Other designations: sinh x, Sh x, cosh x, Ch x, tgh x, tanh x, Th x. Graphs see in fig. 1.
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 = √[ 1]); 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 others 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 of the phenomena studied in natural science are among the so-called processes of organic growth, in which the rates of change of the functions involved 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 common decision as 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, changes in atmospheric pressure depending on the height above the Earth's surface, radioactive decay, cooling or heating of the body in environment 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 the reactant present), 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 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 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 - 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.
, page 611 Basic functions of a complex variable
Recall the definition of the complex exponent - . Then
Maclaurin series expansion. The radius of convergence of this series is +∞, which means that the complex exponent is analytic on the entire complex plane and
(exp z)"=exp z; exp 0=1. (2)
The first equality here follows, for example, from the theorem on term-by-term differentiation of a power series.
11.1 Trigonometric and hyperbolic functions
The sine of a complex variable called a function
Cosine of a complex variable there is a function
Hyperbolic sine of a complex variable is defined like this:
Hyperbolic cosine of a complex variable-- is a function
We note some properties of the newly introduced functions.
A. If x∈ ℝ , then cos x, sin x, ch x, sh x∈ ℝ .
B. There is the following connection between trigonometric and hyperbolic functions:
cos iz=ch z; sin iz=ish z, ch iz=cos z; shiz=isinz.
B. Basic trigonometric and hyperbolic identities:
cos 2 z+sin 2 z=1; ch 2 z-sh 2 z=1.
Proof of the basic hyperbolic identity.
The main trigonometric identity follows from the Ononian hyperbolic identity when the connection between trigonometric and hyperbolic functions is taken into account (see property B)
G Addition Formulas:
In particular,
D. To calculate the derivatives of trigonometric and hyperbolic functions, one should apply the theorem on term-by-term differentiation of a power series. We get:
(cos z)"=-sin z; (sin z)"=cos z; (ch z)"=sh z; (sh z)"=ch z.
E. The functions cos z, ch z are even, while the functions sin z, sh z are odd.
G. (Periodicity) The function e z is periodic with period 2π i. The functions cos z, sin z are periodic with a period of 2π, and the functions ch z, sh z are periodic with a period of 2πi. Moreover,
Applying the sum formulas, we get
W. Decompositions into real and imaginary parts:
If a single-valued analytic function f(z) bijectively maps a domain D onto a domain G, then D is called a domain of univalence.
AND. Domain D k =( x+iy | 2π k≤ y<2π (k+1)} для любого целого k является областью однолистности функции e z , которая отображает ее на область ℂ* .
Proof. Relation (5) implies that the mapping exp:D k → ℂ is injective. Let w be any nonzero complex number. Then, solving the equations e x =|w| and e iy =w/|w| with real variables x and y (we choose y from the half-interval )