Notes 19 Bessel Functions

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1 ECE 638 Fall 17 David R. Jackso Notes 19 Bessel Fuctios Notes are from D. R. Wilto, Dept. of ECE 1

2 Cylidrical Wave Fuctios Helmholtz equatio: ψ + k ψ = I cylidrical coordiates: ψ 1 ψ 1 ψ ψ ρ ρ ρ ρ φ z k ψ = Separatio of variables: ψ ρφ,, z = R ρ Φ( φ) Z( z) let ( ) ( ) Substitute ito previous equatio ad divide by ψ.

3 Cylidrical Wave Fuctios (cot.) ψ 1 ψ 1 ψ ψ ρ ρ ρ ρ φ z k ψ = ( ) ( ) ψ ρφ,, z = R ρ Φ( φ) Z( z) R 1 R 1 Φ Z Φ Z + Φ Z + RZ + RΦ k R Z + Φ = ρ ρ ρ ρ φ z Divide by ψ let R 1 R 1 Φ Z R ρ R ρ Φ Z = k 3

4 Cylidrical Wave Fuctios (cot.) R 1 R 1 Φ Z R ρ R ρ Φ Z = k (1) Therefore Z R 1 R 1 Φ = k Z R ρ R ρ Φ f( z ) g( ρφ, ) Hece, f (z) = costat = - k z 4

5 Cylidrical Wave Fuctios (cot.) Hece Z = k Z z { ± ik } z z Z( z) = hkz ( ) = e,si( kz),cos( kz) z z z Net, to isolate the φ -depedet term, multiply Eq. (1) by ρ : ρ R 1 R 1 Φ + + k + k ρ = R ρ R ρ Φ z 5

6 Cylidrical Wave Fuctios (cot.) Hece Φ 1 R R = ρ k + kz Φ ρ R R () f ( φ ) g( ρ) Hece Φ = costat = Φ so { e ± iφ } Φ= h( φ) =,si( φ),cos( φ) 6

7 Cylidrical Wave Fuctios (cot.) From Eq. () we the have 1 R R = ρ k + kz ρ R R The et goal is to solve this equatio for R(ρ). First, multiply by R ad collect terms: ( ) z ρ R + ρr + ρ k k R R = 7

8 Cylidrical Wave Fuctios (cot.) Defie k k ρ k z The, ( ) ρ R + ρr + kρρ R = Net, defie = k ρ ρ R( ρ) = y ( ) Note that dr dy d R ( ρ) = = = y( k ) dρ d dρ ρ ad R ( p) = y ( ) k ρ 8

9 Cylidrical Wave Fuctios (cot.) The we have + + = y y y Bessel equatio of order Two idepedet solutios: J ( ), Y ( ) Hece y( ) = AJ ( ) + BY ( ) Therefore { } ρ ρ R( ρ) = J ( k ρ), Y ( k ρ) 9

10 Cylidrical Wave Fuctios (cot.) Summary ψ + k ψ = ψ ρφ,, z = R ρ Φ( φ) Z( z) ( ) ( ) { ± ik } z z z z Z( z) = e, si( kz), cos( kz) Φ= { ± i e φ, si( φ), cos( φ) } { } ρ ρ R( ρ) = J ( k ρ), Y ( k ρ) k + ρ k = k z 1

11 Refereces for Bessel Fuctios M. R. Spiegel, Schaum s Outlie Mathematical Hadbook, McGraw-Hill, 4 th Editio, 1. M. Abramowitz ad I. E. Stegu, Hadbook of Mathematical Fuctios with Formulas, Graphs, ad Mathematical Tables, Natioal Bureau of Stadards, Govermet Pritig Office, Teth Pritig, 197. NIST olie Digital Library of Mathematical Fuctios ( N. N. Lebedev, Special Fuctios & Their Applicatios, Dover Publicatios, New York, 197. G. N. Watso, A Treatise o the Theory of Bessel Fuctios ( d Ed.), Cambridge Uiversity Press,

12 Properties of Bessel Fuctios = () J is fiite.6.4 = 1 = J () J( ) J1( ). J(, )

13 Bessel Fuctios (cot.) = = 1 = Y Y( () Y1( ) Y(, ) 3 () Y is ifiite

14 Bessel Fuctios (cot.) Frobeius solutio: J ( ) ( 1) = ( ) k = k! + k! k + k z! =Γ z+ 1 ( ) This series always coverges. 14

15 Bessel Fuctios (cot.) No-iteger order: y ( ) = J( ), J ( ) { } Two liearly idepedet solutios Note: Bessel equatio is uchaged by J ( ) is a always a valid solutio These are liearly idepedet whe is ot a iteger. J ( ) A, J ( ) A as 1 15

16 Bessel Fuctios (cot.) Bessel fuctio of secod kid: J( )cos( π ) J ( ) Y ( ) si( π ) -, -1,, 1, (This defiitio gives a ice asymptotic behavior as.) Y ( ) lim ( ) Y 16

17 Bessel Fuctios (cot.) From the Frobeius solutio we have: ( k ) 1 k 1! 1 Y( ) = J( ) l γ π + π k! k = 1 k 1 ( 1) ( k) ( k) π Φ +Φ + k = k! + k! ( ) k+ (Schaum s Outlie Mathematical Hadbook, Eq. (4.9)) where Φ ( p) = ( p> ) 3 p ( ) Φ = 17

18 Bessel Fuctios (cot.) Iteger order: = Symmetry property: J ( ) = ( 1) J ( ) Y ( ) = ( 1) Y ( ) (They are o loger liearly idepedet.) 18

19 Bessel Fuctios (cot.) Small-Argumet Properties ( ): J ( ) A, =,1, J ( ) A, 1,, 3,... Y ( ) Cl ( ) Y ( ) D, = 1, Y ( ) B, Eamples: J 1 ( ) 1 ( ) J / For order zero, the Bessel fuctio of the secod kid behaves logarithmically rather tha algebraically. 19

20 Bessel Fuctios (cot.) Asymptotic Formulas: J ( ) ~ cos π Y ( ) ~ si π π π 4 π π 4

21 Hakel Fuctios ( 1) H ( ) J ( ) + iy ( ) ( ) H ( ) J ( ) iy ( ) As H e π π π ( ) ( 1 + i ) 4 ( )~ H e π π π ( ) ( i ) 4 ( )~ Icomig wave Outgoig wave These are valid for arbitrary order. 1

22 Hakel Fuctios (cot.) Useful idetity: ( ) ( ) ( ) H ( ) = 1 H ( + ) This is a symmetry property of the Hakel fuctio. N. N. Lebedev, Special Fuctios & Their Applicatios, Dover Publicatios, New York, 197.

23 Geeratig Fuctio The iteger order Bessel fuctio of the first kid ca also be defied through a geeratig fuctio g(,t): 1 t t g( t, ) = e = J ( ) t = The geeratig fuctio defiitio leads to a umber of useful idetities ad represetatios: ik i( kρ ) cosφ t 1 t = kρ, t= iei = ( ) ( ρ ) e = e = e = i J k e φ iφ (plae wave epasio) m π i iρcosφ imφ iφ Jm ( ρ) = e e dφ k = 1 e π (Set i above result, use orthogoality of ) π 1 = cos ( ρ si φ mφ) dφ φ = φ + π /, π (Use φ φ) 3

24 ( ) Recurrece Relatios May recurrece relatios ca be derived from the geeratig fuctio. 1 t t g t, = e = J( t ) = 1 t t 1 g( t, ) = e = 1 + g( t, ) = J( t ) t t t = J ( ) ( t + t ) = [ J 1( ) + J + 1( ) ] t = J ( ) t 1 1 = = = 1 Equatig like powers of t yields: J 1( ) + J+ 1( ) = J( ) 4

25 Recurrece Relatios (cot.) J 1( ) + J+ 1( ) = J( ) This ca be used to geerate other useful recurrece relatios: ( ) 1: 1 J( ) = J 1( ) J ( ) + 1: ( + 1) J( ) = J+ 1( ) J+ ( ) ("upward recursio") ("dowward recursio") 5

26 ( ) Recurrece Relatios (cot.) 1 t t g t, = e = J( t ) = 1 t t 1 t e g t = = t e = t g t t t Also, we have t (, ) (, ) 1 = J( t ) J( t ) = = 1 1 = J t J t J J t = = 1( ) + 1( ) [ 1( ) + 1( )] = = ( ) g t, = J ( t ) = Equatig like powers of t yields: J ( ) = [ J ( ) J ( ) ]

27 Recurrece Relatios (cot.) 1 J ( ) = J 1( ) J+ 1( ) [ ] The use J 1( ) + J+ 1( ) = J( ) This ca be used to replace J +1 or J -1. This yields J ( ) = J 1( ) J( ) J ( ) = J+ 1( ) + J( ) 7

28 Recurrece Relatios (cot.) The same recurrece formulas actually apply to all Bessel fuctios of all orders. If Z () deotes ay Bessel, Neuma, or Hakel fuctio of order, the we have: Z 1( ) + Z+ 1( ) = Z( ) ( ) 1 Z( ) = J 1( ) J ( ) ( + 1) Z( ) = J+ 1( ) J+ ( ) Z = Z Z Z = Z + Z Itegral relatios also follow from these (see the et slide). 8

29 Recurrece Relatios (cot.) Eample of itegral idetity: Z = Z Z Z + Z = Z 1 1 Multiply by 1 1 Z Z Z + = 1 Hece d Z ( ) Z 1( ) Z 1( d ) Z = = ( ) d Z ( ) ( ) 1 d= Z Similarly, we have Z ( ) d Z ( ) + 1 = 9

30 Recurrece Relatios (cot.) Eample ( ) = ( ) J d J 1 ( ) = ( ) J d J 1 (First oe, = 1) (Secod oe, = ) 3

31 Wroskias From the Sturm-Liouville properties, the Wroskias for the Bessel differetial equatio are foud to have the followig form: d l ( ) a a W a a C W( ) = W( a) e = W( a) e = = Recall : W( ) = W( a) e p( ) d a The costat C ca be foud usig the small-argumet approimatios for the Bessel fuctios. siπ W[ J, J ] J ( ) J ( ) J ( ) J ( ) = =!! π ( ) Note: For, the Wroskia is ot idetically zero (i fact, it is ot zero aywhere), ad hece the two fuctios are liearly idepedet. Recall : π Γ( z) Γ(1 z) = siπ z π Γ( ) Γ(1 ) = siπ π ( 1! ) ( )! = siπ π ( )!( )! = siπ 31

32 Wroskias Similarly, we have W[ J, Y] = J ( Y ) ( ) J ( Y ) ( ) = π (1,) (1,) (1,) i W J, H J ( ) H = ( ) J ( ) H ( ) = ± π + for for H H (1) () 3

33 Fourier-Bessel series: Fourier-Bessel Series ρ f( ρ) = cj α, ρ a = 1 a Note: The order is arbitrary here. th where α is the m z ero of J ( ): J ( α ) =. m m To fid the coefficiets, use the orthogoality idetity (derivatio o et slide): a a J ρ ρ α J α ρdρ = δ J ( α ) m m 1 m a a + Multiply both sides of the Fourer - Bessel epasio by J α m ρ ρ ad itegrate from to a. a Note: See Notes 18 for a derivatio of orthogoality whe m. 33

34 Fourier-Bessel Series (cot.) a ρ ρ ρ f( ρ) J αm ρdρ = cj αm J α ρdρ a = 1 a a a = cm J+ 1 ( αm) ρ cm = f( ρ) J αm ρdρ a J a 1 ( ανm ) + a 34

35 Fourier-Bessel Series (cot.) Derivatio of the orthogoality formula Start with this itegral idetity: J v ( p) d = ( J ( p) ) 1 ( p) J p ( ( )) Hece we have a α α α Jv d J a J a a a a α m a a m a m a m = 1 a = ( J ( αm )) a = J+ 1( αm) + J( αm) αm a = ( J ( α )) + 1 m Recall : J ( ) = J+ 1( ) + J( ) 35

Additio Theorems y ρ φ ρ φ y ρ φ Additio theorems allow cylidrical harmoics i oe coordiate system to be epaded i terms of those of a shifted coordiate system.

36 Additio Theorems y ρ φ ρ φ y ρ φ Additio theorems allow cylidrical harmoics i oe coordiate system to be epaded i terms of those of a shifted coordiate system. Shiftig from global origi to local origi: iφ i ( m) φ ( ρ) = m( ρ) m( ρ ) imφ m= J k e J k e J k e H ( kρ) e () iφ m= () m ( ) i mφ m = i ( m) φ () m= imφ H ( kρ ) e J ( kρ ) e, ρ > ρ imφ m J ( kρ ) e H ( kρ ) e, ρ < ρ m 36

37 Additio Theorems (cot.) y ρ φ ρ φ y ρ φ iφ i ( m)( π+ φ ) J ( kρ ) e = J ( kρ ) e J ( kρ) e m m m= () iφ H ( kρ ) e = Shiftig from local origi to global origi: m= m= imφ () i ( m)( π+ φ ) imφ H ( kρ ) e J ( kρ) e, ρ > ρ m m i ( m)( π+ φ ) () imφ J ( kρ ) e H ( kρ) e, ρ < ρ m m 37

38 Additio Theorems (cot.) y ρ φ ρ φ y ρ φ Recall : m ( ) = ( 1) ( ) J J m m ( ) = ( 1) ( ) ( 1) m Y Y e m imπ = m m Special case ( = ): H () ( kρ ) = m= m= () m m ( φ φ ) im m H ( kρ ) J ( kρ) e, ρ > ρ ( φ φ ) im m () J ( kρ ) H ( kρ) e, ρ < ρ 38

ECE Spring Prof. David R. Jackson ECE Dept. Notes 8

ECE Spring Prof. David R. Jackson ECE Dept. Notes 8 ECE 6341 Sprig 16 Prof. David R. Jackso ECE Dept. Notes 8 1 Cylidrical Wave Fuctios Helmholtz equatio: ψ + k ψ = ψ ρφ,, z = A or F ( ) z z ψ 1 ψ 1 ψ ψ ρ ρ ρ ρ φ z + + + + k ψ = Separatio of variables: