperbolic functions are similar to addition theorems for trigonometric functions:
These analogies are easily seen if we take the complex variable r as the argument x. Hyperbolic functions are related to trigonometric functions by the following formulas: sh x = - i sin ix, cosh x = cos ix, where i is one of the values of the root √-1. Hyperbolic functions sh x, as well as ch x: can take as large values as desired (hence, naturally, large units) in contrast to trigonometric ones functions sin x, cos x, which for real values cannot be greater than one in absolute value.
Hyperbolic functions play a role in Lobachevsky geometry (see), they are used in the study of strength of materials, in electrical engineering and other branches of knowledge. There are also notations for hyperbolic functions in the literature such as sinh x; сosh x; tgh x.