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# Class Note for ECE 6341 with Professor Jackson at UH 2

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COURSE
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Class Notes
PAGES
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KARMA
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This 39 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15
Spherical Wave Funotons Consider solving Vzw kzw O in spherical coordinates Spherical Wave Functions cont In spherical coordinates we have Vzw VVw 2 12 a r2 awj 2 a sin awj 2 12 8 r 8r 8r r s1n 986 86 r sun 9 69 Hence we have 2 g am 21 a smearx 2 12 a r 6r 6r r sm686 86 r sm 6 6 Using separation of variables let w RrH6 2 Spherical Wave Functions cont After substituting Eq2 into Eq 1 divide by 11 1 a 28R 1 a 8H 2 r 2 S1116 r R 8r 8r r stH 819 819 1 1 ach r2 sin2 19 CD 8 k20 At this point we cannot yet say that all of the dependence on a given variable is within only one term Spherical Wave Functions cont Next multiply by r2 sin2 6 sin2 9 8 28R sin6 8 8H r sm R 8r 8r H 86 86 1 821 c1gt M2 18 sinz 6 0 3 Since the underlined term is the only one which depends on It must be equal to a constant Hence set i azq m2 4 CD 5 Spherical Wave Functions cont Hence cos m CD 5 m sin m In general r not an integer Now divide Eq 3 by sin2 9 and use Eq 4 to obtain 1 a 2 6R 1 a 6H r sma R 6r 6r Hsinlt9 66 66 2 m k2 20 6 sin26 Spherical Wave Functions cont The underlined terms are the only ones that involve 6 now This time the separation constant is customarily chosen as n n 1 2 1 isini9d Hj m2 nn1 7 P131116 d6 d6 5111 6 In general not an integer To simplify this let dx sini9di9 i sini91 d6 dx and denote yx 1 109 Spherical Wave Functions cont 1 Since sin 6 1 x2 2 we have 1 d dH m2 31119 2 nn1 P131116 d6 d6 sm 6 nn10 I xz Spherical Wave Functions cont Canceling terms 1 x221 x2i11x2 y nnl0 1362 Multiplying by y we have 8 Spherical Wave Functions cont Eq 8 is the associated Legendre equation The solutions are represented as 131mm y x Q5100 If m 0 Eq 8 is called the Legendre equation in which case P300 an x y Q2ltxgt Qnx Spherical Wave Functions cont Hence Relation to Legendre functions when w m integer For m gt w not an integer the associated Legendre function is defined in terms of the hypergeometric function 11 Spherical Wave Functions cont Rodriguez s formula for z n Legendre polynomial a polynomial of order n Spherical Wave Functions cont Note This follows from these two relations Pro 1 x2 PM 1 d Pquot x 2 dxquot x2 1n polynomial of order n Spherical Wave Functions cont Lowestorder Q functions 1 1 x Q0 x Z 5111 The Q functions all tend to infinity as l x x gti1 x 1x x 2 111 1 gt 9 09 Q1 2 1x 3x2 1 1x 3x 1n Q2 4 l x 2 Spherical Wave Functions cont To be as general as possible I l U m gtw Spherical Wave Functions cont Substituting Eq 7 into Eq 6 yields ii rzd R nn1k2 20 Rdr dr Nextet dxzkdr i 1 dr dx and denote yx Rr Spherical Wave Functions cont We then have 1d 261 Rdr r j nn1k2r2 0 dr Spherical Wave Functions cont or spherical Bessel equation Spherical Wave Functions cont Denote and let 9 X x 28100 1 2 3 b39xx 2gnquot x 2gn39Zx 2g 2 1 3 1 1 1 ngn quot 9628 41x zgn2ngn39 x 28 1 x2 nn1x3gnx 0 Hence Spherical Wave Functions cont 1 M ultiply by X 4 A combine these terms 1 or ngnquotxgn39 Zgn x2gn nn1gn 0 Use lnn1j nl2 4 2 ngn quot xgn 39ign ngn 39 gn x2 nn1gnx O Spherical Wave Functions cont We then have ngn quot xgn 39 x2 j2gn 0 This is Bessel s equation of order y 506 Hence gn Y x 7 so that b iJn12xF n J Yn12x 2 added for convenience Spherical Wave Functions cont De ne Then Summary V2wk2w20 Mk1quot yn 0 H J Pn cos 9 Qnmltcosm I cosm sinm J bn x 2 g Bn12 x BTW Q3106 In general Mg l xz m 1 x23 dm dxm dm dxm PM QnOC m gtw I l gtU Properties of Spherical Bessel Functions bnx iBnHZCX V2x Bessel functions of halfinteger order are given by closedform expressions JUx 2 016 gm z rz1 k0 Uk 2 l becomes a simple function Properties of Spherical Bessel Functions cont Examples Jl2 x i sin x xx J12 x 1 cos x xx W J x 1 cos x sin 32 xx x J U x cosu7z J U x Y x E sinu7z q Y12 x J 12x 13205 J 32 x Properties of Spherical Bessel Functions cont Proof for 1 12 00 ijlZ k J12xkk12k 2 Hence x 00 1k i 2k1 EJ12Xkr12kr 2 Properties of Spherical Bessel Functions cont Examine the factorial expression k12k12k 12k 3232 i k12k 12k 3232 2k12k 12k 33 1T 2 NIH 2k1JZ 2k2k 22k 44 2k1T 2k 1kk 1k 22 2k1ZGT 2k 1k Properties of Spherical Bessel Functions cont x 00 1k i 2k1 Hence JEJl2x k If 2 Simplifying we have k39 22k lk39 Properties of Spherical Bessel Functions cont Properties of Legendre Functions 13rltx1 x2 dm m Pn x m integer dx Rodriguez s formula 1 d 12xn n 2 nldx x2 1n polynomial of order n always finite Properties of Legendre Functions QZ x1x2 Qn x m integer Q x infinite series may blow up For Q x the field blows up on the i2 axis xzcos 9 x1 1 gt 192071quot Spherical Wave Functions cont Lowestorder Q functions Q0 x ln1 x l x Q2 x 3x2 11n1x3x Spherical Wave Functions cont Negative index P n1 X P x This Identity also holds for u n Q n1x 7T1 1100 Q X Properties of Legendre Functions cont see Harrington Appendix E P x infinite series Q0 X infinite series P006 lmumll xjmSinEU7r m l Umul1 xjm 2 1 m2 2 N largest integer less than or equal to o QM5Pultxgtcosltmgt PUlt xgt 2 sin U7Z39 linearly independent for u 72 n Properties of Legendre Funct39ons cont Summary of zaxis properties Properties of Legendre Functions oont Z Pquot x allowed PU x allowed y Qnx and Qux are not allowed on 2 axis Properties of Legendre Functions cont POW and PUx are two linearly independent solutions Note the next slide proves that PU x is always a valid solution Valid independent solutions Properties of Legendre Functions cont To see this use the Legendre equation 1 x2y39uuly20 yxPUx d d Let yxft t x 3 Then 6 1t2 1f39uulf 0 I ft P00 Hence a valid solution is PU x Properties of Legendre Functions cont Pn x 1 13406 In this case we must use ECE 6341 Spring 2009 Prof David R Jackson ECE Dept Notes 20

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