Bessel Functions Of The First And Second Kind
Bessel Functions of the First and Second KindReadingProblemsOutlineBackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6Modified Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Kelvin’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Hankel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Orthogonality of Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Assigned Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
BackgroundBessel functions are named for Friedrich Wilhelm Bessel (1784 - 1846), however, DanielBernoulli is generally credited with being the first to introduce the concept of Bessels functions in 1732. He used the function of zero order as a solution to the problem of an oscillatingchain suspended at one end. In 1764 Leonhard Euler employed Bessel functions of both zeroand integral orders in an analysis of vibrations of a stretched membrane, an investigationwhich was further developed by Lord Rayleigh in 1878, where he demonstrated that Besselsfunctions are particular cases of Laplaces functions.Bessel, while receiving named credit for these functions, did not incorporate them into hiswork as an astronomer until 1817. The Bessel function was the result of Bessels study of aproblem of Kepler for determining the motion of three bodies moving under mutual gravitation. In 1824, he incorporated Bessel functions in a study of planetary perturbations wherethe Bessel functions appear as coefficients in a series expansion of the indirect perturbationof a planet, that is the motion of the Sun caused by the perturbing body. It was likelyLagrange’s work on elliptical orbits that first suggested to Bessel to work on the Besselfunctions.The notation Jz,n was first used by Hansen9 (1843) and subsequently by Schlomilch10 (1857)and later modified to Jn(2z) by Watson (1922).Subsequent studies of Bessel functions included the works of Mathews11 in 1895, “A treatiseon Bessel functions and their applications to physics” written in collaboration with AndrewGray. It was the first major treatise on Bessel functions in English and covered topics suchas applications of Bessel functions to electricity, hydrodynamics and diffraction. In 1922,Watson first published his comprehensive examination of Bessel functions “A Treatise onthe Theory of Bessel Functions” 12 .9Hansen, P.A. “Ermittelung der absoluten Strungen in Ellipsen von beliebiger Excentricitt und Neigung,I.” Schriften der Sternwarte Seeberg. Gotha, 1843.10Schlmilch, O.X. “Ueber die Bessel’schen Function.” Z. fr Math. u. Phys. 2, 137-165, 1857.11George Ballard Mathews, “A Treatise on Bessel Functions and Their Applications to Physics,” 189512G. N. Watson , “A Treatise on the Theory of Bessel Functions,” Cambridge University Press, 1922.2
Definitions1. Bessel EquationThe second order differential equation given asx2d2 ydx2 xdydx (x2 ν 2 )y 0is known as Bessel’s equation. Where the solution to Bessel’s equation yields Bessel functionsof the first and second kind as follows:y A Jν (x) B Yν (x)where A and B are arbitrary constants. While Bessel functions are often presented in textbooks and tables in the form of integer order, i.e. ν 0, 1, 2, . . . , in fact they are definedfor all real values of ν .2. Bessel Functionsa) First Kind: Jν (x) in the solution to Bessel’s equation is referred to as a Besselfunction of the first kind.b) Second Kind: Yν (x) in the solution to Bessel’s equation is referred to as aBessel function of the second kind or sometimes the Weber function or theNeumann function.b) Third Kind: The Hankel function or Bessel function of the third kind can bewritten asHν(1) (x) Jν (x) iYν (x)x 0Hν(2) (x) Jν (x) iYν (x)x 0Because of the linear independence of the Bessel function of the first and secondkind, the Hankel functions provide an alternative pair of solutions to the Besseldifferential equation.3
3. Modified Bessel Equation By letting x i x (where i 1) in the Bessel equation we can obtain the modifiedBessel equation of order ν, given asx2d2 ydx2 xdydx (x2 ν 2 )y 0The solution to the modified Bessel equation yields modified Bessel functions of the first andsecond kind as follows:y C Iν (x) D Kν (x)x 04. Modified Bessel Functionsa) First Kind: Iν (x) in the solution to the modified Bessel’s equation is referredto as a modified Bessel function of the first kind.b) Second Kind: Kν (x) in the solution to the modified Bessel’s equation is referred to as a modified Bessel function of the second kind or sometimes theWeber function or the Neumann function.5. Kelvin’s FunctionsA more general form of Bessel’s modified equation can be written asx2d2 ydx2 xdydx (β 2 x2 ν 2 )y 0where β is an arbitrary constant and the solutions is nowy C Iν (βx) D Kν (βx)If we letβ2 iwherei 14
and we noteIν (x) i ν Jν (ix) Jν (i3/2 x)then the solution is written asy C Jν (i3/2 x) DKν (i1/2 x)The Kelvin functions are obtained from the real and imaginary portions of this solution asfollows:berν Re Jν (i3/2 x)beiν Im Jν (i3/2 x)Jν (i3/2 x) berν x i bei xkerν Re i ν Kν (i1/2 x)keiν Im i ν Kν (i1/2 x)i ν Kν (i1/2 x) kerν x i kei x5
TheoryBessel FunctionsBessel’s differential equation, given asx2d2 ydx2 xdydx (x2 ν 2 )y 0is often encountered when solving boundary value problems, such as separable solutionsto Laplace’s equation or the Helmholtz equation, especially when working in cylindrical orspherical coordinates. The constant ν, determines the order of the Bessel functions found inthe solution to Bessel’s differential equation and can take on any real numbered value. Forcylindrical problems the order of the Bessel function is an integer value (ν n) while forspherical problems the order is of half integer value (ν n 1/2).Since Bessel’s differential equation is a second-order equation, there must be two linearlyindependent solutions. Typically the general solution is given as:y AJν (x) BYν (x)where the special functions Jν (x) and Yν (x) are:1. Bessel functions of the first kind, Jν (x), which are finite at x 0 for all real valuesof ν2. Bessel functions of the second kind, Yν (x), (also known as Weber or Neumann functions) which are singular at x 0The Bessel function of the first kind of order ν can be be determined using an infinite powerseries expansion as follows:Jν (x) ( 1)k (x/2)ν 2kk 0 k!Γ(ν k 1)1Γ(1 ν) ν x21 (x/2)21(1 ν)6 1 (x/2)22(2 ν) 1 (x/2)23(3 ν) (1 · · ·
Jn x 10.80.60.40.20-0.2-0.4J0J10J22468x101214Figure 4.1: Plot of the Bessel Functions of the First Kind, Integer Orderor by noting that Γ(ν k 1) (ν k)!, we can writeJν (x) ( 1)k (x/2)ν 2kk 0k!(ν k)!Bessel Functions of the first kind of order 0, 1, 2 are shown in Fig. 4.1.The Bessel function of the second kind, Yν (x) is sometimes referred to as a Weber functionor a Neumann function (which can be denoted as Nν (x)). It is related to the Bessel functionof the first kind as follows:Yν (x) Jν (x) cos(νπ) J ν (x)sin(νπ)where we take the limit ν n for integer values of ν.For integer order ν, Jν , J ν are not linearly independent:J ν (x) ( 1)ν Jν (x)Y ν (x) ( 1)ν Yν (x)7
in which case Yν is needed to provide the second linearly independent solution of Bessel’sequation. In contrast, for non-integer orders, Jν and J ν are linearly independent and Yνis redundant.The Bessel function of the second kind of order ν can be expressed in terms of the Besselfunction of the first kind as follows:Yν (x) 2π Jν (x) ln1 x2 γk 1( 1)π k 01 (ν k 1)!ν 1 π k 0k! 2k νx2 1111 2k ν 1 ··· 1 ··· x2k2k νk!(k ν)!2Bessel Functions of the second kind of order 0, 1, 2 are shown in Fig. 4.2.1Y0Yn x 0.5Y1Y20-0.5-1-1.502468x101214Figure 4.2: Plot of the Bessel Functions of the Second Kind, Integer Order8
Relations Satisfied by the Bessel FunctionRecurrence FormulasBessel functions of higher order be expressed by Bessel functions of lower orders for all realvalues of ν.2νJν 1 (x) x1 (x) Jν 12Jν (x) Jν 1 (x)νddxddxx Yν 1(x) [Jν 1 (x) Jν 1 (x)]Jν (x) Jν 1 (x) Jν (x) Yν 1 (x) νxx1Yν (x) Yν 1 (x)[Yν 1 (x) Yν 1 (x)]2Yν (x) Yν 1 (x) Jν (x)Yν (x) Jν (x) Jν 1 (x)d[xν Jν (x)] xν Jν 1 (x)x ν Jν (x)2νdxd x ν Jν 1 (x)x x ν Yν 1 (x)First KindJn(x) J0 (x) J1 (x) π1π1π1π πcos(nθ x sin θ) dθ 0 πcos(x sin θ) dθ 0 1 ππcos θ sin(x cos θ) dθ09πcos(x sin θ nθ) dθ0πcos(x cos θ) dθcos(θ x sin θ) dθ π0π0 11π πcos(x sin θ θ) dθ0Yν (x)Yν (x) Yν 1 (x)Integral Forms of Bessel Functions for Integer Orders n 0, 1, 2, 3, . . .1x[xν Yν (x)] xν Yν 1 (x)x ν Yν (x)dxνν
from Bowman, pg. 57J0 (x) 2 cos(x sin θ) dθπ2π/20 π/2cos(x cos θ) dθπ0Second Kind for Integer Orders n 0, 1, 2, 3, . . . cos(xt) dt2(x/2) n x 0Yn(x) 2n 1/21 1 (t 1) nπΓ2 1 π1 π ntYn(x) sin(x sin θ nθ) dθ e e nt cos(nπ) exp( x sinh t) dtπ 0π 0x 0Y0 (x) 4cos(x cos θ) γ ln(2x sin2 θ) dθπ2Y0 (x) π/2x 002π cos(x cosh t) dtx 00ApproximationsPolynomial Approximation of Bessel FunctionsFor x 2 one can use the following approximation based upon asymptotic expansions: Jn(x) 2πx 1/2[Pn(x) cos u Qn(x) sin u]where u x (2n 1)π4and the polynomials Pn(x) and Qn(x) are given by10
Pn(x) 1 (4n2 12 )(4n2 32 ) 2 · 1(8x)21 (4n2 52 )(4n2 72 ) 4 · 3(8x)2 (4n2 92 )(4n2 112 )(1 · · · )1 6 · 5(8x)2andQn(x) 4n2 121!(8x) 1 (4n2 32 )(4n2 52 )(1 · · · )3 · 2(8x)2 1 (4n2 72 )(4n2 92 ) 5 · 4(8x)2 The general form of these terms can be written asPn(x) Qn(x) (4n2 (4k 3)2 ) (4n2 (4k 1)2 )2k(2k 1)(8x)2(4n2 (4k 1)2 ) (4n2 (4k 1)2 )2k(2k 1)(8x)2k 1, 2, 3 . . .k 1, 2, 3 . . .For n 01sin u (sin x cos x)21cos u (sin x cos x)21[P0 (x)(sin x cos x) Q0 (x)(sin x cos x)]J0 (x) πx11
or 1/2 ππ Q0 (x) sin x P0 (x) cos x J0 (x) πx44 12 · 3252 · 7292 · 112P0 (x) 1 (1 · · · )1 1 2!(8x)24 · 3(8x)26 · 5(8x)2 32 · 521272 · 921 (1 · · · )1 Q0 (x) 8x3 · 2(8x)25 · 4(8x)2 2For n 11sin u (sin x cos x)21cos u (sin x cos x)21J1 (x) [P1 (x)(sin x cos x) Q1 (x)(sin x cos x)]πxor 1/2 3π3π Q1 (x) sin x P1 (x) cos x J1 (x) πx44 3·521 · 4577 · 117P1 (x) 1 (1 · · · )1 1 2 · 1(8x)24 · 3(8x)26 · 5(8x)2 35345 · 771 (1 · · · )1 Q1 (x) 8x2 · 1(8x)25 · 4(8x)2 212
Asymptotic Approximation of Bessel FunctionsLarge Values of x Y0 (x) 1/22[P0 (x) sin(x π/4) Q0 (x) cos(x π/4)]πx Y1 (x) 1/22πx[P1 (x) sin(x 3π/4) Q1 (x) cos(x 3π/4)]where the polynomials have been defined earlier.Power Series Expansion of Bessel FunctionsFirst Kind, Positive OrderJν (x) ( 1)k (x/2)ν 2kk 0 k!Γ(ν k 1)1 ν xΓ(1 ν)21 (x/2)21(1 ν)The General Term can be written asZk Y Yk(k ν)k 1, 2, 3, . . . (x/2)2whereB0 1B 1 Z1 · B0B2 Z2 · B1.Bk Zk · Bk 113 1 (x/2)22(2 ν) 1 (x/2)23(3 ν) (1 · · ·
The approximation can be written asU(x/2)ν Jν (x) Γ(1 νBkk 0First Kind, Negative OrderJ ν (x) ( 1)k (x/2)2k νk 0 k!Γ(k 1 ν) ν x1Γ(1 ν)21 (x/2)21(1 ν) 1 (x/2)22(2 ν)The General Term can be written asZk Y Yk 1, 2, 3, . . .k(k ν) (x/2)2whereB0 1B 1 Z1 · B0B2 Z2 · B1.Bk Zk · Bk 1The approximation can be written asJ ν (x) U(x/2) ν Γ(1 ν)Bkk 0where U is some arbitrary value for the upper limit of the summation.14 1 (x/2)23(3 ν) (1 · · ·
Second Kind, Positive OrderYν (x) Jν (x) cos νπ J ν (x)sin νπν 0, 1, 2, . . .,Roots of Bessel FunctionsFirst Kind, Order Zero, J0 (x) 0This equation has an infinite set of positive rootsx1 x2 x3 . . . xn xn 1 . . .Note: xn 1 xn π as n The roots of J0 (x) can be computed approximately by Stokes’s approximation which wasdeveloped for large n 262151161255447483686542921 2 xn .4α3α415α6105α8315α10αwith α π(4n 1).An approximation for small n is 26275581 2 xn 4α3α415α6αFirst Kind, Order One, J1 (x) 0This equation has an infinite set of positive, non-zero rootsx1 x2 x3 . . . xn xn 1 . . .15
Note: xn 1 xn π as n These roots can also be computed using Stoke’s approximation which was developed for largen. 66471639024188951673241 2 4 .xn 4ββ5β 635β 835β 10βwith β π(4n 1).An approximation for small n is 6647161 2 4 xn 4ββ10β 6βThe roots of the transcendental equationxnJ1 (xn) BJ0 (xn) 0with 0 B are infinite in number and they can be computed accurately andefficiently using the Newton-Raphson method. Thus the (i 1)th iteration is given by xin xi 1nxinJ1 (xin) BJ0 (xin)xinJ0 (xin) BJ1 (xin)Accurate polynomial approximations of the Bessel functions J0 (·) and J1 (·) may be employed. To accelerate the convergence of the Newton-Raphson method, the first value forthe (n 1)th root can be related to the converged value of the nth root plus π.Aside:Fisher-Yovanovich modified the Stoke’s approximation for roots of J0 (x) 0 and J1 (x) 0. It is based on taking the arithmetic average of the first three and four term expressionsFor Bi roots are solutions of J0 (x) 0δn, α4 1 2α2 623α4 15116 30α616
with α π(4n 1).For Bi 0 roots are solutions of J1 (x) 0δn,0 β4 1 6β2 6β4 4716 10β 6with β π(4n 1).Potential Applications1. problems involving electric fields, vibrations, heat conduction, optical diffraction plusothers involving cylindrical or spherical symmetry2. transient heat conduction in a thin wall3. steady heat flow in a circular cylinder of finite length17
Modified Bessel FunctionsBessel’s equation and its solution is valid for complex arguments of x. Through a simplechange of variable in Bessel’s equation, from x to ix (where i 1), we obtain themodified Bessel’s equation as follows:x2d2 ydx2 xdydx ((ix)2 ν 2 )y 0or equivalentlyx2d2 ydx2 xdydx (x2 ν 2 )y 0The last equation is the so-called modified Bessel equation of order ν. Its solution isy AJν (ix) BYν (ix)x 0ory CIν (x) DKν (x)x 0and Iν (x) and Kν (x) are the modified Bessel functions of the first and second kind of orderν.Unlike the ordinary Bessel functions, which are oscillating, Iν (x) and Kν (x) are exponentially growing and decaying functions as shown in Figs. 4.3 and 4.4.It should be noted that the modified Bessel function of the First Kind of order 0 has a valueof 1 at x 0 while for all other orders of ν 0 the value of the modified Bessel functionis 0 at x 0. The modified Bessel function of the Second Kind diverges for all orders atx 0.18
2.5I x 21.5I01I10.5I200.511.5x22.53Figure 4.3: Plot of the Modified Bessel Functions of the First Kind, Integer Order2.5K x 21.5K2K11K00.500.511.5x22.53Figure 4.4: Plot of the Modified Bessel Functions of the Second Kind, Integer Order19
Relations Satisfied by the Modified Bessel FunctionRecurrence FormulasBessel functions of higher order can be expressed by Bessel functions of lower orders for allreal values of ν.Iν 1 (x) Iν 1 (x) 1Iν (x) 2νddxddxxxIν (x)Kν 1 (x) Kν 1 (x) νxxKν (x) Kν 1 (x) Iν (x)Kν (x) Iν (x) Iν 1 (x)d[xν Iν (x)] xν Iν 1 (x)x ν Iν (x)2νdxd x ν Iν 1 (x)Kν (x)1Kν (x) [Kν 1 (x) Kν 1 (x)]2[Iν 1 (x) Iν 1 (x)]Iν (x) Iν 1 (x) Iν (x) 2νdxνxνxKν (x)Kν (x) Kν 1 (x)[xν Kν (x)] xν Kν 1 (x)x ν Kν (x) x ν Kν 1 (x)Integral Forms of Modified Bessel Functions for Integer Orders n 0, 1, 2, 3, . . .First KindIn(x) I0 (x) I1 (x) 1π1π1π πcos(nθ) exp(x cos θ) dθ0 πexp(x cos θ) dθ0 πcos(θ) exp(x cos θ) dθ020
Alternate Integral Representation of I0 (x) and I1 (x)I0 (x) I1 (x) 1 πcosh(x cos θ) dθπ0dI0 (x)dx 1π πsinh(x cos θ) cos θ dθ0Second KindNote: These are also valid for non-integer values of Kv (x). π(x/2)n sinh2n t exp( x cosh t) dtKn(x) 1 0Γ n 2 Kn(x) cosh(nt) exp( x cosh t) dtx 00 K0 (x) exp( x cosh t) dtx 00 K1 (x) cosh t exp( x cosh t) dt021x 0x 0
ApproximationsAsymptotic Approximation of Modified Bessel Functionsfor Large Values of x ex1 In(x) 2πx ex1 I0 (x) 2πx ex1 I1 (x) 2πx4n2 12 (4n2 32 ) (4n2 52 )1 1 1(8x)2(8x)3(8x) 19251 1 (1 . . . )8x2(8x)3(8x) 52131 1 (1 . . . )8x2(8x)3(8x) (1 . . . )The general term can be written as 4n2 (2k 1)2k(8x)Power Series Expansion of Modified Bessel FunctionsFirst Kind, Positive OrderIν (x) k 0 (x/2)ν 2kk!Γ(ν k 1)1Γ(1 ν) ν x1 2(x/2)21(1 ν)The General Term can be written asZk YYk(k ν)k 1, 2, 3, . . . (x/2)222 1 (x/2)22(2 ν) 1 (x/2)23(3 ν) (1 · · ·
whereB0 1B 1 Z1 · B0B2 Z2 · B1.Bk Zk · Bk 1The approximation can be written asIν (x) (x/2)νΓ(1 ν)U Bkk 0where U is some arbitrary value for the upper limit of the summation.First Kind, Negative OrderI ν (x) (x/2)2k νk 0k!Γ(k 1 ν)1Γ(1 ν) ν x21 (x/2)21(1 ν)The General Term can be written asZk YYk(k ν)k 1, 2, 3, . . . (x/2)223 1 (x/2)22(2 ν) 1 (x/2)23(3 ν) (1 · · ·
whereB0 1B 1 Z1 · B0B2 Z2 · B1.Bk Zk · Bk 1The approximation can be written asI ν (x) U(x/2) ν Γ(1 ν)Bkk 0where U is some arbitrary value for the upper limit of the summation.Second Kind, Positive OrderKν (x) π[I ν (x) Iν (x)]2 sin νπAlternate Forms of Power Series Expansion for Modified Bessel FunctionsFirst Kindzn 1 z2 z2z2 z2 (1 . . . )1 1 2(n 2)3(n 3)4(n 4) z2z2z2z221 1 1 (1 . . . )I0 (x) 1 z 1 2·23·34·45·5 z2z2z2z21 1 1 (1 . . . )I1 (x) z 1 1·22·33·44·5In(x) n!1(n 1)1 24
Second KindWhen n is a positive integer Kn(x) 1/2K0 (x) ππ2x (4n2 32 )(4n2 52 )(4n2 12 )1 1 (1 . . . )1 1(8x)2(8x)3(8x) x 92511 1 1 8x2(8x)3(8x) x 52131 1 1 8x2(8x)3(8x) 1/2e2x xe2x K1 (x) π 1/2eSeries expansions based upon the trapezoidal rule applied to certain forms of the integralrepresentation of the Bessel functions can be developed for any desired accuracy. Severalexpansions are given below.For x 12, 8 decimal place accuracy is obtained by7 15J0 (x) cos x 2cos(x cos jπ/15)j 17 sin(x cos jπ/15) cos(jπ/15)15J1 (x) sin x 2j 1For x 20, 8 decimal place accuracy is obtained by7 cosh(x cos jπ/15)15I0 (x) cosh x 2j 17 sinh(x cos jπ/15) cos(jπ/15)15I1 (x) sinh x 2j 125
For x 0.1, 8 decimal place accuracy is obtained by11 exp[ x cosh(j/2)]4K0 (x) e x e x cosh 6 2j 1 x4K1 (x) e x cosh 6 cosh 6e11 2exp[ x cosh(j/2) cosh(j/2)]j 1Potential Applications1. displacement of a vibrating membrane2. heat conduction in an annular fin of rectangular cross section attached to a circularbase26
Kelvin’s FunctionsConsider the differential equationx2d2 ydx2 xdydx (ik2 x2 n2 )y 0i 1This is of the form of Bessel’s modified equationx2d2 ydx2 xdydx (β 2 x2 n2 )y 0i 1with β 2 ik2 . Since the general solution of Bessel’s modified equation isy AIn(βx) BKn(βx)the general solution of the given equation can be expressed as y AIn( i kx) BKn( i kx)Also, sinceIn(x) i nJn(ix) inIn(x) Jn(ix)we may take the independent solutions of the given equation asy AJn(i3/2 kx) BKn(i1/2 kx)when x is real, Jn(i3/2 x) and Kn(i1/2 x) are not necessarily real. We obtain real functions27
by the following definitions:bern Re Jn(i3/2 x)bein Im Jn(i3/2 x)Jn(i3/2 x) bern x i bei xkern Re i nKn(i1/2 x)kein Im i nKn(i1/2 x)i nKn(i1/2 x) kern x i kei xIt is, however, customary to omit the subscript from the latter definitions when the order nis zero and to write simplyJ0 (i3/2 x) ber x i bei xK0 (i1/2 x) ker x i kei xThe complex function ber x i bei x is often expressed in terms of its modulus and itsamplitude:ber x i bei x M0 (x)eiθ0 (x)whereM0 (x) [(ber x)2 (bei x)2 ]1/2 ,θ0 arc tanSimilarly we can writebern x i bern x Mn(x)eiθn(x)where28bei xber x
Mn(x) [(bern x)2 (bein x)2 ]1/2 ,θn arc tanbein xbern xKelvin’s Functions ber x and bei xThe equation for I0 (t) ist2d2 ydt2 tdydt t2 y 0 Set t x i and the equation becomesx2d2 ydx2 xdydx i x2 y 0 with the solutions I0 (x i) and K0 (x i). The ber and bei functions are defined as follows.SinceI0 (t) 1 2t2(t/2)2 (2!)2 (t/2)6(3!)2 ···we have real and imaginary parts in (x/2)4(x/2)8I0 (x i) 1 ···(2!)2(4!)2 (x/2)6(x/2)102 ··· i (x/2) (3!)2(5!)2 ber x i bei xEquating real and imaginary parts we haveber x 1 (x/2)4(2!)2bei x (x/2) 2 (x/2)8(4!)2(x/2)6(3!)2 ···(x/2)10(5!)229 ···
Both ber x and bei x are real for real x, and it can be seen that both series are absolutelyconvergent for all values of x. Among the more obvious properties areber 0 1bei 0 0and x x ber x dx x bei x,0xx bei x dx x ber x0In a similar manner the functions ker x and kei x are defined to be respectively the realand imaginary parts of the complex function K0 (x i), namely ker x i kei x K0 (x i)From the definition of K0 (x) we can see thatker x [ln(x/2) δ] ber x π4bei x ( 1)r (x/2)4rr 1[(2r)!]2φ(2r)andkei x [ln(x/2) δ] bei x π4ber x r 1whereφ(r) r 1s 1 ( 1)r (x/2)4r 2s30[(2r 1)!]2φ(2r 1)
Potential Applications1. calculation of the current distribution within a cylindrical conductor2. electrodynamics of a conducting cylinder31
Hankel FunctionsWe can define two new linearly dependent functionsHn(1) (x) Jn(x) iYn(x)x 0Hn(2) (x) Jn(x) iYn(x)x 0which are obviously solutions of Bessel’s equation and therefore the general solution can bewritten asy AHn(1) (x) BHn(2) (x)where A and B are arbitrary constants. The functions Hn(1) (x) and Hn(2) (x) are calledHankel’s Bessel functions of the third kind. Both are, of course, infinite at x 0, theirusefulness is connected with their behavior for large values of x.Since Hankel functions are linear combinations of Jn and Yn, they satisfy the same recurrencerelationships.32
Orthogonality of Bessel FunctionsLet u Jn(λx) and v Jn(µx) with λ µ be two solutions of the Bessel equationsx2 u xu (λ2 x2 n2 )u 0andx2 v xv (µ2 x2 n2 )v 0where the primes denote differentiation with respect to x.Multiplying the first equation by v and the second by u, and subtracting, we obtainx2 (vu uv ) x(vu uv ) (µ2 λ2 )x2 uvDivision by x givesx(vu uv ) (vu uv ) (µ2 λ2 )xuvorddx[x(vu uv )] (µ2 λ2 )xuvThen by integration and omitting the constant of integration we have (µ λ )22xuv dx x(vu uv )or making the substitutions for u, u , v and v we have (µ λ )22xJn(λx) Jn(µx) dx x[Jn(µx) Jn (λx) Jn(λx) Jn (µx)]33
This integral is the so-called Lommel integral. The right hand side vanishes at the lowerlimit zero. It also vanishes at some arbitrary upper limit, say x b, providedJn(µb) 0 Jn(λb)orJn (λb) 0 Jn (µb)In the first case this means that µb and λb are two roots of Jn(x) 0, and in the secondcase it means that µb and λb are two roots of Jn (x) 0. In either case we have thefollowing orthogonality property bxJn(λx)Jn(µx) dx 0aThis property is useful in Bessel-Fourier expansions of some arbitrary function f (x) overthe finite interval 0 x b. Further the functions Jn(λx) and Jn(µx) are said to beorthogonal in the interval 0 x b with respect to the weight function x.As µ λ, it can be shown by the use of L’Hopital’s rule that 0xxJn2 (λx)dx x22 2Jn (λx) 1 whereJn (λx) dJn(r)drwith r λx34n2(λx)2 2[Jn(λx)]
Assigned ProblemsProblem Set for Bessel Equations and Functions1. By means of power series, asymptotic expansions, polynomial approximations or Tables, compute to 6 decimal places the following Bessel functions:a)J0 (x)e)Y0 (x)b) J1 (y)f)Y1 (y)I0 (z)g)K0 (z)c)d) I1 (z)h) K1 (z)givenx 3.83171y 2.40482z 1.757552. Compute to 6 decimal places the first six roots xn of the transcendental equationx J1 (x) B J0 (x) 0B 0when B 0.1, 1.0, 10, and 100.3. Compute to 4 decimal places the coefficients An(n 1, 2, 3, 4, 5, 6) givenAn 2B(x2n B 2 ) J0 (xn)for B 0.1, 1.0, 10, and 100. The xn are the roots found in Problem 2.35
4. Compute to 4 decimal places the coefficients Bn(n 1, 2, 3, 4) givenBn 2AnJ1 (xn)xnfor B 0.1, 1.0, 10, and 100. The xn are the roots found in Problem 2 and theAn are the coefficients found in Problem 3.5. The fin efficiency of a longitudinal fin of convex parabolic profile is given as 1 I2/3 (4 γ/3)η γ I 2/3 (4 γ/3)Compute η for γ 3.178.6. The fin efficiency of a longitudinal fin of triangular profile is given by 1 I1 (2 γ)η γ I0 (2 γ)Compute η for γ 0.5, 1.0, 2.0, 3.0, and 4.0.7. The fin efficiency of a radial fin of rectangular profile is given byη 2ρ x(1 ρ2 )I1 (x)K1 (ρx) K1 (x)I1 (ρx)I0 (ρx)K1 (x) I1 (x)K0 (ρx)Compute the efficiency for x 2.24, ρx 0.894.Ans: η 0.5378. Show thati)ii)J0 (x) J1 (x)ddx[xJ1 (x)] xJ0 (x)36
9. Given the functionf (x) xJ1 (x) B J0 (x)with B constant 0determine f , f , and f . Reduce all expressions to functions of J0 (x), J1 (x) andB only.10. Show that 22i)x J0 (x) dx x J1 (x) xJ0 (x) J0 (x) dx x3 J0 (x) dx x(x2 4) J1 (x) 2x2 J0 (x)ii)11. Show that 1xJ0 (λx) dx i)0 1λ13ii)x J0 (λx) dx 0J1 (λ)λ2 4λ3J1 (λ) 2λ2J0 (λ)12. If δ is any root of the equation J0 (x) 0, show that 1i)J1 (δx) dx 0 1δδJ1 (x) dx 1ii)013. If δ( 0) is a root of the equation J1 (x) 0 show that 1xJ0 (δx) dx 0037
14. Given the Fourier-Bessel expansion of f (x) of zero order over the interval 0 x 1f (x) A1 J0 (δ1 x) A2 J0 (δ2 x) A3 J0 (δ3 x) . . .where δn are the roots of the equation J0 (x) 0. Determine the coefficients Anwhen f (x) 1 x2 .15. Show that over the interval 0 x 1x 2 J1 (δn)n 1δn J2 (δn)where δn are the positive roots of J1 (x) 0.16. Obtain the solution to the following second order ordinary differential equations:i)y xy 0ii)y 4x2 y 0iii)y e2x y 0iv)xy y k2 y 0v)x2 y x2 y vi) y 1xHint: let u ex14y 0 k 0y 1 4 x2y 0vii) xy 2y xy 017. Obtain the solution for the following problem:xy y m2 by 00 x b, m 0withy(0) andy(b) y038
18. Obtain the solution for the following problem:x2 y 2xy m2 xy 00 x b, m 0withy(0) andy(b) y019. Show that I3/2 (x) 1/2 cosh x πx I 3/2 (x) 22πx 1/2 sinh x 39sinh x xcosh xx
Selected References1. Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, Dover,New York, 1965.2. Bowman, F., Introduction to Bessel Functions, Dover New York, 1958.3. Gray A, Mathews, G. B. and MacRobert, T. M., A Treatise on Bessel Functions,Basingstoke MacMillan, 1953.4. Luke, Y. L., Integrals of Bessel Functions, McGraw-Hill, New York, 1962.5. Magnus W, Oberhettinger, F. and Soni, R. P. Formulas and Theorems for theSpecial Functions of Mathematical Physics, Springer, Berlin, 1966.6. McLachlan, N.W., Bessel Functions for Engineers, 2nd Edition, Oxford UniversityPress, London, 1955.7. Rainville, E.D., Special Functions, MacMillan, New York, 1960.8. Relton, F.E., Applied Bessel Functions, Blackie, London, 1946.9. Sneddon, I.N., Special Functions of Mathematical Physics and Chemistry, 2nd Edition, Oliver and Boyd, Edinburgh, 1961.10. Watson, G.N., A Treatise on the Theory of Bessel Functions, 2nd Edition, CambridgeUniversity Press, London, 1931.11. Wheelon, A. D., Tables of Summable Series and Integrals Involving Bessel Functions,Holden-Day, San Francisco, 1968.40
The Bessel function of the second kind, Y ν(x) is sometimes referred to as a Weber function or a Neumann function (which can be denoted as N ν(x)). It is related to the Bessel function of the first kind as follows: Y ν(x) J ν(x)cos(νπ) J ν(x) sin(νπ) where we take the limit ν n for integer values of ν. Forinteger order ν,J
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Spherical Bessel Functions of the first and second kinds (5.1.1) jm the Spherical Bessel Function of the first kind nm the Spherical Bessel Function of the second kind The solution to the problem that we are currently considering, the interior of a spherical shell, must be analytic (f
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