DOCSLIB.ORG

Bessel Functions and Their Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bessel Functions and Their Applications Bessel Functions and Their Applications Jennifer Niedziela∗ University of Tennessee - Knoxville (Dated: October 29, 2008) Bessel functions are a series of solutions to a second order differential equation that arise in many diverse situations. This paper derives the Bessel functions through use of a series solution to a differential equation, develops the different kinds of Bessel functions, and explores the topic of zeroes. Finally, Bessel functions are found to be the solution to the Schroedinger equation in a situation with cylindrical symmetry. 1 INTRODUCTION 0 0 X 2 n+s−1 (xy ) = an(n + s) x n=0 While special types of what would later be known as 1 0 0 X 2 n+s Bessel functions were studied by Euler, Lagrange, and (xy ) = an(n + s) x the Bernoullis, the Bessel functions were first used by F. n=0 W. Bessel to describe three body motion, with the Bessel functions appearing in the series expansion on planetary When the coefficients of the powers of x are organized, we perturbation [1]. This paper presents the Bessel func- find that the coefficient on xs gives the indicial equation tions as arising from the solution of a differential equa- s2 − ν2 = 0; =) s = ±ν, and we develop the general tion; an equation which appears frequently in applica- formula for the coefficient on the xs+n term: tions and solutions to physical situations [2] [3]. Fre- quently, the key to solving such problems is to recognize an−2 the form of this equation, thus allowing employment of an = − (3) (n + s)2 − ν2 the Bessel functions as solutions. The subject of Bessel Functions and applications is a very rich subject; never- In the case s = ν: theless, due to space and time restrictions and in the in- terest of studying applications, the Bessel function shall an−2 an = − (4) be presented as a series solution to a second order dif- n(n + 2ν) ferential equation, and then applied to a situation with cylindrical symmetry. Appropriate development of ze- and since a1 = 0, an = 0 for all n = odd integers. Coeffi- roes, modified Bessel functions, and the application of cients for even powers of n are found: boundary conditions will be briefly discussed. a a = − 2n−2 (5) 2n 22n(n + ν) THE BESSEL EQUATION Recalling that for the gamma function: Bessel's equation is a second order differential equation Γ(ν + 2) = (ν + 1)Γ(ν + 1); of the form Γ(ν + 3) = (ν + 2)Γ(ν + 2) = (ν + 2)(ν + 1)Γ(ν + 1); x2y00 + xy0 + (x2 − ν2)y = 0 (1) we can write the coefficients: By re-writing this equation as: a0 Γ(1 + ν) a2 = − 2 = − 2 x(xy0)0 + (x2 − ν2)y = 0 (2) 2 (1 + ν) 2 Γ(2 + ν) a0Γ(1 + ν) and employing the use of a generalized power series, we a2n = − n!22nΓ(n + 1 + ν) re-write the terms of (2) in terms of the series: 1 Which allows us to write the terms of the series : X n+s y = anx 1 n n=0 X (−1) x2n+ν y = J (x) = (6) 1 ν Γ(n + 1)Γ(n + ν + 1) 2 0 X n+s−1 n=0 y = an(n + s)x n=0 1 Where Jν (x) is the Bessel function of the first kind, order 0 X n+s ν. The first five Bessel functions of this kind are shown xy = an(n + s)x n=0 in figure 1. 2 g the gravitational constant of acceleration. Eq. (9) is a form of eq. (1), and solution is: 1 1 u = AJ0(2kz 2 ) + BY0(2kz 2 ); (10) where the A and B are determined by the boundary con- ditions. Modified Bessel Functions Modified Bessel functions are found as solutions to the modified Bessel equation FIG. 1: The Bessel Functions of orders ν = 0 to ν = 5 x2y00 + xy0 − (x2 − ν2)y = 0 (11) DIFFERENT ORDERS OF BESSEL FUNCTIONS which transforms into eq. (1) when x is replaced with ix. However, this leaves the general solution of eq. (1) In the preceding section, the form of Bessel functions a complex function of x. To avoid dealing with complex were obtained are known as "Bessel functions of the first solutions in practical applications [2], the solutions to kind."[3]. Different kinds of Bessel functions are obtained (11) are expressed in the form: with negative values of ν, or with complex arguments. This section briefly explores these different kinds of func- νπi iπ I (x) = e 2 J (xe 2 ) (12) tions ν ν The Iν (x) are a set of functions known as the modified Neumann Functions Bessel functions of the first kind. The general solution of the modified Bessel function is expressed as a combina- tion of I (x) and a function I (x): Bessel functions of the second kind are known as Neu- ν −ν mann functions, and are developed as a linear combina- y = AI−ν (x) − BIν (x) (13) tion of Bessel functions of the first order described: where again A and B are determined from the boundary cosνπJ (x) − J (x) conditions. N (x) = ν −ν (7) ν sinνπ A solution for non-integer orders of ν is found: For integral values of ν, the expression of Nν (x) has an π I−ν (x) − Iν (x) Kν (x) = (14) indeterminate form, and Nν (x)jx=0 = ±∞. Nevertheless 2 sinνπ the limit of this function for x 6= 0, the expression for Nν is valid for any value of ν, allowing the general solution The functions Kν (x) are known as modified Bessel func- to Bessel's equation to be written: tions of the second kind. A plot of the Neumann Func- tions (Nν (x)) and Modified Bessel functions (Iν (x))is y = AJν (x) + BNν (x) (8) shown in figure (2). A plot of the Modified Second Kind functions (Kn(x)) is shown in fig. (3). with A and B as arbitrary constants determined from Modified Bessel functions appear less frequently in ap- boundary conditions. plications, but can be found in transmission line studies, Bessel functions of the first and second kind are the non-uniform beams, and the statistical treatment of a most commonly found forms of the Bessel function in ap- relativistic gas in statistical mechanics. plications. Many applications in hydrodynamics, elastic- ity, and oscillatory systems have solutions that are based on the Bessel functions. One such example is that of a Zeroes of Bessel Functions uniform density chain fixed at one end undergoing small oscillations. The differential equation of this situation is: The zeroes of Bessel functions are of great importance d2u 1 du k2u in applications [5]. The zeroes, or roots, of the Bessel + + = 0 (9) dz2 z dz z functions are the values of x where value of the Bessel function goes to zero (Jν (x) = 0). Frequently, the ze- 2 p2 where z references a point on the chain, k = g , with p roes are found in tabulated formats, as they must the be as the frequency of small oscillations at that point, and numerically evaluated [5]. Bessel function's of the first 3 APPLICATION - SOLUTION TO SCHROEDINGER'S EQUATION IN A CYLINDRICAL WELL Consider a particle of mass m placed into a two- dimensional potential well, where the potential is zero inside of the radius of the disk, infinite outside of the radius of the disk. In polar coordinates using r, φ as representatives of the system, the Laplacian is written: 1 @ @Ψ 1 @2Ψ r2Ψ = r + : (15) r @r @r r2 @φ2 Which in the Schroedinger equation presents: FIG. 2: The Neumann Functions (black) and the Modified Bessel Functions (blue) for integer orders ν = 0 to ν = 5 ¯h2 1 @ @Ψ 1 @2Ψ − r + = EΨ: (16) 2m r @r @r r2 @φ2 Using the method of separation of variables with a pro- posed solution Ψ = R(r)T (φ) in (16), produces ¯h2 1 @ @R(r) 1 @2T − T (φ) r + R(r) = ER(r)T (φ) 2m R(r) @r @r r2 @φ2 (17) and then dividing by Ψ: 1 1 @ @R 1 1 @2T −2mE r + = (18) R r @r @r r2 T @φ2 ¯h2 2mE 2 2 setting h¯2 = k and multiplying through by r pro- duces 2 r d dR 2 2 1 d T FIG. 3: The Modified Bessel Functions of the second kind for r + k r + 2 = 0 (19) orders ν = 0 to ν = 5 [4] R dr dr T dφ which is fully separated in r and φ. To solve, the φ dependent portion is set to −m2, yielding the harmonic oscillator equation in T (φ), which presents the solution: and second kind have an infinite number of zeros as the T (φ) = Aeimφ (20) value of x goes to 1. The zeroes of the functions can be seen in the crossing points of the graphs in figure ( 1), Where A is a constant determined via proper normaliza- and figure ( 2). The modified Bessel functions of the first tion in φ: kind (Iν (x)) have only one zero at the point x = 0, and the modified Bessel equations of the second kind (Kν (x)) functions do not have zeroes. Z 2π r 1 A2T (φ)T (φ) dφ = 1 =) A = (21) 0 2π Bessel function zeros are exploited in frequency modu- lated (FM) radio transmission. FM transmission is math- q Leaving the φ dependent portion T (φ) = 1 eimφ ematically represented by a harmonic distribution of a 2π sine wave carrier modulated by a sine wave signal which Working now with the r dependent portion of the sep- arated equation, multiplying the r dependent portion of can be represented with Bessel Functions.
Load more

Recommended publications
  • Cylindrical Waves Guided Waves
    Cylindrical Waves Guided Waves Cylindrical Waves Daniel S. Weile Department of Electrical and Computer Engineering University of Delaware ELEG 648—Waves in Cylindrical Coordinates D. S. Weile Cylindrical Waves 2 Guided Waves Cylindrical Waveguides Radial Waveguides Cavities Cylindrical Waves Guided Waves Outline 1 Cylindrical Waves Separation of Variables Bessel Functions TEz and TMz Modes D. S. Weile Cylindrical Waves Cylindrical Waves Guided Waves Outline 1 Cylindrical Waves Separation of Variables Bessel Functions TEz and TMz Modes 2 Guided Waves Cylindrical Waveguides Radial Waveguides Cavities D. S. Weile Cylindrical Waves Separation of Variables Cylindrical Waves Bessel Functions Guided Waves TEz and TMz Modes Outline 1 Cylindrical Waves Separation of Variables Bessel Functions TEz and TMz Modes 2 Guided Waves Cylindrical Waveguides Radial Waveguides Cavities D. S. Weile Cylindrical Waves Separation of Variables Cylindrical Waves Bessel Functions Guided Waves TEz and TMz Modes The Scalar Helmholtz Equation Just as in Cartesian coordinates, Maxwell’s equations in cylindrical coordinates will give rise to a scalar Helmholtz Equation. We study it first. r2 + k 2 = 0 In cylindrical coordinates, this becomes 1 @ @ 1 @2 @2 ρ + + + k 2 = 0 ρ @ρ @ρ ρ2 @φ2 @z2 We will solve this by separating variables: = R(ρ)Φ(φ)Z(z) D. S. Weile Cylindrical Waves Separation of Variables Cylindrical Waves Bessel Functions Guided Waves TEz and TMz Modes Separation of Variables Substituting and dividing by , we find 1 d dR 1 d2Φ 1 d2Z ρ + + + k 2 = 0 ρR dρ dρ ρ2Φ dφ2 Z dz2 The third term is independent of φ and ρ, so it must be constant: 1 d2Z = −k 2 Z dz2 z This leaves 1 d dR 1 d2Φ ρ + + k 2 − k 2 = 0 ρR dρ dρ ρ2Φ dφ2 z D.
    [Show full text]
  • Appendix a FOURIER TRANSFORM
    Appendix A FOURIER TRANSFORM This appendix gives the definition of the one-, two-, and three-dimensional Fourier transforms as well as their properties. A.l One-Dimensional Fourier Transform If we have a one-dimensional (1-D) function fix), its Fourier transform F(m) is de­ fined as F(m) = [ f(x)exp( -2mmx)dx, (A.l.l) where m is a variable in the Fourier space. Usually it is termed the frequency in the Fou­ rier domain. If xis a variable in the spatial domain, miscalled the spatial frequency. If x represents time, then m is the temporal frequency which denotes, for example, the colour of light in optics or the tone of sound in acoustics. In this book, we call Eq. (A.l.l) the inverse Fourier transform because a minus sign appears in the exponent. Eq. (A.l.l) can be symbolically expressed as F(m) =,r 1{Jtx)}. (A.l.2) where ~-~denotes the inverse Fourier transform in Eq. (A.l.l). Therefore, the direct Fourier transform in this case is given by f(x) = r~ F(m)exp(2mmx)dm, (A.1.3) which can be symbolically rewritten as fix)= ~{F(m)}. (A.l.4) Substituting Eq. (A.l.2) into Eq. (A.l.4) results in 200 Appendix A ftx) = 1"1"-'{.ftx)}. (A.l.5) Therefore we have the following unity relation: ,.,... = l. (A.l.6) It means that performing a direct Fourier transform and an inverse Fourier transform of a function.ftx) successively leads to no change. Using the identity exp(ix) = cosx + isinx and Eq.
    [Show full text]
  • The Laplace Transform of the Modified Bessel Function K( T± M X) Where
    THE LAPLACE TRANSFORM OF THE MODIFIED BESSEL FUNCTION K(t±»>x) WHERE m-1, 2, 3, ..., n by F. M. RAGAB (Received in revised form 13th August 1963) 1. Introduction In the present paper we determine the Laplace transforms of the modified ±m Bessel function of the second kind Kn(t x), where m is any positive integer. The Laplace transforms are given in (2) and (4) below, p being the transform parameter and having positive real part. The formulae to be established are as follows (l)-(4): f Jo 2m-1 2\-~ -i-,lx 2m 2m 2m J k + v - 2 2m (2m)2mx2 v+1 v + 2 v + 2m (1) 2m 2m 2m where R(k±nm)>0 and x is taken for simplicity to be real and positive" When m = 1, x may be taken to be complex with real part greater than 1. The asterisk in the generalised hypergeometric function denotes that the . 2m . ., v+1 v + 2 v + 2m . factor =— m th e parametert s , , ..., is omitted; m is a positive 2m 2m 2m 2m integer. m m 1 e~"'Kn(t x)dt = 2 ~ V f V J ±l - 2m-l 2 2m 2m 2m 2m J 2m v+1 n v+1. 4p ' 2 2m 2 1m"' (2m)2mx2 2r2m+l v+1 v+2 v + 2m (2) _ 2m 2m ~lm~ E.M.S.—Y Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 05:34:47, subject to the Cambridge Core terms of use. 326 F. M. RAGAB where m is a positive integer, R(± mn +1)>0 and R(p) > 0; x is taken to be rea and positive and the asterisk has the same meaning as before.
    [Show full text]
  • Non-Classical Integrals of Bessel Functions
    J. Austral. Math. Soc. (Series E) 22 (1981), 368-378 NON-CLASSICAL INTEGRALS OF BESSEL FUNCTIONS S. N. STUART (Received 14 August 1979) (Revised 20 June 1980) Abstract Certain definite integrals involving spherical Bessel functions are treated by relating them to Fourier integrals of the point multipoles of potential theory. The main result (apparently new) concerns where /,, l2 and N are non-negative integers, and r, and r2 are real; it is interpreted as a generalized function derived by differential operations from the delta function 2 fi(r, — r^. An ancillary theorem is presented which expresses the gradient V "y/m(V) of a spherical harmonic function g(r)YLM(U) in a form that separates angular and radial variables. A simple means of translating such a function is also derived. 1. Introduction Some of the best-known differential equations of mathematical physics lead to Fourier integrals that involve Bessel functions when, for instance, they are solved under conditions of rotational symmetry about a centre. As long as the integrals are absolutely convergent they are amenable to the classical analysis of Watson's standard treatise, but a calculus of wider serviceability can be expected from the theory of generalized functions. This paper identifies a class of Bessel-function integrals that are Fourier integrals of derivatives of the Dirac delta function in three dimensions. They arise in the practical context of manipulating special functions-notably when changing the origin of the polar coordinates, for example, in order to evaluate so-called two-centre integrals and ©Copyright Australian Mathematical Society 1981 368 Downloaded from https://www.cambridge.org/core.
    [Show full text]
  • Radial Functions and the Fourier Transform
    Radial functions and the Fourier transform Notes for Math 583A, Fall 2008 December 6, 2008 1 Area of a sphere The volume in n dimensions is n n−1 n−1 vol = d x = dx1 ··· dxn = r dr d ω. (1) Here r = |x| is the radius, and ω = x/r it a radial unit vector. Also dn−1ω denotes the angular integral. For instance, when n = 2 it is dθ for 0 ≤ θ ≤ 2π, while for n = 3 it is sin(θ) dθ dφ for 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. The radial component of the volume gives the area of the sphere. The radial directional derivative along the unit vector ω = x/r may be denoted 1 ∂ ∂ ∂ ωd = (x1 + ··· + xn ) = . (2) r ∂x1 ∂xn ∂r The corresponding spherical area is ωdcvol = rn−1 dn−1ω. (3) Thus when n = 2 it is (1/r)(x dy−y dx) = r dθ, while for n = 3 it is (1/r)(x dy dz+ y dz dx + x dx dy) = r2 sin(θ) dθ dφ. The divergence theorem for the ball Br of radius r is thus Z Z div v dnx = v · ω rn−1dn−1ω. (4) Br Sr Notice that if one takes v = x, then div x = n, while x · ω = r. This shows that n n times the volume of the ball is r times the surface areaR of the sphere. ∞ z −t dt Recall that the Gamma function is defined by Γ(z) = 0 t e t . It is easy to show that Γ(z + 1) = zΓ(z).
    [Show full text]
  • Unified Bessel, Modified Bessel, Spherical Bessel and Bessel-Clifford Functions
    Unified Bessel, Modified Bessel, Spherical Bessel and Bessel-Clifford Functions Banu Yılmaz YAS¸AR and Mehmet Ali OZARSLAN¨ Eastern Mediterranean University, Department of Mathematics Gazimagusa, TRNC, Mersin 10, Turkey E-mail: [email protected]; [email protected] Abstract In the present paper, unification of Bessel, modified Bessel, spherical Bessel and Bessel- Clifford functions via the generalized Pochhammer symbol [ Srivastava HM, C¸etinkaya A, Kıymaz O. A certain generalized Pochhammer symbol and its applications to hypergeometric functions. Applied Mathematics and Computation, 2014, 226 : 484-491] is defined. Several potentially useful properties of the unified family such as generating function, integral representation, Laplace transform and Mellin transform are obtained. Besides, the unified Bessel, modified Bessel, spherical Bessel and Bessel- Clifford functions are given as a series of Bessel functions. Furthermore, the derivatives, recurrence relations and partial differential equation of the so-called unified family are found. Moreover, the Mellin transform of the products of the unified Bessel functions are obtained. Besides, a three-fold integral representation is given for unified Bessel function. Some of the results which are obtained in this paper are new and some of them coincide with the known results in special cases. Key words Generalized Pochhammer symbol; generalized Bessel function; generalized spherical Bessel function; generalized Bessel-Clifford function 2010 MR Subject Classification 33C10; 33C05; 44A10 1 Introduction Bessel function first arises in the investigation of a physical problem in Daniel Bernoulli's analysis of the small oscillations of a uniform heavy flexible chain [26, 37]. It appears when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates.
    [Show full text]
  • Special Functions
    Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS table of content Chapter 5 Special Functions 5.1 Heaviside step function - filter function 5.2 Dirac delta function - modeling of impulse processes 5.3 Sine integral function - Gibbs phenomena 5.4 Error function 5.5 Gamma function 5.6 Bessel functions 1. Bessel equation of order ν (BE) 2. Singular points. Frobenius method 3. Indicial equation 4. First solution – Bessel function of the 1st kind 5. Second solution – Bessel function of the 2nd kind. General solution of Bessel equation 6. Bessel functions of half orders – spherical Bessel functions 7. Bessel function of the complex variable – Bessel function of the 3rd kind (Hankel functions) 8. Properties of Bessel functions: - oscillations - identities - differentiation - integration - addition theorem 9. Generating functions 10. Modified Bessel equation (MBE) - modified Bessel functions of the 1st and the 2nd kind 11. Equations solvable in terms of Bessel functions - Airy equation, Airy functions 12. Orthogonality of Bessel functions - self-adjoint form of Bessel equation - orthogonal sets in circular domain - orthogonal sets in annular fomain - Fourier-Bessel series 5.7 Legendre Functions 1. Legendre equation 2. Solution of Legendre equation – Legendre polynomials 3. Recurrence and Rodrigues’ formulae 4. Orthogonality of Legendre polynomials 5. Fourier-Legendre series 6. Integral transform 5.8 Exercises Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS Introduction In this chapter we summarize information about several functions which are widely used for mathematical modeling in engineering. Some of them play a supplemental role, while the others, such as the Bessel and Legendre functions, are of primary importance. These functions appear as solutions of boundary value problems in physics and engineering.
    [Show full text]
  • Circular Waveguides Circular Waveguides Introduction
    Circular Waveguides Circular waveguides Introduction Waveguides can be simply described as metal pipes. Depending on their cross section there are rectangular waveguides (described in separate tutorial) and circular waveguides, which cross section is simply a circle. This tutorial is dedicated to basic properties of circular waveguides. For properties visualisation the electromagnetic simulations in QuickWave software are used. All examples used here were prepared in free CAD QW-Modeller for QuickWave and the models preparation procedure is described in separate documents. All examples considered herein are included in the QW-Modeller and QuickWave STUDENT Release installation as both, QW-Modeller and QW-Editor projects. Table of Contents CUTOFF FREQUENCY ........................................................................................................................................ 2 PROPAGATION MODES IN CIRCULAR WAVEGUIDE ........................................................................................... 5 MODE TE11........................................................................................................................................................ 5 MODE TM01 ...................................................................................................................................................... 9 1 www.qwed.eu Circular Waveguides Cutoff frequency Similarly as in the case of rectangular waveguides, propagation in circular waveguides is determined by a cutoff frequency. The cutoff frequency
    [Show full text]
  • The Laplace Transform of the Bessel Functions <Inlineequation ID="IE1
    The Laplace T,'ansform of the Bessel Functions where n ~ 1, 2, 3~ ... by F. M. RAhaB (a Cairo, Egitto - U.A.R.) (x) oo Summary. - The integrals / e--PtI(~ (1)xt~-n tl~--~dt and f e--pttk--iJ f 2a~t=~ni 1] di, n=l, 2, 3, .. o o are evaluated in term of MacRoberts' E--Functious and iu terms of generalized hyper. geometric functions. By talcing k~l, the Laplace transform of the four functions in the title is obtained. § 1. Introduction. - Nothing is known in the literature ([1] and [2]) of the Laplace transform of the functions where n is any posit.ire integer, except the case where the argument of the Bessel functions is equal to [~cti/2] (see [1] p. 185 and p. 199). In this paper the following four formulae, which give the Laplace transform of the above functions, will be established: / co 1 (1) e-Pttk--iKp xt dt=p--k2~--~(2u)~-'~x ~ o ](2n)2"p2]~_k nk ~ h n; : e :ei~ [~4x-~--]2 E A n; ~ + 2 , 2 2: (2n)~"p ~ 1 [[27~2n~2~ i . 1 E ~ n; 2 ~- , A n; 2 : 2: (2n)2"p 2 e:~ where R(k + U) ~ o, R{pI ~ o and n is any positive integer. The symbol h(n; a) represents the set of 318 F.M. RAGAB: The Laptace Tra.ns]orm o~ the Bessel Functions, etc. parameters o~ a+l a+n--1 8 (2~ e-ptt~-I xt dt = 2 k-~ r: o [ 1 1 1 ( 1 ) ( ~, 1 I, k, k+ h n; ~ h n ~, -~i ~ 2 )' ~ , ; --~.
    [Show full text]
  • Chapter on Diffraction
    Chapter 3 Diffraction While the particle description of Chapter 1 works amazingly well to understand how ra- diation is generated and transported in (and out of) astrophysical objects, it falls short when we investigate how we detect this radiation with our telescopes. In particular the diffraction limit of a telescope is the limit in spatial resolution a telescope of a given size can achieve because of the wave nature of light. Diffraction gives rise to a “Point Spread Function”, meaning that a point source on the sky will appear as a blob of a certain size on your image plane. The size of this blob limits your spatial resolution. The only way to improve this is to go to a larger telescope. Since the theory of diffraction is such a fundamental part of the theory of telescopes, and since it gives a good demonstration of the wave-like nature of light, this chapter is devoted to a relatively detailed ab-initio study of diffraction and the derivation of the point spread function. Literature: Born & Wolf, “Principles of Optics”, 1959, 2002, Press Syndicate of the Univeristy of Cambridge, ISBN 0521642221. A classic. Steward, “Fourier Optics: An Introduction”, 2nd Edition, 2004, Dover Publications, ISBN 0486435040 Goodman, “Introduction to Fourier Optics”, 3rd Edition, 2005, Roberts & Company Publishers, ISBN 0974707724 3.1 Fresnel diffraction Let us construct a simple “Camera Obscura” type of telescope, as shown in Fig. 3.1. We have a point-source at location “a”, and we point our telescope exactly at that source. Our telescope consists simply of a screen with a circular hole called the aperture or alternatively the pupil.
    [Show full text]
  • Bessel Functions and Equations of Mathematical Physics Markel Epelde García
    Bessel Functions and Equations of Mathematical Physics Final Degree Dissertation Degree in Mathematics Markel Epelde García Supervisor: Judith Rivas Ulloa Leioa, 25 June 2015 Contents Preface 3 1 Preliminaries 5 1.1 The Gamma function . .6 1.2 Sturm-Liouville problems . .7 1.3 Resolution by separation of variables . 10 1.3.1 The Laplacian in polar coordinates . 11 1.3.2 The wave equation . 12 1.3.3 The heat equation . 13 1.3.4 The Dirichlet problem . 14 2 Bessel functions and associated equations 15 2.1 The series solution of Bessel’s equation . 15 2.2 Bessel functions of the second and third kind . 20 2.3 Modified Bessel functions . 25 2.4 Exercises . 29 3 Properties of Bessel functions 33 3.1 Recurrence relations . 33 3.2 Bessel coefficients . 37 3.3 Bessel’s integral formulas . 38 3.4 Asymptotics of Bessel functions . 40 3.5 Zeros of Bessel functions . 42 3.6 Orthogonal sets . 47 3.7 Exercises . 54 4 Applications of Bessel functions 57 4.1 Vibrations of a circular membrane . 58 4.2 The heat equation in polar coordinates . 61 4.3 The Dirichlet problem in a cylinder . 64 Bibliography 67 1 2 Preface The aim of this dissertation is to introduce Bessel functions to the reader, as well as studying some of their properties. Moreover, the final goal of this document is to present the most well- known applications of Bessel functions in physics. In the first chapter, we present some of the concepts needed in the rest of the dissertation. We give definitions and properties of the Gamma function, which will be used in the defini- tion of Bessel functions.
    [Show full text]
  • (Everything a Physicist Needs to Know About) Bessel Functions Jn(X) of Integer Order (And Also Hankel Functions
    (Everything a physicist needs to know about) Bessel functions Jn(x) of integer order (1;2) (and also Hankel functions Hn ) Nikolai G. Lehtinen December 18, 2020 Abstract Some properties of integer-order Bessel functions Jn(x) are derived from their definition using the generating function. The results may be of use in such areas as plasma physics. Now with a Section on Hankel (1;2) functions Hn (x)! We assume that the reader knows some complex analysis (e.g., can integrate in the complex plane using residues). 1 Basic properties 1.1 Generating function We derive everything else from here, which will serve us the definition of the integer-order Bessel functions (of the first kind): +1 x (t− 1 ) X n g(x; t) = e 2 t = Jn(x)t (1) n=−∞ We immediately make our first observations that since g(x; t) is real for real x; t then Jn(x) must be real for real x. From now on, +1 X X X or just will mean n n=−∞ Z dθ Z 2π dθ will mean 2π 0 2π 1 1.2 Some useful formulas − x (t− 1 ) P n n P n 1. t ! −t in (1) =) e 2 t = (−1) Jn(x)t = Jn(−x)t =) (equate powers of t) =) n Jn(−x) = (−1) Jn(x) x 1 1 − 2 (t− t ) P n P −n 2. t ! t in (1) =) e = Jn(−x)t = Jn(x)t =) (equate powers of t and use previous expression) =) n J−n(x) = (−1) Jn(x) = Jn(−x) (2) iθ 1 1 3.
    [Show full text]