Coursework: Special properties of the Euler gamma function. Gamma radiation and its properties

Gamma rays are electromagnetic oscillations of a very high frequency, propagating in space at the speed of light. These radiations are emitted by the nucleus in the form of separate portions, called gamma quanta or photons.

The energy of gamma quanta lies in the range from 0.05 to 5 MeV. Gamma radiation with an energy of less than 1 MeV is conditionally called soft radiation, and with an energy of more than 1 MeV - hard radiation.

Gamma radiation is not an independent type of radiation. Usually gamma radiation accompanies beta decay, less often alpha decay. By ejecting alpha or beta particles, the nucleus is freed from excess energy, but can still remain in an excited state. The transition from the excited state to the ground state is accompanied by the emission of gamma rays, while the composition of the nucleus does not change.

In the air, gamma rays propagate over long distances, measured in tens and hundreds of meters.

The penetrating power of gamma rays is 50-100 times greater than the penetrating power of beta particles and thousands of times greater than the penetrating power of alpha particles.

Ionize the medium during the passage of gamma rays through it: only with secondary electrons that arise as a result of the interaction of gamma rays with atoms of matter. The ionizing ability of gamma quanta is determined by their energy. In general, one gamma quantum gives as many pairs of ions as there are beta or alpha particles of the same energy. However, due to the lower absorption of gamma rays, the ions they form are distributed over a greater distance. Therefore, the specific ionizing power of gamma rays is hundreds of times less than the specific ionizing power of beta particles, thousands of times less than the specific ionizing power of alpha particles and amounts to several pairs of ions in the air per 1 cm of the path.

Conclusion. Gamma radiation has the highest penetrating power compared to the penetrating power of other types of radioactive radiation. At the same time, gamma radiation has a very low specific ionizing capacity, amounting to several pairs of ions in air per 1 cm of the path of gamma rays.

Neutron radiation and its main properties

Neutron radiation is a corpuscular radiation that occurs in the process of fission or fusion of nuclei.

Neutrons have a strong damaging effect, since they, having no electric charge, easily penetrate into the nuclei of atoms that make up living tissues and are captured by them.

More than 99% of the total number of neutrons in a nuclear explosion is released within 10 -14 s. These neutrons are called prompt. The rest (about 1%) of neutrons are emitted later by some fission fragments during their beta decay. These neutrons are called delayed.

The neutron propagation speed reaches 20,000 km/h. The time required for all neutrons to travel the distance from the point of explosion to the place where they pose a threat of destruction is about one second after the moment of explosion.

Depending on the energy, neutrons are classified as follows:

slow neutrons 0-0.1 keV;

intermediate energy neutrons 0.1-20 keV;

fast neutrons 20 keV-10 MeV;

high energy neutrons over 10 MeV.

Thermal neutrons - neutrons that are in thermal equilibrium with the environment (with an energy not exceeding 1 eV), are included in the region of slow neutrons.

The passage of neutrons through matter is accompanied by a weakening of their intensity. This weakening is due to the interaction of neutrons with the nuclei of atoms of matter.

x-ray radiation

X-rays are produced when fast electrons bombard solid targets. The X-ray tube is an evacuated balloon with several electrodes (Fig. 1.2). The current-heated cathode K serves as a source of free electrons emitted due to thermionic emission. Cylindrical electrode Z is designed for focusing the electron beam.

The target is the anode A, which is also called the anticathode. It is made from heavy metals (W, C. Pt, etc.). Electrons are accelerated by a high voltage generated between the cathode and anticathode. Almost all the energy of electrons is released at the anticathode in the form of heat (only 1-3% of the energy is converted into radiation).

Once in the substance of the anticathode, the electrons experience strong deceleration and become a source of electromagnetic waves.

At a sufficiently high electron velocity, in addition to bremsstrahlung (i.e., radiation caused by electron deceleration), characteristic radiation is also excited (caused by excitation of the inner electron shells of anticathode atoms).

The intensity of X-ray radiation can be measured both by the degree of photographic action and by the ionization it produces in gaseous media, in particular in air. * The more intense the radiation, the more ionization it produces. According to the mechanism of interaction with matter, x-rays are similar to y-radiation. The wavelength of X-ray radiation is 10 -10 -10 -6 cm, gamma radiation -10-9 cm and below.

Currently, x-rays are used as a control tool. With the help of x-rays, they control the quality of welding, the uniformity of the corresponding products, etc. In medicine, x-rays are widely used for diagnosis, and in some cases as a means of influencing cancer cells.

Lecture No. 11 (2 lectures can be done)

The GAMMA FUNCTION, the G-function, is a transcendental function T(z) that propagates the values of the factorial z! for the case of any complex z ≠ 0, -1, -2, .... G.-f. introduced by L. Euler [(L. Euler), 1729, letter to Ch. Goldbach] using the infinite product

from which L. Euler obtained an integral representation (an Euler integral of the second kind)

true for Re z > 0. The polysemy of the function x z-1 is eliminated by the formula x z-1 = e (z-1)ln x with real ln x. Designation Г(z) and names. G.-f. were proposed by A. M. Legendre (A. M. Legendre, 1814).

On the entire z-plane with ejected points z = 0, -1, -2, ... for the G.-f. the Hankel integral representation is valid:

where s z-1 = e (z-1)ln s , and ln s is a branch of the logarithm, for which 0

Basic relationships and properties of G.-f.

1) Euler functional equation:

zГ(z) = Г(z + 1),

G(1) = 1, G(n + 1) = n!, if n > 0 is an integer, while counting 0! = Г(1) = 1.

2) Euler's complement formula:

Г(z)Г(1 - z) = π/sin πz.

In particular,

if n > 0 is an integer, then

y is real.

3) Gauss multiplication formula:

For m = 2, this is the Legendre doubling formula.

4) When Re z ≥ δ > 0 or |Im z| ≥ δ > 0, the asymptotic expansion of ln Г(z) in a Stirling series:

where B 2n are Bernoulli numbers. What does equality imply?

In particular,

More accurate is Sonin's formula:

5) In the real area G(x) > 0 for x > 0 and takes the sign (-1) k + 1 in the sections -k - 1

ГГ "" > Г" 2 ≥ 0,

i.e., all branches both |Г(x)| and ln |Г(х)| are convex functions. The property is logarithmic. convexity determines G.-f. among all solutions of the functional equation

G(1 + x) = xG(x)

up to a constant factor.

Rice. 2. Graph of the function y \u003d G (x).

For positive x H.-f. has a single minimum at x = 1.4616321... equal to 0.885603... . Local minima of the function |Г(х)| as x → -∞ they form a sequence tending to zero.

Rice. 3. Graph of the function 1/Г(x).

6) In the complex domain, for Re z > 0, G.-f. decreases rapidly as |Im z| → -∞

7) The function 1/Г(z) (see Fig. 3) is an entire function of the 1st order of maximum type, and asymptotically as Г → ∞

log М(r) ~ r log r,

It can be represented by an infinite Weierstrass product:

absolutely and uniformly convergent on any compact set of the complex plane (here the C-Euler constant). The integral Hankel representation is valid:

where the circuit C * is shown in fig. 4.

Integral representations for degrees of G.-f. were obtained by G. F. Vorony.

In applications, the so-called polygamma functions that are kth derivatives of ln Г(z). Function (ψ-Gauss function)

is meromorphic, has simple poles at the points z = 0,-1,_-2, ... and satisfies the functional equation

ψ(z + 1) - ψ(z) = 1/z.

From the representation ψ(z) for |z|

this formula is useful for calculating Г(z) in the vicinity of the point z = 1.

For other polygamma functions, see . The incomplete gamma function is defined by the equality

The functions Г(z), ψ(z) are transcendental functions that do not satisfy any linear differential equation with rational coefficients (Hölder's theorem).

The exclusive role of G.-f. in math. analysis is determined by the fact that with the help of G.-f. a large number of definite integrals, infinite products, and sums of series are expressed (see, for example, the Beta function). Besides, G.-f. finds wide applications in the theory of special functions (hypergeometric functions, for which the G.-f. is the limiting case, cylindrical functions, etc.), in analytic. number theory, etc.

Lit .: Whittaker E. T., Watson J. N., A course in modern analysis, trans. from English, vol. 2, 2nd ed., M., 1963; Bateman G., Erdeyi A., Higher transcendental functions Hypergeometric function. Legendre functions, trans. from English, M., 1965; Bourbaki N., Functions of a real variable. Elementary Theory, trans. from French, Moscow, 1965; Mathematical analysis. Functions, Limits, Series, Continued Fractions, (Reference Mathematical Library), M., 1961; Nielsen N. Handbuch der Theorie der Gamma-funktion, Lpz., 1906; Sonin N. Ya., Studies on cylindrical functions and special polynomials, Moscow, 1954; Voronoi G.F., Sobr. soch., vol. 2, K., 1952, p. 53-62; Janke E., Emde F., Lesh F., Special functions. Formulas, graphs, tables, trans. from German, 2nd ed., M., 1968; Ango A., Mathematics for electrical and radio engineers, trans. from French, 2nd ed., M., 1967.

L. P. Kuptsov.

Sources:

Mathematical Encyclopedia. T. 1 (A - D). Ed. collegium: I. M. Vinogradov (chief editor) [and others] - M., "Soviet Encyclopedia", 1977, 1152 stb. from ill.

The explanatory note to the course work is made in the amount of 36 sheets. It contains a table of gamma function values for some values of variables and program texts for calculating the gamma function values and for plotting, as well as 2 figures.

7 sources were used to write the term paper.

Introduction

Allocate a special class of functions, representable in the form of proper or improper integral, which depends not only on the formal variable, but also on the parameter.

Such functions are called parameter dependent integrals. These include the Euler gamma and beta functions.

The beta functions are represented by the Euler integral of the first kind:

The gamma function is represented by the Euler integral of the second kind:

The gamma function is one of the simplest and most significant special functions, the knowledge of the properties of which is necessary for the study of many other special functions, for example, cylindrical, hypergeometric and others.

Thanks to its introduction, our capabilities in the calculation of integrals are significantly expanded. Even in cases where the final formula does not contain functions other than elementary ones, deriving it often makes it easier to use the function Г, at least in intermediate calculations.

Euler integrals are well-studied non-elementary functions. The problem is considered solved if it is reduced to the calculation of Euler integrals.

1. Beta functions i euler

The beta functions are determined by the Euler integral of the first kind:

=(1.1)

It represents a function of two variable parameters

and : function B. If these parameters satisfy the conditions and , then the integral (1.1) will be an improper integral depending on the parameters and , and the singular points of this integral will be the points and

Integral (1.1) converge for

.Assuming we get: = -

i.e. argument

and enter symmetrically. Taking into account the identity

by the integration formula we have

Where do we get

For integer b = n, successively applying (1.2)

for integer

= m,= n, we have

but B(1,1) = 1, so:

We put in (1.1)

.Since the graph of the function

symmetrical with respect to a straight line, then

and as a result of the substitution

, we get

setting in (1.1)

, whence , we get

dividing the integral by two ranging from 0 to 1 and from 1 to

and applying the substitution to the second integral, we get

2. Gamma function

2.1 Definition

An exclamation point in mathematical works usually means taking the factorial of some non-negative integer:

n! = 1 2 3 ... n.

The factorial function can also be written as a recursion relation:

(n+1)! = (n+1) n!.

This relation can be considered not only for integer values of n.

Consider the difference equation

Despite the simple notation, this equation cannot be solved in elementary functions. Its solution is called the gamma function. The gamma function can be written as a series or as an integral. To study the global properties of the gamma function, the integral representation is usually used.

2.2 integral representation

Let's move on to solving this equation. We will look for a solution in the form of the Laplace integral:

In this case, the right side of equation (2.1) can be written as:

This formula is valid if there are limits for the non-integral term. We do not know beforehand the behavior of the image [(G)\tilde](p) as p®±¥. Let us assume that the image of the gamma function is such that the term outside the integral is equal to zero. After the solution is found, it will be necessary to check whether the assumption about the non-integral term is true, otherwise we will have to look for G(z) in some other way.

abstract

The purpose of this course work is to study the special properties of the Euler Gamma function. In the course of the work, the Gamma function, its main properties were studied, and a calculation algorithm was compiled with varying degrees of accuracy. The algorithm was written in a high-level language - C. The result of the program is compared with the table. No discrepancies were found in the values.

7 sources were used to write the term paper.

Introduction

Allocate a special class of functions, representable in the form of proper or improper integral, which depends not only on the formal variable, but also on the parameter.

Such functions are called parameter dependent integrals. These include the Euler gamma and beta functions.

The beta functions are represented by the Euler integral of the first kind:

The gamma function is represented by the Euler integral of the second kind:

Euler integrals are well-studied non-elementary functions. The problem is considered solved if it is reduced to the calculation of Euler integrals.

1. Beta functions i euler

The beta functions are determined by the Euler integral of the first kind:

It represents a function of two variable parameters and : a function B. If these parameters satisfy the conditions and , then the integral (1.1) will be an improper integral depending on the parameters and , and the singular points of this integral will be the points and

Integral (1.1) converge at . Assuming we get:

= - =

i.e. argument and enter symmetrically. Taking into account the identity

by the integration formula we have

Where do we get

For integer b = n, successively applying (1.2)

for integers = m,= n, we have

but B(1,1) = 1, so:

We put in (1.1) .Since the graph of the function symmetrical with respect to a straight line, then

and as a result of substitution, we get

assuming in (1.1) , whence , we get

dividing the integral by two in the range from 0 to 1 and from 1 to and applying the substitution integral to the second integral, we get

2. Gamma function

2.1 Definition

An exclamation point in mathematical works usually means taking the factorial of some non-negative integer:

n! = 1 2 3 ... n.

The factorial function can also be written as a recursion relation:

(n+1)! = (n+1) n!.

This relation can be considered not only for integer values of n.

Consider the difference equation

2.2 integral representation

Let's move on to solving this equation. We will look for a solution in the form of the Laplace integral:

In this case, the right side of equation (2.1) can be written as:

The left side of equality (2.1) is written as follows:

Then equation (2.1) for the image of the gamma function has the form:

This equation is easy to solve:

It is easy to see that the found function [(Γ)\tilde](p) is in fact such that the non-integral term in formula (2.2) is equal to zero.

Knowing the image of the gamma function, it is easy to obtain an expression for the preimage:

This is a non-canonical formula, in order to bring it to the form obtained by Euler, it is necessary to change the integration variable: t = exp(-p), then the integral will take the form:

The constant C is chosen so that for integer values of z the gamma function coincides with the factorial function: Г(n+1) = n!, then:

hence C = 1. Finally, we obtain the Euler formula for the gamma function:

This function is very common in mathematical texts. When working with special functions, perhaps even more often than an exclamation mark.

You can check that the function defined by formula (2.3) really satisfies equation (2.1) by integrating the integral on the right side of this formula by parts:

2.3 Domain and poles

In the integrand of the integral (2.3) at , the exponent exp( -tz) for R( z) > 0 decreases much faster than the algebraic function grows t(z-1) . The singularity at zero is integrable, so the improper integral in (2.3) converges absolutely and uniformly for R (z) > 0. Moreover, by successive differentiation with respect to the parameter z it is easy to verify that G( z) is a holomorphic function for R ( z) > 0. However, the unsuitability of the integral representation (2.3) for R ( z) 0 does not mean that the gamma function itself is not defined there - the solution of equation (2.1).

Let us consider the behavior of Г(z) in a neighborhood of zero. To do this, let's imagine:

where is a holomorphic function in the neighborhood z = 0. From formula (2.1) it follows:

that is, Г(z) has a first-order pole at z = 0.

It's also easy to get:

that is, in a neighborhood of the point, the function Г( z) also has a first-order pole.

In the same way, you can get the formula:

It follows from this formula that the points z = 0,-1,-2,... are simple poles of the gamma function and this function has no other poles on the real axis. It is easy to calculate the residue at the point z = -n, n = 0,1,2,...:

2.4 Hankel representation via loop integral

Find out if the gamma function has zeros. To do this, consider the function

The poles of this function are the zeros of the function Г(z).

Difference equation for I( z) is easy to obtain using the expression for Г( z):

The expression for solving this equation in the form of an integral can be obtained in the same way as the integral expression for the gamma function was obtained - through the Laplace transform. Below are the calculations. Neither are the same as in paragraph 1). And  the integral will be points ____________________________________________________________________________

After separating the variables, we get:

After integrating we get:

Passing to the Laplace preimage gives:

In the resulting integral, we make a change of the integration variable:

Then

It is important to note here that the integrand for noninteger values z has a branch point t= 0. On the complex plane of the variable t Let us draw a cut along the negative real semiaxis. We represent the integral along this semiaxis as the sum of the integral along the upper side of this section from to 0 and the integral from 0 to along the lower side of the section. So that the integral does not pass through the branch point, we arrange a loop around it.

Fig1: Loop in integral Hankel representation.

As a result, we get:

To figure out the value of the constant, remember that I(1) = 1, on the other hand:

integral representation

is called the Hankel representation with respect to the loop.

It is easy to see that the function 1/Γ( z) has no poles in the complex plane, hence the gamma function has no zeros.

Using this integral representation, one can obtain a formula for the product of gamma functions. To do this, in the integral we will make a change of variable , then:

2.5 Euler limit form

The gamma function can be represented as an infinite product. This can be seen if in the integral (2.3) we represent

Then the integral representation of the gamma function is:

In this formula, we can change the limits - the limit of integration in the improper integral and the limit for inside the integral. Here is the result:

Let's take this integral by parts:

If we carry out this procedure n times, we get:

Passing to the limit, we obtain the Euler limit form for the gamma function:

2.6 Formula for product

Below we need a formula in which the product of two gamma functions is represented through one gamma function. We derive this formula using the integral representation of the gamma functions.

We represent the iterated integral as a double improper integral. This can be done using Fubini's theorem. As a result, we get:

The improper integral converges uniformly. It can be considered, for example, as an integral over a triangle bounded by the coordinate axes and a straight line x + y = R at R. In the double integral, we make a change of variables:

Jacobian of this replacement

Integration limits: u changes from 0 to ∞, v while changing from 0 to 1. As a result, we get:

We rewrite this integral again as a repeated one, as a result we get:

where R p> 0, R v > 0.

2. Derivative of the gamma function

Integral

converges for each , because , and the integral at converges.

In the region where is an arbitrary positive number, this integral converges uniformly, since and we can apply the Weirstrass test. The whole integral is also convergent for all values since the second term on the right-hand side is an integral that certainly converges for any. It is easy to see that the integral converges over any domain where arbitrary. Valid for all specified values and for all , and since converges, then the conditions of the Weierstrass criterion are satisfied. Thus, in the area integral converges evenly.

This implies the continuity of the gamma function at. Let us prove the differentiability of this function at . Note that the function is continuous for and, and we show that the integral:

converges uniformly on each segment, . Let's choose a number so that ; then for . Therefore, there exists a number such that and for. But then the inequality holds for

and since the integral converges, the integral converges uniformly with respect to . Similarly, for there exists a number such that for all the inequality . With such and all we get , whence, by virtue of the criterion of comparison, it follows that the integral converges uniformly with respect to . Finally, the integral

in which the integrand is continuous in the domain

Obviously, converges uniformly with respect to . Thus, for the integral

converges uniformly, and, consequently, the gamma function is infinitely differentiable for any and the equality

Regarding the integral, we can repeat the same reasoning and conclude that

It is proved by induction that the Γ-function is infinitely differentiable and its i-th derivative satisfies the equality

Let us now study the behavior - functions and construct a sketch of its graph. (See Appendix 1)

It can be seen from the expression for the second derivative of the -function that for all . Therefore, it increases. Since , then, by the Role theorem on the segment, the derivative for and for , i.e., decreases monotonically on and monotonically increases on . Further, since , then at . For , it follows from the formula that for .

Equality , valid for , can be used when extending the -function to a negative value.

Let's put for that . The right side of this equality is defined for from (-1,0) . We get that the function continued in this way takes on (-1,0) negative values and at , as well as at the function .

Having defined in this way on , we can continue it to the interval (-2,-1) using the same formula. On this interval, the continuation will be a function that takes positive values and such that for and . Continuing this process, we define a function that has discontinuities at integer points (See Appendix 1.)

Note again that the integral

defines the Γ-function only for positive values of , continuation to negative values is carried out by us formally using the reduction formula .

4. Calculation of some integrals.

Stirling formula

Let's apply the gamma function to the calculation of the integral:

where m > -1,n > -1. Assuming that , we have

and based on (2.8) we have

In integral

Where k > -1,n > 0, it suffices to put

Integral

Where s > 0, expand in series

where is the Riemann zetta function

Consider incomplete gamma functions (Prim functions)

bound by inequality

Expanding, in a row we have

Turning to the derivation of the Stirling formula, which gives, in particular, an approximate value of n! for large values of n , consider first the auxiliary function

(4.2)

Continuous on the interval (-1,) monotonically increases from to when changing from to and turns to 0 at u = 0. Since

And so the derivative is continuous and positive in the entire interval, satisfies the condition

It follows from the above that there is an inverse function defined on an interval that is continuous and monotonically increasing in this interval,

Turning to 0 at v=0 and satisfying the condition

We derive the Stirling formula from the equality

assuming we have

assuming at the end, we get

in the limit at i.e. at (see 4.3)

where does Stirling's formula come from

which can be taken in the form

where , at

for sufficiently large suppose

the calculation is made using logarithms

if a positive integer, then (4.5) also turns into an approximate formula for calculating factorials for large values of n

we give without derivation a more precise formula

where in parentheses is a non-converging series.

5. Examples of calculating integrals

Formulas are needed to calculate:

G()

Calculate Integrals

PRACTICAL PART

To calculate the gamma function, an approximation of its logarithm is used. To approximate the gamma function on the interval x>0, the following formula is used (for complex z):

Г(z+1)=(z+g+0.5) z+0.5 exp(-(z+g+0.5))

This formula is similar to Stirling's approximation, but it has a correction series. For the values g=5 and n=6, it is checked that the error ε does not exceed 2*10 -10 . Moreover, the error does not exceed this value on the entire right half of the complex plane: z > 0.

To obtain the (real) gamma function on the interval x>0, the recursive formula Г(z+1)=zГ(z) and the above approximation Г(z+1) are used. In addition, it can be seen that it is more convenient to approximate the logarithm of the gamma function than the gamma function itself. Firstly, this will require calling only one mathematical function - the logarithm, and not two - the exponent and the degree (the latter still uses the call of the logarithm), and secondly, the gamma function is rapidly growing for large x, and its approximation by the logarithm removes overflow issues.

To approximate Ln(Г(х) - the logarithm of the gamma function - the formula is obtained:

log(G(x))=(x+0.5)log(x+5.5)-(x+5.5)+

log(C 0 (C 1 +C 2 /(x+1)+C 3 /(x+2)+...+C 7 /(x+8))/x)

Coefficient values C k- tabular data (see in the program).

The gamma function itself is obtained from its logarithm by taking the exponent.

Conclusion

Gamma functions are a convenient tool for calculating some integrals, in particular many of those integrals that are not representable in elementary functions.

Because of this, they are widely used in mathematics and its applications, in mechanics, thermodynamics, and other branches of modern science.

Bibliography

1. Special functions and their applications:

Lebedev I.I., M., Gostekhterioizdat, 1953

2. Mathematical analysis part 2:

Ilyin O.A., Sadovnichiy V.A., Sendov Bl.Kh., M.,”Moscow University”,1987

3. Collection of problems in mathematical analysis:

Demidovich B.P., M., Nauka, 1966

4. Integrals and series of special functions:

Prudnikov A.P., Brychkov Yu.A., M., Nauka, 1983

5. Special Features:

Kuznetsov, M.,”High School”,1965

6. Asymptotics and special functions

F. Olver, M., Nauka, 1990.

7.Monster zoo or introduction to special features

O.M. Kiselev,

APPS

Appendix 1 - Graph of the gamma function of a real variable

Appendix 2 - Graph of the Gamma Function

Table - a table of gamma function values for some values of the argument.

Appendix 3 is a program listing that draws a table of gamma function values for some argument values.

Appendix 4 - listing of a program that draws a graph of the gamma function

Abstract................................................. ............ ...................................3

Introduction ................................................ .......... ...................................4

Theoretical part…………………………………………………….5

Euler Beta Function……………………………………………….5

Gamma function................................................... . ...................................eight

2.1. Definition…………………………………………………...8

2.2. Integral representation………………………………8

2.3. Domain of definition and poles…………………………..10

2.4. Hankel representation in terms of loop integral………..10

2.5. Euler limit form………………………………...12

2.6. The formula for the product………………………………..13

Derivative of the gamma function .......................................................... ...........fifteen

Calculation of integrals. Stirling Formula..............................18

Examples of calculation of integrals ............................................................... ......23

Practical part…………………………………………………….24

Conclusion................................................. ...... .................................25

References………………………………………………..............26

Applications……………………………………………………………..27

APPENDIX 1

Graph of the gamma function of a real variable

APPENDIX 2

Graph of the gamma function

TABLE

X	g(x)

APPENDIX 3

#include

static double cof=(

2.5066282746310005,

1.0000000000190015,

76.18009172947146,

86.50532032941677,

24.01409824083091,

1.231739572450155,

0.1208650973866179e-2,

0.5395239384953e-5,

double GammLn(double x) (

lg1=log(cof*(cof+cof/(x+1)+cof/(x+2)+cof/(x+3)+cof/(x+4)+cof/(x+5)+cof /(x+6))/x);

lg=(x+0.5)*log(x+5.5)-(x+5.5)+lg1;

double Gamma(double x) (

return(exp(GammLn(x)));

cout<<"vvedite x";

printf("\n\t\t\t| x |Gamma(x) |");

printf("\n\t\t\t_________________________________________");

for(i=1;i<=8;i++)

x=x[i]+0.5;

g[i]=Gamma(x[i]);

printf("\n\t\t\t| %f | %f |",x[i],g[i]);

printf("\n\t\t\t_________________________________________");

printf("\n Dlia vuhoda iz programmu najmite lybyiy klavishy");

APPENDIX 4

#include

double gam(double x, double eps)

Int I, j, n, nb;

Double dze=(1.6449340668422643647,

1.20205690315959428540,

1.08232323371113819152,

1.03692775514336992633,

1.01734306198444913971};

Double a=x, y, fc=1.0, s, s1, b;

Printf("You entered incorrect data, please try again\n"); return -1.0;

If(a==0) return fc;

For(i=0;i<5;i++)

S=s+b*dze[i]/(i+2.0);

Nb=exp((i.0/6.0)*(7.0*log(a)-log(42/0)-log(eps)))+I;

For(n=1;n<=nb;n++)

For(j=0; j<5; j++)

Si=si+b/(j+1.0);

S=s+si-log(1.0+a/n);

Double dx,dy, xfrom=0,xto=4, yto=5, h, maxy, miny;

Int n=100, I, gdriver=DETECT, gmode, X0, YN0, X, Y, Y0,pr=0;

Initgraph(&gdriver,&gmode, “ ”);

YN0=getmaxy()-20;

Line(30, getmaxy()-10,30,30);

Line(20, getmaxy()-30, getmaxx()-20, getmaxy()-30);

)while (Y>30);

)while (X<700);

)while (X<=620);

)while (y>=30);

X=30+150.0*0.1845;

For9i=1;i

Dy=gam(dx,1e-3);

X=30+(600/0*i)/n;

If(Y<30) continue;

X=30+150.0*308523;

line(30,30,30,10);

Line(620,450,640,450);

Line(30,10,25,15);

Line(640,450,635,445);

Line(640,450,635,455);

Line(170,445,170,455);

Line(320,445,320,455);

Line(470,445,470,455);

Line(620,445,620,455);

Line(25,366,35,366);

Line(25,282,35,282);

Line(25,114,35,114);

Line(25,30,35,30);

Outtexty(20,465,"0");

Outtexty(165,465, "1";

Outtexty(315,465, "2";

Outtexty(465,465, "3";

Outtexty(615,465, "4";

Outtexty(630,465, "x";

Outtexty(15,364, "1";

Outtexty(15,280, "2";

Outtexty(15,196, "3";

Outtexty(15,112, "4";

Outtexty(15,30, "5";

It has been experimentally established that g-radiation (see § 255) is not an independent form of radioactivity, but only accompanies a- and b-decays and also arises during nuclear reactions, during the deceleration of charged particles, their decay, etc. g-spectrum is lined. The g-Spectrum is the energy distribution of the number of g-quanta (the same interpretation of the b-spectrum is given in §258). The discreteness of the g-spectrum is of fundamental importance, since it is proof of the discreteness of the energy states of atomic nuclei.

It is now firmly established that g-radiation is emitted by the daughter (rather than the parent) nucleus. The daughter nucleus at the moment of its formation, being excited, passes into the ground state with the emission of g-radiation in a time of approximately 10 -13 - 10 -14 s, which is much shorter than the lifetime of an excited atom (approximately 10 -8 s). Returning to the ground state, the excited nucleus can pass through a number of intermediate states, so g-radiation of the same radioactive isotope can contain several groups of g-quanta, differing from each other in their energy.

With g-radiation BUT and Z of the kernel do not change, so it is not described by any displacement rules. The g-radiation of most nuclei is of such short wavelength that its wave properties are very weakly manifested. Here, corpuscular properties come to the fore, so g-radiation is considered as a stream of particles - g-quanta. During radioactive decays of various nuclei, g-quanta have energies from 10 keV to 5 MeV.

The nucleus, which is in an excited state, can go into the ground state not only by emitting a g-quantum, but also by directly transferring the excitation energy (without prior emission of a g-quantum) to one of the electrons of the same atom. In this case, the so-called conversion electron is emitted. The phenomenon itself is called internal conversion. Internal conversion is a process that competes with g-radiation.

Conversion electrons correspond to discrete values of energy, depending on the work function of the electron from the shell from which the electron escapes, and on the energy E , given by the nucleus during the transition from the excited state to the ground state. If all the energy E is released in the form of a y-quantum, then the radiation frequency v is determined from the known relation E=hv . If they emit L electrons of internal conversion, then their energies are equal to E-A K, E-A L, ..., where A k, A L, ... is the work function of an electron from K - and L-shells. The monoenergetic nature of conversion electrons makes it possible to distinguish them from b-electrons, whose spectrum is continuous (see § 258). The vacancy on the inner shell of the atom that has arisen as a result of the emission of an electron will be filled with electrons from the overlying shells. Therefore, internal conversion is always accompanied by characteristic X-ray emission.

G-quanta, having zero rest mass, cannot slow down in a medium, therefore, when g-radiation passes through matter, they are either absorbed or scattered by it. g-quanta do not carry an electric charge and thus do not experience the influence of Coulomb forces. When a beam of y-quanta passes through a substance, their energy does not change, but as a result of collisions, the intensity is weakened, the change of which is described by the exponential law x, m - absorption coefficient). Since g-radiation is the most penetrating radiation, m for many substances is a very small value; m depends on the properties of matter and on the energy of g-quanta.

g-quanta, passing through the substance, can interact both with the electron shell of the atoms of the substance, and with their nuclei. In quantum electrodynamics, it is proved that the main processes accompanying the passage of g-radiation through matter are the photoelectric effect, the Compton effect (Compton scattering), and the formation of electron-positron pairs.

The photoelectric effect, or photoelectric absorption of g-rays, is a process in which an atom absorbs a g-quantum and emits an electron. Since the electron is knocked out of one of the inner shells of the atom, the vacated space is filled with electrons from the overlying shells, and the photoelectric effect is accompanied by characteristic X-ray radiation. The photoelectric effect is the predominant absorption mechanism in the region of low energies of g-quanta (E g< 100 кэВ). Фотоэффект может идти только на связанных электронах, так как свободный электрон не может поглотить g-квант, при этом одновременно не удовлетворяются законы сохранения энергии и импульса.

As the energy of g-quanta increases (E g » 0.5 MeV), the probability of the photoelectric effect is very small, and the main mechanism for the interaction of g-quanta with matter is Compton scattering (see § 206).

When E g >1.02 MeV = 2m e c 2 (m e is the rest mass of an electron), the process of formation of electron-positron pairs in the electric fields of nuclei becomes possible. The probability of this process is proportional to Z 2 and increases with E g. Therefore, at E g » 10 MeV, the main process of g-radiation interaction in any substance is the formation of electric-positron pairs.

If the energy of a g-quantum exceeds the binding energy of nucleons in the nucleus (7-8 MeV), then as a result of the absorption of a g-quantum, a nuclear photoelectric effect can be observed - the emission of one of the nucleons from the nucleus, most often a neutron.

The large penetrating power of g-radiation is used in gamma flaw detection - a flaw detection method based on the different absorption of g-radiation when it propagates over the same distance in different media. The location and size of defects (cavities, cracks, etc.) are determined by the difference in the intensities of the radiation that has passed through different parts of the translucent product.

The impact of g-radiation (as well as other types of ionizing radiation) on a substance is characterized by a dose of ionizing radiation. Differ:

The absorbed dose of radiation is a physical quantity equal to the ratio of the radiation energy to the mass of the irradiated substance.

The unit of the absorbed radiation dose is gray (Gy) *: 1 Gy \u003d 1 J / kg - radiation dose at which the energy of any ionizing radiation of 1 J is transferred to an irradiated substance weighing 1 kg.

The exposure dose of radiation is a physical quantity equal to the ratio of the sum of electric charges of all ions of the same sign, created by electrons released in irradiated air (under the condition of full use of the ionizing ability of electrons), to the mass of this air.

The unit of the exposure dose of radiation is a pendant per kilogram (C/kg); the dark unit is the roentgen (R): 1 R=2.58×10 -4 C/kg.

Biological dose - a value that determines the effect of radiation on the body.

The biological dose unit is the biological equivalent of a roentgen (rem): 1 rem is a dose of any type of ionizing radiation that produces the same biological effect as a dose of x-ray or g radiation in 1 R (1 rem = 10 -2 J / kg).

Portal for the student. Self-training

Introduction

Introduction

2. Derivative of the gamma function

Let's apply the gamma function to the calculation of the integral:

Conclusion

Bibliography

1. Special functions and their applications:

Lebedev I.I., M., Gostekhterioizdat, 1953

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