写在前面: 下面, 所有的 m m m ν \nu ν ( − ) k (-)^k ( − ) k ( − 1 ) k (-1)^k ( − 1 ) k
Bessel function
在柱坐标系中对Hemholtz方程 ∇ 2 u + k 2 u = 0 \nabla^2 u + k^2 u = 0 ∇ 2 u + k 2 u = 0 J ( x ) J(x) J ( x ) x x x 宗量 , 反映在Bessel方程上, 就是
1 r d d r [ r R ′ ( r ) ] + [ k 2 − λ − μ r 2 ] 中的 ( k 2 − λ ) r \frac{1}{r} \frac{ \text{d}}{ \text{d} r } \left[ r R'(r) \right] + \left[ k^2 -\lambda - \frac{\mu}{r^2} \right] \text{中的} \sqrt{(k^2 - \lambda )} r
r 1 d r d [ r R ′ ( r ) ] + [ k 2 − λ − r 2 μ ] 中的 ( k 2 − λ ) r
使用级数解法可以得到Bessel方程之解.
The expression of Bessel function is:
J ± ν ( x ) = ∑ k = 0 ∞ ( − ) k k ! Γ ( k + ν + 1 ) ( x 2 ) 2 k ± ν (0) \boxed{J_{\pm\nu}(x) = \sum_{k=0}^{\infty} \frac{(-)^k}{k! \Gamma(k + \nu + 1)} \left( \frac{x}{2} \right) ^{2k \pm \nu}\tag0}
J ± ν ( x ) = k = 0 ∑ ∞ k ! Γ ( k + ν + 1 ) ( − ) k ( 2 x ) 2 k ± ν ( 0 )
Properties of Bessel Equation
Parity
By using eq(0), we can verify that
∀ m ∈ Z , J m ( − x ) = ( − ) m J m ( x ) \boxed{\forall m \in \Z, J_{m}(-x) = (-)^m J_m(x)}
∀ m ∈ Z , J m ( − x ) = ( − ) m J m ( x )
Note: Here, ∣ m ∣ |m| ∣ m ∣
Generating function
exp [ x 2 ( t − 1 t ) ] = ∑ n = − ∞ + ∞ J n ( x ) t n \boxed{\text{exp} \left[ \frac{x}{2}(t - \frac{1}{t} ) \right] = \sum_{n=-\infty}^{+\infty} J_n(x) t^n}
exp [ 2 x ( t − t 1 ) ] = n = − ∞ ∑ + ∞ J n ( x ) t n
The generating function can be used to prove the iteration relation of Bessel function / integral expression .
Iteration relation
Here are two important iteration relations(which can be used to define the so-called cylindrical runction ).
By using the relation
J n ( x ) = ∑ k = 0 ∞ J ± ν ( x ) = ∑ k = 0 ∞ ( − ) k k ! Γ ( k + ν + 1 ) ( x 2 ) 2 k ± ν J_n(x) = \sum_{k=0}{\infty} J_{\pm\nu}(x) = \sum_{k=0}^{\infty} \frac{(-)^k}{k! \Gamma(k + \nu + 1)} \left( \frac{x}{2} \right) ^{2k \pm \nu}
J n ( x ) = k = 0 ∑ ∞ J ± ν ( x ) = k = 0 ∑ ∞ k ! Γ ( k + ν + 1 ) ( − ) k ( 2 x ) 2 k ± ν
we can get the following ralations:
d d x [ x m J m ( x ) ] = x m J m − 1 ( x ) (rel1) \boxed{\frac{ \text{d}}{ \text{d} x } [x^m J_m(x)]= x^m J_{m-1}(x) \tag{rel1}}
d x d [ x m J m ( x )] = x m J m − 1 ( x ) ( rel1 )
d d x [ x − m J m ( x ) ] = − x − m J m + 1 ( x ) (rel2) \boxed{\frac{ \text{d}}{ \text{d} x } [x^{-m} J_m(x)] = - x^{-m} J_{m+1} (x) \tag{rel2}}
d x d [ x − m J m ( x )] = − x − m J m + 1 ( x ) ( rel2 )
You can also derive more iteration relation by applying the operator d k d x k \frac{ \text{d}^k}{ \text{d} x^k } d x k d k x ± m J m ( x ) x^{\pm m}J_m(x) x ± m J m ( x )
Definition : Any function that satisfies the iteration relation ( r e l 1 ) , ( r e l 2 ) (rel1), (rel2) ( re l 1 ) , ( re l 2 )
Think : How about
d d x [ x ± m J m ( a x ) ] ? ( a = constant ∈ C ) \frac{ \text{d}}{ \text{d} x } [x^{\pm m} J_m(ax)]? \quad(a = \text{constant} \in \Complex)
d x d [ x ± m J m ( a x )]? ( a = constant ∈ C )
Integral expression
The integral expression of Bessel function can be derived by the generating function. Below is the deriviation.
Let t = e i θ t = e^{i\theta} t = e i θ
e i x sin θ = ∑ n = − ∞ ∞ J n ( x ) e i n θ (1) e^{ix\sin\theta} = \sum_{n=-\infty}^{\infty} J_n(x) e^{in\theta} \tag1
e i x s i n θ = n = − ∞ ∑ ∞ J n ( x ) e in θ ( 1 )
eq.(1) has the same form of Fourier Series. Take the FT(Fourier Transformation) of e i x sin θ e^{ix\sin\theta} e i x s i n θ integreal expression of Bessel equation:
J n ( x ) = 1 2 π ∫ − π π [ cos ( x sin θ − n θ ) + i sin ( x sin θ − n θ ) ] d θ J_n(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} [\cos(x\sin\theta - n\theta) + i\sin(x\sin\theta - n\theta)] \text{d}\theta
J n ( x ) = 2 π 1 ∫ − π π [ cos ( x sin θ − n θ ) + i sin ( x sin θ − n θ )] d θ
Because of the 2nd item inside the integral sign is an odd function, so the integral expression can be further simplified to
J n ( x ) = 1 2 π ∫ − π π [ cos ( x sin θ − n θ ) ] d θ \boxed{J_n(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} [\cos(x\sin\theta - n\theta)] \text{d}\theta}
J n ( x ) = 2 π 1 ∫ − π π [ cos ( x sin θ − n θ )] d θ
Cylindrical function
As mentioned in Iteration relation , any function that satisfies the ( r e l 1 ) , ( r e l 2 ) (rel1), (rel2) ( re l 1 ) , ( re l 2 )
Bessel function, J ν ( x ) = cos ν π J ν ( x ) − J − ν ( x ) sin ν π J_\nu(x) = \frac{ \cos\nu\pi J_\nu(x) - J_{-\nu}(x) }{ \sin\nu\pi } J ν ( x ) = s i n ν π c o s ν π J ν ( x ) − J − ν ( x )
Neumann function, N ν ( x ) = J ν ( x ) + i N ν ( x ) N_\nu(x) = J_\nu(x) + i N_\nu(x) N ν ( x ) = J ν ( x ) + i N ν ( x )
Hankel function, H ν ( x ) = J ν ( x ) − i N ν ( x ) H_\nu(x) = J_\nu(x) - i N_\nu(x) H ν ( x ) = J ν ( x ) − i N ν ( x )
Note: A problem occured in the definition of Neumann function: ν ∈ N ⟹ N ν ( x ) do not exits \nu\in\N \implies N_\nu(x)\text{do not exits} ν ∈ N ⟹ N ν ( x ) do not exits
A understanding of this phenomenon is: N ν ( x ) N_\nu(x) N ν ( x ) N m ( x ) = lim ν → m J ν ( x ) N_m(x) = \lim_{\substack{\nu \to m}}J_\nu(x) N m ( x ) = lim ν → m J ν ( x )
Asympototic relation
lim x → 0 J m ( x ) = { 0 m ≠ 0 1 m ∈ N & m > 0 lim x → ∞ J ν ( x ) ∼ 2 π x cos ( x − ν x 2 − π 4 ) lim x → 0 N ν ( x ) ∼ − Γ ( ν ) π ( x 2 ) − ν lim x → ∞ N ν ( x ) ∼ 2 π x sin ( x − ν x 2 − π / 4 ) \begin{aligned}
\lim_{\substack{x \to 0 }} J_m (x) &= \begin{cases} 0 & m \neq 0 \\
1 & m \in N \And m > 0\end{cases} \newline
\lim_{\substack{x\to\infty} } J_\nu(x) &\sim \sqrt{ {2 \over \pi x} } \cos (x - {\nu x \over 2 } - {\pi \over 4})\newline
\lim_{\substack{x\to 0}} N_\nu(x) &\sim -{ \Gamma(\nu) \over \pi } \big( {x \over 2 } \big)^{ -\nu } \newline
\lim_{\substack{x\to \infty}} N_\nu(x) &\sim \sqrt{ {2 \over \pi x } }\sin( x - {\nu x \over 2 } - \pi/4 )\end{aligned}
x → 0 lim J m ( x ) x → ∞ lim J ν ( x ) x → 0 lim N ν ( x ) x → ∞ lim N ν ( x ) = { 0 1 m = 0 m ∈ N & m > 0 ∼ π x 2 cos ( x − 2 νx − 4 π ) ∼ − π Γ ( ν ) ( 2 x ) − ν ∼ π x 2 sin ( x − 2 νx − π /4 )
Some use of Bessel equation
The example below demostrates how to expand plane wave by the Bessel function.
Let t = i e i θ t = ie^{i\theta} t = i e i θ
e i x cos θ = ∑ − ∞ ∞ J n ( x ) i n e i n θ = J 0 ( x ) + ∑ n = 1 ∞ i n J n ( x ) e i n θ + J − n ( x ) i n e − i n θ = J 0 ( x ) + 2 ∑ n = 0 ∞ i n J n ( x ) cos ( n θ ) (2) \begin{aligned}
e^{ix\cos\theta} &= \sum_{-\infty}^{\infty} J_n(x) i^n e^{in\theta} \\
&= J_0(x) + \sum_{n=1}^{\infty} i^n J_n(x)e^{in\theta} + J_{-n}(x) i^n e^{-in\theta} \\
&= J_0(x) + 2 \sum_{n=0}^{\infty} i^n J_n(x) \cos(n\theta)
\end{aligned} \tag2
e i x c o s θ = − ∞ ∑ ∞ J n ( x ) i n e in θ = J 0 ( x ) + n = 1 ∑ ∞ i n J n ( x ) e in θ + J − n ( x ) i n e − in θ = J 0 ( x ) + 2 n = 0 ∑ ∞ i n J n ( x ) cos ( n θ ) ( 2 )
In the last step of the deriviarion, we used the fact that J − n ( x ) = ( − 1 ) n J n ( x ) J_{-n}(x) = (-1)^n J_n(x) J − n ( x ) = ( − 1 ) n J n ( x ) e i x + e − i x = 2 cos x e^{ix} + e^{-ix} = 2 \cos x e i x + e − i x = 2 cos x
If we let x = k r x = kr x = k r k k k wave vector , and r r r plane wave .
Others
Bessel functions with virtual viriables
Spherical Bessel function
