1 Bessel, Neumann, and Hankel Functions: J ν (x), N ν (x), H (1)

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1 Bessel, Neumann, and Hankel Functions: J ν x), N ν x), H ) ν x), H ) ν x) Bessel functions are solutions of the following differential equation: x y +xy +x ν )y =.) Any two of the following functions are linearly independent solutions of.) J ν x) N ν x) H ) ν x) H ) ν x) when ν is not an integer, J ν x) and J ν x) are also linearly independent principal solutions of.). The Neumann function N ν x) is related to J ν and J ν : N ν x) = cosνπj νx) J ν x) sinνπ cosνπj ν x) J ν x) N n x) = lim ν n sinνπ in some ooks Neumann functions are denoted y Y ν x) instead of N ν x). Hankel functions of the first and second kind are related to Bessel and Neumann functions:.).3) H ν ) z) J ν z)+jn ν z) = j e jνπ J ν z) J ν z) sinνπ H ν ) z) J ν z) jn ν z) = ejνπ J ν z) J ν z) jsinνπ.4).5) With a variale transformation x = κρ equation.) can e transformed into: ρ y +ρy +κ ρ ν )y =.6) When ν = n is an integer J n and J n are not independent anymore and we have: J n x) = ) n J n x) N n x) = ) n N n x).7) Plots of the first three Bessel and Neumann functions are shown in Fig.. and Fig.., respectively..8 J x).5 N x) N x) N x).6 J x) J x) Figure.: Bessel functions of the first kind Figure.: Bessel functions of the second kind In general for aritrary ν we have J ν e ±jπ x) = e ±jνπ J ν x) N ν e ±jπ x) = e jνπ N ν x)±jcosνπj ν x).8) Sharif University of Technology

2 . Asymptotic Approximations.. Small Argument Limit x J x) x 4.9) x ν J ν x) ) Γν +) x ) n.) n! N ν x) ) ν π Γν) n )! ) n n.) x π x N x) γx ln γ = Euler s constant.) π.. Large Argument Limit x ν, π < argx) < π J ν x) x πx cos π 4 νπ ) N ν x) x πx sin π 4 νπ ).3).4)..3 Wronskian relations The wronskian etween two functions is defined y which is independent of ν W{f,g} fx)g x) f x)gx).5) W{J ν,n ν } = J ν+ N ν J ν N ν+ = πx.6). Integral Representations When n is an integer: ) J n x) = e jnα+π π ) J n x) = e jnα+π π J n x) = In general for aritrary ν J ν x) = π J ν x) = π N ν x) = π J n x) = π J x) = π J x) = π nπ e j π π e jxcosφ α) e jnφ dφ = π e jxcosφ α) e jnφ dφ = π π e jxsinφ e jnφ dφ.7) e jxsinφ e jnφ dφ.8) cosnφ)e jxcosφ dφ.9) cosxsinφ) nφ) dφ.) cosxsinα)) dα = π cosxsinα)) dα = π cosx cosα)) dα.) cosx cosα)) dα.) cosxsinφ) νφ) dφ sinνπ e xsinht νt dt R{x} >.3) π sin xcosht νπ ) cosh νt dt.4) sinxsinφ) νφ) dφ π e νt +e νt cosνπ ) e xsinht dt R{x} >.5) Sharif University of Technology

3 .3 Orthogonality Relationships and Fourier-Bessel Series Bessel equation.6) can e written in the following form: ρy ) + ν ρ y κ ρy = = L[y] = κ y L = ρ ρ ρ )+ ν ρ ρ.6) This is a Sturm-Liouville equation with pρ) = ρ, qρ) = n ρ, wρ) = ρ, and λ = κ. With appropriate oundary conditions on a finite interval such as ρ [a,] we can have a Sturm-Liouville eigenvalue prolem regular if a > and irregular if a = ). Here n can e any real and non-negative numer. Case I: Consider.6) on the interval ρ with oundary condition y) =. At ρ we require the solution to e ounded. Thus, we can not have N n κρ) and eigenfunctions must e in the form of J n κρ). Eigenvalues are: J n κ) = = κ m = ν = λ m = κ m ) ν ) =.7) in which ν is the m th root of the Bessel function J n x) =, i.e. J n ν ) =. The following orthogonality property exists:, m k ν ) J n ρ νnk ) J n ρ ρdρ =.8) [J n+ν )], m = k For any piecewise continuous function fρ) we have: fρ) m= F m J n ν ρ ) in which the coefficients F m are otained y using the orthogonality property.8) F m = [J n+ ν )].9) fρ)j n ν ρ ) ρdρ.3) Expression.9) is called the Fourier-Bessel Series expansion of fρ). Note that the series always converges to zero at ρ =. Case II: Consider.6) on the interval ρ with oundary condition y ) =. At ρ we require the solution to e ounded. Again eigenfunctions must e in the form of J n κρ) and since y ) = we otain the eigenvalues: J nκ) = dj nκρ) = = κ m = ν = λ m = κ m ) = dρ ρ= ) ν.3) in which ν is the m th root of the derivative of Bessel function J nx) =, i.e. J nν ) =. The following orthogonality property holds:, m k ) ν J ν ) n ρ J nk n ρ ρdρ = ).3) n [J n ν )], m = k m= ν For any piecewise continuous function fρ) we can write: ) ν fρ) F m J n ρ.33) in which the coefficients F m are otained y using the orthogonality property.3) ) ν F m = ) fρ)j n n [J ν n ν )] ρ ρdρ.34) Sharif University of Technology 3

4 Again this is called Fourier-Bessel expansion of fρ). Note that the derivative of the series always converges to zero at ρ =. If the interval is [a,] and a >, then the SLP is regular and the general form of eigenfunctions would e A m J n κ m ρ) + B m N n κ m ρ). The oundary conditions at ρ = a and ρ = will determine the eigenvalues κ m and we have similar orthogonality property etween the eigenfunctions as well..4 Recursion Relationships Consider Z ν x) to e J ν x) or N ν x) or H ) ν x) or H ) ν x) or any linear comination of these functions. Then, the following recursive formulas are applicale ν can e any numer): Z ν x)+z ν+ x) = ν x Z νx).35) Z ν x) Z ν+ x) = Z νx).36) Z νx)+ ν x Z νx) = Z ν x).37) Z νx) ν x Z νx) = Z ν+ x).38) [x ν Z ν x)] = x ν Z ν x).39) [ x ν Z ν x) ] = x ν Z ν+ x).4) inparticularz x) = Z x). Equation.39)and.4)areveryusefulwhenintegratingoverBesselfunctions..5 Series and Integral Relationships e jkρcosφ = e jkρsinφ = j) n J n kρ)e jnφ e jkρcosφ = ) n J n kρ)e jnφ e jkρsinφ = j n J n kρ)e jnφ.4) J n kρ)e jnφ.4) In the following expressions Z n x) and B n x) can e any of J n x), N n x), H ) n x), H ) n x) or linear cominations of them. m, n, α, β are aritrary real numers. Z n αx)b n βx)xdx = x βz nαx)b n βx) αz n αx)b n βx) α β.43) = x αz n+αx)b n βx) βz n αx)b n+ βx) α β.44) Znαx)xdx = x [ Z n αx) Z n αx)z n+ αx) ].45) x n+ Z n x)dx = x n+ Z n+ x).46) x n+ Z n x)dx = x n+ Z n+ x).47) x Z nαx)b m αx)dx = αx Z nαx)b m+ αx) Z n+ αx)b m αx) n m + Z nαx)b m αx) n+m.48) Z x)dx = Z x).49) xz x)dx = xz x).5) Modified I ν x) and K ν x) Modified Bessel functions are solutions of the following differential equation: x y +xy x +ν )y =.) Sharif University of Technology 4

5 which is called the modified Bessel s differential equation. The general solution of.) can e written as a linear comination of the modified Bessel functions of the first and second kind: AI ν x)+bk ν x) When ν is not an integer ν n) I ν and I ν are linearly independent principal) solutions of.), however, we usually use I ν x) and K ν x) also called Kelvin function) which is related to I ν and I ν : π K ν x) = sinνπ [I νx) I ν x)].) With a variale transformation x = κρ equation.) can e transformed into: ρ y +ρy κ ρ +ν )y =.3) whose independent solutions are I ν κρ) and K ν κρ). When ν is an integer I n and I n are not independent anymore and we have I n x) = I n x). Furthermore, for aritrary ν we always have: I ν e ±jπ x) = e ±jνπ I ν x) K ν e ±jπ x) = jπi ν x)+e jνπ K ν x).4) Plots of the first three modified Bessel functions of the first and second kind are shown in Fig.. and Fig.., respectively I x) I x) 3 K x) K x) I x) K x) Figure.: Modified Bessel functions of the first kind Figure.: Modified Bessel functions of the second kind. Small and Large Argument Approximations.. Small Argument Limit x.. Large Argument Limit x I + x 4.5) x ν I ν x) ) Γν +) x ) n.6) n! K ν x) Γν) ) ν n )! ) n ν.7) x x K x) ln γx γ = Euler s constant.8) I ν x) e x.9) πx π K ν x) x e x.) Sharif University of Technology 5