Bessel Functions of the First and Second Kind

1 Bessel Functions of the First and Second Bessel 18 Kelvin s 32 Orthogonality of Bessel Functions are named for Friedrich Wilhelm Bessel (1784 - 1846), however, DanielBernoulli is generally credited with being the First to introduce the concept of Bessels func-tions 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 , while receiving named credit for these Functions , did not incorporate them into hiswork as an astronomer until 1817.

2 The Bessel function was the result of Bessels study of aproblem of Kepler for determining the motion of three bodies moving under mutual gravita-tion. 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 notationJz,nwasfirst used by Hansen9(1843) and subsequently by Schlomilch10(1857)and later modified toJn(2z)byWatson (1922).Subsequent studies of Bessel Functions included the works of Mathews11in 1895, A treatiseon Bessel Functions and their applications to physics written in collaboration with AndrewGray.

3 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 , Ermittelung der absoluten Strungen in Ellipsen von beliebiger Excentricitt und Neigung,I. Schriften der Sternwarte Seeberg. Gotha, , Ueber die Bessel schen Function. Z. fr Math. u. Phys. 2, 137-165, 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, Bessel EquationThe Second order differential equation given asx2d2ydx2+xdydx+(x2 2)y=0is known as Bessel s equation.

4 Where the solution to Bessel s equation yields Bessel functionsof the First and Second kind as follows:y=AJ (x)+BY (x)whereAandBare arbitrary constants. While Bessel Functions are often presented in textbooks and tables in the form of integer order, =0,1,2,..,infact they are definedfor all real values of < < .2. Bessel Functionsa) First Kind:J (x)in the solution to Bessel s equation is referred to as a Besselfunction of the First ) Second Kind:Y (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 ) Third Kind:The Hankel function or Bessel function of the third kind can bewritten asH(1) (x)=J (x)+iY (x)x>0H(2) (x)=J (x) iY (x)x>0 Because 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 Modified Bessel EquationBy lettingx=ix(wherei= 1)inthe Bessel equation we can obtain the modifiedBessel equation of order ,given asx2d2ydx2+xdydx (x2+ 2)y=0 The solution to the modified Bessel equation yields modified Bessel Functions of the First andsecond kind as follows:y=CI (x)+DK (x)x>04.

5 Modified Bessel Functionsa) First Kind:I (x)in the solution to the modified Bessel s equation is referredto as a modified Bessel function of the First ) Second Kind:K (x)in the solution to the modified Bessel s equation is re-ferred to as a modified Bessel function of the Second kind or sometimes theWeberfunction or the Neumann Kelvin s FunctionsAmore general form of Bessel s modified equation can be written asx2d2ydx2+xdydx ( 2x2+ 2)y=0where is an arbitrary constant and the solutions is nowy=CI ( x)+DK ( x)If we let 2=iwherei= 14and we noteI (x)=i J (ix)=J (i3/2x)then the solution is written asy=CJ (i3/2x)+DK (i1/2x)The Kelvin Functions are obtained from the real and imaginary portions of this solution asfollows.

6 Ber =Re J (i3/2x)bei =Im J (i3/2x)J (i3/2x)=ber x+ibei xker =Re i K (i1/2x)kei =Im i K (i1/2x)i K (i1/2x)=ker x+ikeix5 TheoryBessel FunctionsBessel s differential equation, given asx2d2ydx2+xdydx+(x2 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 ,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 ( =n)while forspherical problems the order is of half integer value ( =n+1/2).

7 Since Bessel s differential equation is a Second -order equation, there must be two linearlyindependent solutions. Typically the general solution is given as:y=AJ (x)+BY (x)where the special functionsJ (x)andY (x)are:1. Bessel Functions of the First kind,J (x),which are finite atx=0for all real valuesof 2. Bessel Functions of the Second kind,Y (x), (also known as Weber or Neumann func-tions) which are singular atx=0 The Bessel function of the First kind of order can be be determined using an infinite powerseries expansion as follows:J (x)= k=0( 1)k(x/2) +2kk! ( +k+1)=1 (1 + ) x2 1 (x/2)21(1 + ) 1 (x/2)22(2 + ) 1 (x/2)23(3 + )(1 x J0J1J2 Figure : Plot of the Bessel Functions of the First Kind, Integer Orderor by noting that ( +k+1)=( +k)!)

8 ,wecan writeJ (x)= k=0( 1)k(x/2) +2kk!( +k)! Bessel Functions of the First kind of order0,1,2are shown in Fig. Bessel function of the Second kind,Y (x)is sometimes referred to as a Weber functionor a Neumann function (which can be denoted asN (x)). It is related to the Bessel functionof the First kind as follows:Y (x)=J (x)cos( ) J (x)sin( )where we take the limit nfor integer values of .Forinteger order , J ,J are not linearly independent:J (x)=( 1) J (x)Y (x)=( 1) Y (x)7in which caseY is needed to provide the Second linearly independent solution of Bessel sequation. In contrast, for non-integer orders,J andJ are linearly independent andY is Bessel function of the Second kind of order can be expressed in terms of the Besselfunction of the First kind as follows:Y (x)=2 J (x) lnx2+ 1 1 k=0( k 1)!

9 K! x2 2k ++1 k=0( 1)k 1 1+12+ +1k + 1+12+ +1k+ k!(k+ )! x2 2k+ Bessel Functions of the Second kind of order0,1,2are shown in Fig. x Y0Y1Y2 Figure : 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 .J +1(x)=2 xJ (x) J 1(x)Y +1(x)=2 xY (x) Y 1(x)J +1(x)=12[J 1(x) J +1(x)]Y +1(x)=12[Y 1(x) Y +1(x)]J (x)=J 1(x) xJ (x)Y (x)=Y 1(x) xY (x)J (x)= xJ (x) J +1(x)Y (x)= xY (x) Y +1(x)ddx[x J (x)] =x J 1(x)ddx[x Y (x)] =x Y 1(x)ddx x J (x) = x J +1(x)ddx x Y (x) = x Y +1(x)Integral Forms of Bessel Functions for Integer Ordersn=0,1,2,3.

10 First KindJn(x)=1 0cos(n xsin )d =1 0cos(xsin n )d J0(x)=1 0cos(xsin )d =1 0cos(xcos )d J1(x)=1 0cos( xsin )d =1 0cos(xsin )d =1 0cos sin(xcos )d 9from Bowman, pg. 57J0(x)=2 /20cos(xsin )d =2 /20cos(xcos )d Second Kind for Integer Ordersn=0,1,2,3,..Yn(x)= 2(x/2) n 12 n 1cos(xt)dt(t2 1)n+1/2x>0Yn(x)=1 0sin(xsin n )d 1 0 ent+e ntcos(n ) exp( xsinht)dtx>0Y0(x)=4 2 /20cos(xcos ) +ln(2xsin2 ) d x >0Y0(x)= 2 0cos(xcosht)dtx >0 ApproximationsPolynomial Approximation of Bessel FunctionsForx 2one can use the following approximation based upon asymptotic expansions:Jn(x)= 2 x 1/2[Pn(x)cosu Qn(x)sinu]whereu x (2n+1) 4and the polynomialsPn(x)andQn(x)are given by10Pn(x)=1 (4n2 12)(4n2 32)2 1(8x)2 1 (4n2 52)(4n2 72)4 3(8x)2 1 (4n2 92)(4n2 112)6 5(8x)2(1 ) andQn(x)=4n2 121!