BESSEL EQUATION ad BESSEL FUNCTIONS
Bessel s Equatio Summary of Bessel Fuctios d y dy y d + d + =. If is a iteger, the two idepedet solutios of Bessel s Equatio are J J, Bessel fuctio of the first kid, ( ) p p+ p= ( ) = + p! p! Geeratig fuctio for J ( ) e ( t t ) = t J = ( ) Y, Bessel fuctio of the secod kid p = ( p )! p l Y = J π π p! p p p+ p + ( ) ( ) γ π p! p! + m m p= m= m=
Other relatios for J ( ), Z. J ( ) J ( ) =. J ( ) = J ( ) J ( ) + J ( ) = J ( ) + J ( ) J ( ) + J ( ) J ( ) ( ) p p+ = + p! p! p= p p 4 6 ( ) 4 6 ( p! ) (! ) (! ) ( 3! ) p= = = + +... p p+ 3 5 7 ( ) ( p ) = = + +... +! p!!!!!!3! 3!4! 3 5 7 p=
Questio Show that J d y dy y. d + d + =, is a solutio of the above differetial equatio. proof
Questio Startig from the geeratig fuctio of the Bessel fuctio of the first kid show that e ( t ) t J ( ), Z, t = = J = J. proof
Questio 3 Startig from the series defiitio of the Bessel fuctio of the first kid show that J ( ) r r+ ( ), Z, ( + r)! r! r= = J J =. proof
Questio 4 Startig from the geeratig fuctio of the Bessel fuctio of the first kid e ( t ) t J ( ), Z, t = = show that J = J ( ) J+ ( ). proof
Questio 5 Startig from the geeratig fuctio of the Bessel fuctio of the first kid e ( t ) t J ( ), Z, t = = show that J ( ) = J ( ) + J + ( ). proof
Questio 6 The geeratig fuctio of the Bessel fuctio of the first kid is e ( t ) t J ( ), Z. t = = a) By differetiatig the geeratig fuctio relatio with respect to, show that J J + ( ) = J ( ). b) By differetiatig the geeratig fuctio relatio with respect to t, show that J ( ) = J ( ) + J+ ( ). c) Hece fid a simplified epressio for ( + ) J ( ) d d. d d MM3-A, ( + ) J ( ) = J ( ) J ( ) +
Questio 7 Startig from the geeratig fuctio of the Bessel fuctio of the first kid e ( t ) t J ( ), Z, t = = determie the series epasio of J 4 6, ad hece show that J = + +... 4 6!! 3! 3 5 7 J = + +... 3 5 7!!!!!3! 3!4! J ( ) p p+ p= ( ) = + p! p!
Questio 8 The geeratig fuctio of the Bessel fuctio of the first kid is e ( t ) t J ( ), Z. t = = Use the geeratig fuctio relatio, to show that for a) J ( ) J ( ) = b) J ( ) J ( ) J ( ) + =. + =. c) J ( ) J ( ) J ( ) + Use parts (b) ad (c) to fid simplified epressios for d J d d). d J d e) f) Use parts (d) ad (e) to show that the positive zeros of J those of J ( ) +. iterlace with d J = J d d J J + d, = [solutio overleaf]
Questio 9 The Bessel fuctio of the first kid is defied by the series J ( ) Use the above defiitio to show r r+ ( ), Z. ( + r)! r! r= = e ( t ) t = t J ( ), Z. = MM3-B, proof
Questio The geeratig fuctio of the Bessel fuctio of the first kid is e ( t ) t J ( ), Z. t = = Use the geeratig fuctio relatio, to show that J ( + y) = Jm ( y) J m ( ). m= MM3-C, proof
Questio The Bessel fuctio of the first kid is defied by the series J ( ) Use the above defiitio to show r r+ ( ), Z. ( + r)! r! r= = ( ) J lim =, Z.! MM3-D, proof
Questio d y dy y d + d + =. The above differetial equatio is kow as modified Bessel s Equatio. Use the Frobeius method to show that the geeral solutio of this differetial equatio, for =, is y = Acosh + Bsih. [ ] proof
Questio 3 Fid the two idepedet solutios of Bessel s equatio ( ν ) d y dy + + y =, ν Z. d d Give the aswer as eact simplified summatios. r ( ) ν r or ν ν r= r= ( )! r r+ ν ν y = A J = r! + r! r! ν + r! r ( ν ) r or ν ν r= r= r! r ν ν y B = J = r! r! r! r ν!
Questio 4 Fid the two idepedet solutios of Bessel s equatio d y dy y d + d + =, Z. Give the aswer as eact simplified summatios. r ( ) r or r= r= ( )! r r+ y = A J = r! + r! r! + r! r r r r! r! r y = B l + B + r! + r! r! + r! m m + r= r= m= m=
Questio 5 Fid the two idepedet solutios of Bessel s equatio d y dy y d + d + =, =. Give the aswer as eact simplified summatios. r r or ( ) ( r ) r= r= ( ) ( r ) r r y = A J = ( )!! r r r r r ( ) ( ) ( ) y = B( l ) + B ( r! ) ( r! ) m r= r= m=
Questio 6 The geeratig fuctio of the Bessel fuctio of the first kid is e ( t ) t J ( ), Z. t = = a) Use the geeratig fuctio, to show that for i. J ( ) J ( ) = ii. J ( ) J ( ) J ( ) + =. + =. iii. J ( ) J ( ) J ( ) + b) Use part (a) deduce that d J J d i. = d J J d ii. = c) Use part (b) to show further that. J + J + J =. proof [solutio overleaf]
Questio 7 d y dy t + + ty, t >. dt dt The Bessel fuctio of order zero, J equatio. It is further give that J ( t) lim =. t t, is a solutio of the above differetial By takig the Laplace trasform of the above differetial equatio, show that L J ( t) =. s + proof
Questio 8 It ca be show that for N i t t e dt =. ( m)! Γ ( m + + ) m= m ( ) Γ ( m + ) Γ ( + ) Use Legedre s duplicatio formula for the Gamma Fuctio to show ( ) cos J ( ) = t t dt. π Γ ( + ) MM3-E, proof
Questio 9 Legedre s duplicatio formula for the Gamma Fuctio states ( ) Γ Γ + ( ) π Γ a) Prove the validity of the above formula. b) Hece show that, N. J = cos. π c) Determie a eact simplified epressio for J ( ) + J ( ). π
Questio a) By usig techiques ivolvig the Beta fuctio ad the Gamma fuctio, show that k k+ ( k! ) ( cosθ ) dθ =. ( k + )! π The series defiitio of the Bessel fuctio of the first kid J ( ) r r+ ( ), Z. ( + r)! r! r= = b) Use the above defiitio ad the result of part (a), to show that π cosθ J ( cosθ ) dθ =, proof
Questio The Bessel fuctio J ( ) where α is a o zero costat. α satisfies the differetial equatio ( α ) d y dy + + y =, Z, d d If J ( α ) ad J ( α ) satisfy J ( α ) J ( α ) = =, with α α, show that J ( α ) J ( α ) d =. proof
Questio The series defiitio of the Bessel fuctio of the first kid J ( ) Use the above defiitio to show that r r+ ( ), Z. ( + r)! r! r= = J ( ) m m+ where = ( ) I =! ( m ) m m, m I t t J t dt > >., proof
Questio 3 I ( t ) i t = e dt a) By usig the series defiitio of the epoetial fuctio ad covertig the itegrad ito a Beta fuctio, show that I = m m Γ + Γ m +. ( m)! Γ ( + m + ) m= Legedre s duplicatio formula for the Gamma Fuctio states ( m ) Γ m Γ + m ( ) π Γ( m), m N. b) Use the above formula ad the result of part (a) to show further ( ) cos J ( ) = t t dt. π Γ ( + ) proof
Questio 4 The Bessel fuctio of the first kid J J ( ), satisfies p p+ ( ). ( + p)! p! p= = Show that J ( ) I = π Γ ( + ), π I = cos siθ cos θ dθ. where proof
Questio 5 The geeratig fuctio of the Bessel fuctio of the first kid is e ( t ) t J ( ), Z. t = = a) Use the geeratig fuctio, to show that for i. J ( ) J ( ) = ii. J ( ) J ( ) J ( ) + =. + =. iii. J ( ) J ( ) J ( ) + b) Give that y J ( λ) = satisfies the differetial equatio ( λ ) d y dy + + y =, =,,, 3,... d d verify that d dy d + ( λ ) ( y ) =, d d d ad hece show that if i λ is a o zero root of J ( λ ) = J ( λ ) d = J ( λ ) = J + ( λ ). i i i proof [solutio overleaf]