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 is a certain constant coefficient of proportionality.

Integrating this equation leads to general 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 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.

An increase in the amount of money due to accrual on it 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 - is an even function, and - 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.

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11 Basic functions of a complex variable

Let us recall the definition of a complex exponent – ​​. Then

Maclaurin series expansion. The radius of convergence of this series is +∞, which means that the complex exponential 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

Sine of a complex variable called 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-- this is a function

Let us note some properties of the newly introduced functions.

A. If x∈ ℝ, then cos x, sin x, cosh x, sh x∈ ℝ.

B. The following connection exists between trigonometric and hyperbolic functions:

cos iz=ch z; sin iz=ish z, ch iz=cos z; sh iz=isin z.

B. Basic trigonometric and hyperbolic identities:

cos 2 z+sin 2 z=1; ch 2 z-sh 2 z=1.

Proof of the main hyperbolic identity.

Basics trigonometric identity follows from the basic hyperbolic identity when taking into account the connection between trigonometric and hyperbolic functions (see property B)

G Addition formulas:

In particular,

D. To calculate derivatives of trigonometric and hyperbolic functions, one should apply the theorem on term-by-term differentiation of 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, and the functions sin z, sin z are odd.

J. (Frequency) 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, sin z are periodic with a period of 2πi. Moreover,

Applying the sum formulas, we get

Z. Expansion 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 univalence domain.

AND. Region 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. From relation (5) it follows that the mapping exp:D k → ℂ is injective. Let w be any non-zero complex number. Then, solving the equations e x =|w| and e iy =w/|w| with real variables x and y (y is chosen from the half-interval); sometimes introduced into consideration... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Ephron

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

Functions inverse to hyperbolic. functions; expressed by formulas... Natural science. Encyclopedic Dictionary

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

Books

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

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 cot 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

,
,
,

,
,
.

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

,
,
.

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:
,

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.