Dynamics at the Horsetooth, Volume 2A, Focussed Issue: Asymptotics and Perturbations Asymptotic Expansion of Bessel Functions; Applications to Electromagnetics Nada Sekeljic Department of Electrical Engineering Colorado State University [email protected] Report submitted to Prof. I. Oprea for Math 676, Fall 2010 Abstract. Bessel function is de¯ned as particular solution of a linear di®erential equation of the second order known as Bessel's equation. This equation is often used as model of real physical problems. For instance, separation of the wave equation (wave equation in complex domain is called Helmholtz equation) in cylindrical coordinate system leads to Bessel's equation. Besides regular series expansion of the Bessel functions, this paper extends to asymptotic analysis based on contour integral representation of Hankel function. Keywords: Bessel functions, Asymptotic expansion, Electromagnetics 1 Introduction Although, there are di®erent approaches, Bessel functions of the ¯rst kind are introduced in Section 2 by means of a generating function. In Section 3, Bessel's equation is derived based on ¯eld analysis inside a circular waveguide. To de¯ne the general system of solutions of the Bessel's equation, we present Bessel functions of the second order known as Neumann functions in Section 4. In section 5, we specify Hankel functions. Section 6 summarize all relations between Bessel functions. Finally, Section 7 explains asymptotic forms of the functions using contour integral de¯nition of Hankel function. 2 Bessel Function of the First Kind, Jº(z) One very convenient and instructive way to introduce Bessel functions is due to generating function. This approach provides useful properties of the functions because of its advantage of focusing on the functions themselves rather than on the di®erential equation they satisfy. Generating function is given in form g(z; t) = e(z=2)(t¡1=t) (1) This function is expanded in a Laurent series as function of t and complex variable z: X+1 (z=2)(t¡1=t) n e = Jn(z) ¢ t (2) n=¡1 Asymptotic Expansion of Bessel Functions; Applications to Electromagnetics Nada Sekeljic where Jn(z) are Bessel functions of the ¯rst kind, of order n (n is an integer). Expanding the exponentials, we have a product of two absolutely convergent series in zt=2 and ¡z=(2t), respectively: 1 1 X ³z ´r tr X ³z ´m t¡m ezt=2 ¢ e¡z=(2t) = ¢ (¡1)m 2 r! 2 m! r=0 m=0 1 1 X X ³z ´(r+m) t(r¡m) = (¡1)m (n = r ¡ m, r = n + m) 2 r!m! r=0 m=0 1 1 X X ³z ´(n+2m) tn = (¡1)m 2 m!(n + m)! n=¡1 m=0 1 X (¡1)m ³z ´(n+2m) J (z) = (3) n m!(n + m)! 2 m=0 This is regular expansion for Bessel functions of the ¯rst kind, of an integral order n, and it is valid for small argument z (jzj). For n < 0, Eq.(3) gives: 1 X (¡1)m ³z ´(2m¡n) J (z) = (4) ¡n m!(m ¡ n)! 2 m=0 Because n is an integer, (m ¡ n)! goes to in¯nity for m = 0; 1; 2; 3; :::n ¡ 1 and these terms in the expansion go to zero, so the series may start with m = n. Replacing m by m + n, we get: 1 X (¡1)(n+m) ³z ´(n+2m) J (z) = (5) ¡n m!(n + m)! 2 m=0 From Eqs. (3) and (5), we conclude that Jn(z) and J¡n(z) are not independent but are related by: n J¡n(z) = (¡1) Jn(z) (6) This property, Eq. (6), is only valid for an integral order n. These series expansions, Eqs. (3) and (5) are also valid for n replaced by º to de¯ne Jº(z) and J¡º(z) for nonintegral order º. Bessel functions Jn(z), functions of two variables { unrestricted z and restricted n (integer), are also called Bessel coe±cients. General representation of Bessel functions of the ¯rst kind, of nonintegral order º is de¯ned by equation: 1 X (¡1)m ³z ´(º+2m) J (z) = (7) º m!¡(º + m + 1) 2 m=0 where ¡ is called Gamma function. The Gamma function is an extension of the factorial function with its argument shifted down by one, to real or complex number. If the argument of the function is positive integer: ¡(n) = (n ¡ 1)! (8) Also, function exists for all complex numbers with a positive real part except non-positive integers and it is de¯ned throughout complex integral as follows: Z 1 ¡(z) = t(z¡1)e¡t dt (9) 0 Dynamics at the Horsetooth 2 Volume 2A Asymptotic Expansion of Bessel Functions; Applications to Electromagnetics Nada Sekeljic The recurrence formulas for Jº(z) 2º J (z) + J (z) = J (z) º¡1 º+1 z º 0 Jº¡1(z) ¡ Jº+1(z) = 2Jº(z) 0 zJº(z) + ºJº(z) = zJº¡1(z) 0 zJº(z) ¡ ºJº(z) = ¡zJº+1(z) (10) These relations are also valid for integral order n. 3 Cylindrical Waveguide In this section, we analyze wave propagation along metallic waveguide of circular cross section of radius a; the waveguide is located along the z axis of circular coordinate sys- tem (longitudinal axis of the guide coincides with z axis) as indicated in Fig. (1). z a y x Figure 1: Cylindrical waveguide We assume that waveguide conductor is perfect, and dielectric inside the guide is homogeneous and without losses. We propose to study electromagnetic ¯eld inside the guide. Field expressions for TE and TM waves are derived based on ¯rst two Maxwell's equations { coupled system which gives relation between electric and magnetic ¯eld: r £ E = ¡j!¹H (11) r £ H = j!"E (12) where E is electric ¯eld intensity vector, and H is magnetic ¯eld intensity vector [note equations are given in complex domain, so both vectors are complex]; ! is angular frequency, ! = 2¼f (f is Dynamics at the Horsetooth 3 Volume 2A Asymptotic Expansion of Bessel Functions; Applications to Electromagnetics Nada Sekeljic operating frequency of the guide), " and ¹ are relative permittivity and permeability of the guide dielectric , respectively. Here, we show that for TE waves transverse components of electric and magnetic ¯eld can be expressed in terms of longitudinal component of magnetic ¯eld, and on the other hand, for TM waves, transverse components of electric and magnetic ¯eld can be expressed in terms of longitudinal component of electric ¯eld. In our case, transverse components are r and Á, and z is longitudinal component (in cylindrical coordinate system). Total electric and magnetic ¯eld can be written as: E(r; Á; z) = Er(r; Á; z)^r + EÁ(r; Á; z) uÁ^ + Ez(r; Á; z) ^z H(r; Á; z) = Hr(r; Á; z)^r + HÁ(r; Á; z) uÁ^ + Hz(r; Á; z) ^z Assume that wave propagates in positive z direction, therefore, the electric and magnetic ¯eld dependance on z coordinate is given by multiplicative factor e¡j¯z, where ¯ is phase coe±cient of the wave. Electric ¯eld components are expressed as follows: ¡j¯z Er(r; Á; z) = Er(r; Á)e ¡j¯z EÁ(r; Á; z) = EÁ(r; Á)e ¡j¯z Ez(r; Á; z) = Ez(r; Á)e (13) Magnetic ¯eld components are written in the same way. Cross product between r operator and any vector in cylindrical coordinate system A (A = Ar ^r + AÁ uÁ^ + Az ^z) is given by following formula: · ¸ · ¸ · ¸ 1 @A @A @A @A 1 @(A r) @A r £ A = z ¡ Á ^r + r ¡ z uÁ^ + Á ¡ r ^z (14) r @Á @z @z @r r @r @Á After we apply Eq. (14) to the ¯rst Maxwell's equation, Eq. (11), and equal the same vector components from left and right side, we obtain three scalar equations: µ ¶ 1 @E z + j¯rE = ¡j!¹H (15) r @Á Á r @E j¯E + z = j!¹H (16) r @r Á µ ¶ 1 @(rE ) @E Á ¡ r = ¡j!¹H (17) r @r @Á z In the similar fashion, we decompose the second Maxwell's equation, Eq. (12) on three scalar equations: µ ¶ 1 @H z + j¯rH = j!"E (18) r @Á Á r @H j¯H + z = ¡j!"E (19) r @r Á µ ¶ 1 @(rH ) @H Á ¡ r = j!"E (20) r @r @Á z Now, we have system of six scalar equations and six unknown ¯eld components. First step is to express transverse electric and magnetic ¯eld components, Er, Hr, EÁ, and HÁ, in terms of longitudinal components Ez and Hz. If combine Eqs. (16) and (18), we get: µ ¶ ¡j @E 1 @H E (r; Á) = ¯ z + !¹ z (21) r K2 @r r @Á Dynamics at the Horsetooth 4 Volume 2A Asymptotic Expansion of Bessel Functions; Applications to Electromagnetics Nada Sekeljic where K2 = !2"¹¡¯2. Similarly, after combination Eqs. (15) and (19) we obtain radial component of magnetic ¯eld expressed due to Ez and Hz: µ ¶ ¡j 1 @H @E H (r; Á) = ¯r z ¡ !" z (22) r K2 r @r @Á Substituting the Eq. (22) into Eq. (19), and Eq. (21) into Eq. (16) we have Á component of electric and magnetic ¯eld, respectively: µ ¶ ¡j ¯ @E @H E (r; Á) = z ¡ !¹ z (23) Á K2 r @Á @r µ ¶ ¡j @E ¯ @H H (r; Á) = !" z + z (24) Á K2 @r r @Á Finally, after substitute Eqs. (22) and (24) into Eq. (20), we have wave equation for longitudinal component of electric ¯eld: @2E 1 @E 1 @2E z + z + z + K2E = 0 (25) @r2 r @r r2 @Á2 z If substitute Eqs.