Bessel functions (Bessel or cylindrical functions). Bessel functions with positive integer sign Zeros of first order Bessel functions

In order to move on to solving the problem of oscillations of a circular membrane, we must first become familiar with the Bessel functions. Bessel functions are solutions to a second-order linear differential equation with variable coefficients

This equation is called Bessel's equation. Both the equation itself and its solutions are found not only in the problem of oscillations of a circular membrane, but also in very large number other tasks.

The parameter k included in equation (10.1) can, generally speaking, take any positive value. Solutions of the equation for a given k are called Bessel functions of order k (sometimes called cylindrical functions). We will consider in detail only the most simple cases, when and since in the further presentation we will encounter only Bessel functions of the zeroth and first orders.

For general study Bessel functions we refer the reader to special manuals (see, for example, / n, (4.5)

where Δ 1, Δ 2, …, Δ n – random errors;

n – number of measurements.

The root mean square error is a reliable criterion for assessing the accuracy of measurements. Even with a small number of measurements, it is quite stable and well reflects the presence of large random errors, which essentially determine the quality of measurements.

Formula (4.5) is used to calculate the mean square error when the true value of the measured value is known. These cases are very rare in practice. As a rule, the true value of the measured value is unknown, but from measurements the most reliable result can be obtained - the arithmetic mean. Let us obtain a formula for calculating the mean square error using the deviation of individual results from the arithmetic mean using the so-called most probable errors V.

Let l 1, l 2, ..., l n be the results of equivalent measurements of the same quantity, the true value of which is X, and the arithmetic mean is L. Then n random or true errors can be calculated

Δ i = l i – X (4.6)

and n most probable errors

V i = l i – L. (4.7)

Sum n to equality (4.7)

[V] = [l] – nL. (4.8)

But, according to equality (4.4) nL = [l], therefore

that is, the sum of the most probable errors should always be equal to zero.

Subtracting equality (4.7) from equality (4.6), we obtain

Δ i – V i = L – X. (4.10)

On the right side of equality (4.10) we have a random error in the arithmetic mean. Let us denote it by ε. Then

Δi = V i + ε. (4.11)

Let us square equality (4.11), take their sum and divide it by n:

[Δ 2 ] / n = / n + nε 2 / n + 2ε[V] / n. (4.12)

The left side of this equality is nothing more than m 2 . Due to equality (4.9), the last term on the right-hand side is equal to zero.

m 2 = / n + ε 2. (4.13)

Let us replace the random error ε with its average value, i.e., the mean square error of the arithmetic mean. It will be proven below that mean square error of the arithmetic mean

M 2 = ε 2 = m 2 / n. (4.14)

m 2 – m 2 / n = / n or m 2 (n – 1) / n = / n,

where ___________

m 2 = / (n – 1), or m = √ / (n – 1). (4.15)

Formula (4.15) is called Bessel's formula and has great practical significance. It allows you to calculate the root mean square error based on the most likely deviations of measurement results from the arithmetic mean.

In addition to the mean square error, there are also average, probable and relative errors.

The average error (Θ) is the arithmetic mean of absolute values random errors i.e.

Θ = (|Δ 1 | + |Δ 2 | + … + |Δ n |) / n = [|Δ|] / n. (4.16)

In error theory it is proven that when n → ∞ Θ = 0.8 m, or m = 1.25Θ.

Sometimes in applied questions they use probable error r. Probable error is the value of a random error in one series of equally accurate measurements, in relation to which an error is equally possible both greater and less than this value, according to absolute value. To find r, all errors in a given series are arranged in ascending order in absolute value and the value that occupies the middle position is selected, i.e., there are as many errors less than it as there are more. The probable error is related to the mean square error by the ratio r = 2/3 m = 0.67 m or m = 1.5 r.

As can be seen, m > Θ and m > r, which shows that the mean square error better characterizes the accuracy of measurements than the average and probable errors.

The accuracy of measured quantities such as lines, areas, and volumes is often assessed using relative error. Relative error is the ratio of the absolute error to the value of the measured quantity. The relative error is written as a fraction, the numerator of which is one, and the denominator is a number indicating what proportion of the measured value should be the permissible error. For example, the length of a side D = 150 m is measured with an absolute error of m d = 0.05 m. Then the relative error of the measurement result will be m d / D = 0.05 m / 150 m = 1 / 3000.

The value 1/3000 means that at 3000 m of distance an error of 1 m can be allowed. The larger the denominator of the relative error, the higher the accuracy of the measurements. The accuracy of all linear measurements in geodesy is always specified by the relative error, which is given in the relevant instructions and manuals for the production of this type of geodetic work.

Introduction

Cylindrical functions are solutions to a second-order linear differential equation

where is a complex variable,

A parameter that can take any real or complex values.

The term “cylindrical functions” owes its origin to the fact that equation (1) occurs when considering boundary value problems of potential theory for a cylindrical domain.

Special classes of cylindrical functions are known in the literature as Bessel functions, and sometimes this name is assigned to the entire class of cylindrical functions.

The well-developed theory of the functions under consideration, the availability of detailed tables and a wide range of applications provide sufficient reason to classify cylindrical functions as one of the most important special functions.

The Bessel equation arises when finding solutions to the Laplace equation and the Helmholtz equation in cylindrical and spherical coordinates. Therefore, Bessel functions are used in solving many problems about wave propagation, static potentials, etc., for example:

1) electromagnetic waves in a cylindrical waveguide;

2) thermal conductivity in cylindrical objects;

3) vibration modes of a thin round membrane;

4) the speed of particles in a cylinder filled with liquid and rotating around its axis.

Bessel functions are also used in solving other problems, for example, in signal processing.

Cylindrical Bessel functions are the most common of all special functions. They have numerous applications in all natural and technical sciences(especially in astronomy, mechanics and physics). In a number of problems in mathematical physics, there are cylindrical functions in which the argument or index (sometimes both) take complex values. For numerical solution For such problems, it is necessary to develop algorithms that allow one to calculate Bessel functions with high accuracy.

Purpose of the course work: study of Bessel functions and application of their properties in solving differential equations.

1) Study the Bessel equation and modified equation Bessel.

2) Consider the basic properties of Bessel functions, asymptotic representations.

3) Decide differential equation using the Bessel function.

Bessel functions with positive integer sign

To consider many problems associated with the use of cylindrical functions, it is enough to confine ourselves to studying a special class of these functions, which corresponds to the case when the parameter in equation (1) is equal to zero or a positive integer.

The study of this class is more elementary than the theory relating to arbitrary values, and may serve as a good introduction to this general theory.

Let us show that one of the solutions to the equation

0, 1, 2, …, (1.1)

is the Bessel function of the first kind of order, which for any values ​​is defined as the sum of the series

Using d'Alembert's test, it is easy to verify that the series under consideration converges on the entire plane of a complex variable and, therefore, represents an entire function of.

If we denote the left side of equation (1.1) by and introduce an abbreviated notation for the coefficients of series (1.2), putting

then as a result of substitution we get

from which it follows that the expression in curly brackets is equal to zero. Thus, the function satisfies equation (1.1), i.e., it is a cylindrical function.

The simplest functions of the class under consideration are the Bessel functions of order zero and one:

Let us show that Bessel functions of other orders can be expressed in terms of these two functions. To prove this, assume that a is a positive integer, multiply series (1.2) by and differentiate by. We'll get it then

Similarly, multiplying the series by we find

Having differentiated in equalities (1.4 - 1.1) and divided by a factor, we arrive at the formulas:

which directly follows:

The resulting formulas are known as recurrence relations for Bessel functions.

The first of the relations makes it possible to express a function of an arbitrary order through functions of orders zero and one, which significantly reduces the work of compiling tables of Bessel functions.

The second relation allows one to represent derivatives of Bessel functions through Bessel functions. For this relation to be replaced by the formula

directly following from the definition of these functions.

Bessel functions of the first kind are simply related to the coefficients of the Laurent series expansion of the function):

The coefficients of this expansion can be calculated by multiplying power series:

and associations of members containing the same degrees. Having done this, we get:

whence it follows that the expansion under consideration can be written in the form

The function is called the generating function for Bessel functions with an integer sign; the found relation (1.12) plays an important role in the theory of these functions.

To obtain the general integral of equation (1.1), which gives an expression for an arbitrary cylindrical function with an integer sign, it is necessary to construct a second solution to the equation, linearly independent of c. As such a solution, the Bessel function of the second kind can be taken, based on the definition of which it is easy to obtain an analytical expression for it in the form of a series

(- Euler's constant) and, in the case, the first of the sums should be set equal to zero.

The function is regular in the plane with a cut. An essential feature of the solution under consideration is that it goes to infinity when. The general cylindrical function expression for represents linear combination constructed solutions

where and are arbitrary constants,