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Bessel Functions Summary
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Bessel functions and Legendre polynomials Bessel’s equation of order p x 2 y + xy + (x 2 − p 2 )y = 0 Solution: y(x) = aJ p (x) + bY p (x) If p is not an integer, Y p (x) = J − p (x). 0 10 20 x −0.5 0 0.5 1 J n ( x ) J 0 (x) J 1 (x) J 2 (x) 0 10 20 x −1 −0.5 0 0.5 Y n ( x ) Y 0 (x) Y 1 (x) Y 2 (x) Behavio r at zero: J 0 (0) = 1, J p (0) = 0 for p = 1, 2, 3,..., Y p (0) = −∞ for all p Beha vior at infini ty: J p (x) → 0 and Y p (x) → 0 as x → ∞ Zeros J p (σ pm ) = 0 for all p and m = 1, 2, 3,... . (Each J p , Y p has infinite nu mbe r of zer os labe led σ pm .) Derivatives • d dx [x p J p (x)] = x p J p−1 (x), • d dx [x − p J p (x)] = −x − p J p+1 (x) • d dx [x p Y p (x)] = x p Y p−1 (x), • d dx [x − p Y p (x)] = −x − p Y p+1 (x) Orthogonality o xJ p (σ pm x/)J p (σ pn x/) dx = 0 n = m 2 2 J 2 p+1 (σ pm ) n = m Idea: Besse l functio ns are like sine and cos ine. Modified Bessel’s equation of order p x 2 y + xy − (x 2 + p 2 )y = 0 Solution: y(x) = aI p (x) + bK p (x) If p is not an integer, K p (x) = I − p (x). 0 2 4 x 0 1 2 I n ( x I 0 (x) I 1 (x) I 2 (x) 0 2 4 x 0 1 2 K n ( x K 0 (x) K 1 (x) K 2 (x) Beha vior at zero : I 0 (0) = 1, I p (0) = 0, p = 1, 2, 3,... K p (0) = ∞ for all p Behavio r at infinity: I p (x) → ∞, K p (x) → 0 as x →∞
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Document summarizing Bessel Functions
Transcript of Bessel Functions Summary
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Bessel functions and Legendre polynomials
Bessels equation of order p
x2y + xy + (x2 p2)y = 0
Solution: y(x) = aJp(x) + bYp(x)If p is not an integer, Yp(x) = Jp(x).
0 10 20
x
0.5
0
0.5
1
Jn(x
)
J0(x)
J1(x)
J2(x)
0 10 20
x
1
0.5
0
0.5
Yn(x
)
Y0(x)
Y1(x)
Y2(x)
Behavior at zero: J0(0) = 1, Jp(0) = 0 forp = 1, 2, 3, . . ., Yp(0) = for all p
Behavior at infinity: Jp(x) 0 and Yp(x) 0as x
Zeros Jp(pm) = 0 for all p and m = 1, 2, 3, . . ..(Each Jp, Yp has infinite number of zeros labeledpm.)
Derivatives
ddx [xpJp(x)] = xpJp1(x),
ddx [xpJp(x)] = xpJp+1(x)
ddx [xpYp(x)] = xpYp1(x),
ddx [xpYp(x)] = xpYp+1(x)
Orthogonality
`o
xJp(pmx/`)Jp(pnx/`) dx
=
{0 n 6= m`2
2 J2p+1(pm) n = m
Idea: Bessel functions are like sine and cosine.
Modified Bessels equation of order p
x2y + xy (x2 + p2)y = 0
Solution: y(x) = aIp(x) + bKp(x)If p is not an integer, Kp(x) = Ip(x).
0 2 4
x
0
1
2
3I n
(x)
I0(x)
I1(x)
I2(x)
0 2 4
x
0
1
2
3
Kn(x
)
K0(x)
K1(x)
K2(x)
Behavior at zero: I0(0) = 1, Ip(0) = 0, p =1, 2, 3, . . . Kp(0) = for all p
Behavior at infinity: Ip(x) , Kp(x) 0 asx
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Derivatives
ddx [xpIp(x)] = xpIp1(x), ddx [xpIp(x)] = xpIp+1(x) ddx [xpKp(x)] = xpKp1(x), ddx [xpKp(x)] = xpKp+1(x)
Idea: Modified Bessel functions are like epx andepx.
Legendres equation:
(1 x2)y 2xy + n(n+ 1)y = 0,where n is an integer.Solution: y(x) = aPn(x) + bQn(x)
1 0 1x
1
0
1
Pn(x
)
1 0 1x
2.5
0
2.5
Qn(x
) Pn(x) = 12nn! d
n
dxn (x2 1)n
P0 = 1, P1 = x, P2 = 12 (3x2 1), . . .
Qn(x) = 12Pn(x) ln(1+x1x
)nm=1 1nPm1Pnm
for n = 1, 2, 3, . . .
Q0(x) = 12 ln(1+x1x
)Note: Qn(x) only finite on 1 < x < 1
Properties on 1 < x < 1 n even: Pn(1) = 1, Pn(1) = 1 n odd: Pn(1) = 1, Pn(1) = 1 Qn(1) = , Qn(1) =
Zeros:
Pn(x) has n zeros. Pn(0) = 0 if n odd. Qn(x) has n+ 1 zeros.
Orthogonality: 11Pn(x)Pm(x) dx =
{0 n 6= m
22n+1 n = m
pi0
Pn(cos)Pm(cos) sind
=
{0 n 6= m
22n+1 n = m
Idea: Pn(x) is a specific polynomial in x of ordern. Qn(x) is a polynomial times a log.
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