# Bessel Functions¶

This module provides symbolic Bessel Functions. These functions use the mpmath library for numerical evaluation and Maxima, GiNaC, Pynac for symbolics.

The main objects which are exported from this module are:

• bessel_J – The Bessel J function
• bessel_Y – The Bessel Y function
• bessel_I – The Bessel I function
• bessel_K – The Bessel K function
• Bessel – A factory function for producing Bessel functions of various kinds and orders
• Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessel’s differential equation:

$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \nu^2\right)y = 0,$

for an arbitrary complex number $$\nu$$ (the order).

• In this module, $$J_\nu$$ denotes the unique solution of Bessel’s equation which is non-singular at $$x = 0$$. This function is known as the Bessel Function of the First Kind. This function also arises as a special case of the hypergeometric function $${}_0F_1$$:

$J_\nu(x) = \frac{x^n}{2^\nu \Gamma(\nu + 1)} {}_0F_1(\nu + 1, -\frac{x^2}{4}).$
• The second linearly independent solution to Bessel’s equation (which is singular at $$x=0$$) is denoted by $$Y_\nu$$ and is called the Bessel Function of the Second Kind:

$Y_\nu(x) = \frac{ J_\nu(x) \cos(\pi \nu) - J_{-\nu}(x)}{\sin(\pi \nu)}.$
• There are also two commonly used combinations of the Bessel J and Y Functions. The Bessel I Function, or the Modified Bessel Function of the First Kind, is defined by:

$I_\nu(x) = i^{-\nu} J_\nu(ix).$

The Bessel K Function, or the Modified Bessel Function of the Second Kind, is defined by:

$K_\nu(x) = \frac{\pi}{2} \cdot \frac{I_{-\nu}(x) - I_n(x)}{\sin(\pi \nu)}.$

We should note here that the above formulas for Bessel Y and K functions should be understood as limits when $$\nu$$ is an integer.

• It follows from Bessel’s differential equation that the derivative of $$J_n(x)$$ with respect to $$x$$ is:

$\frac{d}{dx} J_n(x) = \frac{1}{x^n} \left(x^n J_{n-1}(x) - n x^{n-1} J_n(z) \right)$
• Another important formulation of the two linearly independent solutions to Bessel’s equation are the Hankel functions $$H_\nu^{(1)}(x)$$ and $$H_\nu^{(2)}(x)$$, defined by:

$H_\nu^{(1)}(x) = J_\nu(x) + i Y_\nu(x)$
$H_\nu^{(2)}(x) = J_\nu(x) - i Y_\nu(x)$

where $$i$$ is the imaginary unit (and $$J_*$$ and $$Y_*$$ are the usual J- and Y-Bessel functions). These linear combinations are also known as Bessel functions of the third kind; they are also two linearly independent solutions of Bessel’s differential equation. They are named for Hermann Hankel.

EXAMPLES:

Evaluate the Bessel J function symbolically and numerically:

sage: bessel_J(0, x)
bessel_J(0, x)
sage: bessel_J(0, 0)
bessel_J(0, 0)
sage: bessel_J(0, x).diff(x)
-1/2*bessel_J(1, x) + 1/2*bessel_J(-1, x)

sage: N(bessel_J(0, 0), digits = 20)
1.0000000000000000000
sage: find_root(bessel_J(0,x), 0, 5)
2.404825557695773


Plot the Bessel J function:

sage: f(x) = Bessel(0)(x); f
x |--> bessel_J(0, x)
sage: plot(f, (x, 1, 10))
Graphics object consisting of 1 graphics primitive


Visualize the Bessel Y function on the complex plane (set plot_points to a higher value to get more detail):

sage: complex_plot(bessel_Y(0, x), (-5, 5), (-5, 5), plot_points=20)
Graphics object consisting of 1 graphics primitive


Evaluate a combination of Bessel functions:

sage: f(x) = bessel_J(1, x) - bessel_Y(0, x)
sage: f(pi)
bessel_J(1, pi) - bessel_Y(0, pi)
sage: f(pi).n()
-0.0437509653365599
sage: f(pi).n(digits=50)
-0.043750965336559909054985168023342675387737118378169


Symbolically solve a second order differential equation with initial conditions $$y(1) = a$$ and $$y'(1) = b$$ in terms of Bessel functions:

sage: y = function('y', x)
sage: a, b = var('a, b')
sage: diffeq = x^2*diff(y,x,x) + x*diff(y,x) + x^2*y == 0
sage: f = desolve(diffeq, y, [1, a, b]); f
(a*bessel_Y(1, 1) + b*bessel_Y(0, 1))*bessel_J(0, x)/(bessel_J(0,
1)*bessel_Y(1, 1) - bessel_J(1, 1)*bessel_Y(0, 1)) -
(a*bessel_J(1, 1) + b*bessel_J(0, 1))*bessel_Y(0, x)/(bessel_J(0,
1)*bessel_Y(1, 1) - bessel_J(1, 1)*bessel_Y(0, 1))


For more examples, see the docstring for Bessel().

AUTHORS:

• Benjamin Jones (2012-12-27): initial version
• Some of the documentation here has been adapted from David Joyner’s original documentation of Sage’s special functions module (2006).

REFERENCES:

sage.functions.bessel.Bessel(*args, **kwds)

A function factory that produces symbolic I, J, K, and Y Bessel functions. There are several ways to call this function:

• Bessel(order, type)
• Bessel(order) – type defaults to ‘J’
• Bessel(order, typ=T)
• Bessel(typ=T) – order is unspecified, this is a 2-parameter function
• Bessel() – order is unspecified, type is ‘J’

where order can be any integer and T must be one of the strings ‘I’, ‘J’, ‘K’, or ‘Y’.

See the EXAMPLES below.

EXAMPLES:

Construction of Bessel functions with various orders and types:

sage: Bessel()
bessel_J
sage: Bessel(1)(x)
bessel_J(1, x)
sage: Bessel(1, 'Y')(x)
bessel_Y(1, x)
sage: Bessel(-2, 'Y')(x)
bessel_Y(-2, x)
sage: Bessel(typ='K')
bessel_K
sage: Bessel(0, typ='I')(x)
bessel_I(0, x)


Evaluation:

sage: f = Bessel(1)
sage: f(3.0)
0.339058958525936
sage: f(3)
bessel_J(1, 3)
sage: f(3).n(digits=50)
0.33905895852593645892551459720647889697308041819801

sage: g = Bessel(typ='J')
sage: g(1,3)
bessel_J(1, 3)
sage: g(2, 3+I).n()
0.634160370148554 + 0.0253384000032695*I
sage: abs(numerical_integral(1/pi*cos(3*sin(x)), 0.0, pi)[0] - Bessel(0, 'J')(3.0)) < 1e-15
True


Symbolic calculus:

sage: f(x) = Bessel(0, 'J')(x)
sage: derivative(f, x)
x |--> -1/2*bessel_J(1, x) + 1/2*bessel_J(-1, x)
sage: derivative(f, x, x)
x |--> 1/4*bessel_J(2, x) - 1/2*bessel_J(0, x) + 1/4*bessel_J(-2, x)


Verify that $$J_0$$ satisfies Bessel’s differential equation numerically using the test_relation() method:

sage: y = bessel_J(0, x)
sage: diffeq = x^2*derivative(y,x,x) + x*derivative(y,x) + x^2*y == 0
sage: diffeq.test_relation(proof=False)
True


Conversion to other systems:

sage: x,y = var('x,y')
sage: f = maxima(Bessel(typ='K')(x,y))
sage: f.derivative('_SAGE_VAR_x')
%pi*csc(%pi*_SAGE_VAR_x)*('diff(bessel_i(-_SAGE_VAR_x,_SAGE_VAR_y),_SAGE_VAR_x,1)-'diff(bessel_i(_SAGE_VAR_x,_SAGE_VAR_y),_SAGE_VAR_x,1))/2-%pi*bessel_k(_SAGE_VAR_x,_SAGE_VAR_y)*cot(%pi*_SAGE_VAR_x)
sage: f.derivative('_SAGE_VAR_y')
-(bessel_k(_SAGE_VAR_x+1,_SAGE_VAR_y)+bessel_k(_SAGE_VAR_x-1,_SAGE_VAR_y))/2


Compute the particular solution to Bessel’s Differential Equation that satisfies $$y(1) = 1$$ and $$y'(1) = 1$$, then verify the initial conditions and plot it:

sage: y = function('y', x)
sage: diffeq = x^2*diff(y,x,x) + x*diff(y,x) + x^2*y == 0
sage: f = desolve(diffeq, y, [1, 1, 1]); f
(bessel_Y(1, 1) + bessel_Y(0, 1))*bessel_J(0, x)/(bessel_J(0,
1)*bessel_Y(1, 1) - bessel_J(1, 1)*bessel_Y(0, 1)) - (bessel_J(1,
1) + bessel_J(0, 1))*bessel_Y(0, x)/(bessel_J(0, 1)*bessel_Y(1, 1)
- bessel_J(1, 1)*bessel_Y(0, 1))
sage: f.subs(x=1).n() # numerical verification
1.00000000000000
sage: fp = f.diff(x)
sage: fp.subs(x=1).n()
1.00000000000000

sage: f.subs(x=1).simplify_full() # symbolic verification
1
sage: fp = f.diff(x)
sage: fp.subs(x=1).simplify_full()
1

sage: plot(f, (x,0,5))
Graphics object consisting of 1 graphics primitive


Plotting:

sage: f(x) = Bessel(0)(x); f
x |--> bessel_J(0, x)
sage: plot(f, (x, 1, 10))
Graphics object consisting of 1 graphics primitive

sage: plot([ Bessel(i, 'J') for i in range(5) ], 2, 10)
Graphics object consisting of 5 graphics primitives

sage: G = Graphics()
sage: G += sum([ plot(Bessel(i), 0, 4*pi, rgbcolor=hue(sin(pi*i/10))) for i in range(5) ])
sage: show(G)


A recreation of Abramowitz and Stegun Figure 9.1:

sage: G  = plot(Bessel(0, 'J'), 0, 15, color='black')
sage: G += plot(Bessel(0, 'Y'), 0, 15, color='black')
sage: G += plot(Bessel(1, 'J'), 0, 15, color='black', linestyle='dotted')
sage: G += plot(Bessel(1, 'Y'), 0, 15, color='black', linestyle='dotted')
sage: show(G, ymin=-1, ymax=1)

class sage.functions.bessel.Function_Bessel_I

The Bessel I function, or the Modified Bessel Function of the First Kind.

DEFINITION:

$I_\nu(x) = i^{-\nu} J_\nu(ix)$

EXAMPLES:

sage: bessel_I(1, x)
bessel_I(1, x)
sage: bessel_I(1.0, 1.0)
0.565159103992485
sage: n = var('n')
sage: bessel_I(n, x)
bessel_I(n, x)
sage: bessel_I(2, I).n()
-0.114903484931900


Examples of symbolic manipulation:

sage: a = bessel_I(pi, bessel_I(1, I))
sage: N(a, digits=20)
0.00026073272117205890528 - 0.0011528954889080572266*I

sage: f = bessel_I(2, x)
sage: f.diff(x)
1/2*bessel_I(3, x) + 1/2*bessel_I(1, x)


Special identities that bessel_I satisfies:

sage: bessel_I(1/2, x)
sqrt(2)*sqrt(1/(pi*x))*sinh(x)
sage: eq = bessel_I(1/2, x) == bessel_I(0.5, x)
sage: eq.test_relation()
True
sage: bessel_I(-1/2, x)
sqrt(2)*sqrt(1/(pi*x))*cosh(x)
sage: eq = bessel_I(-1/2, x) == bessel_I(-0.5, x)
sage: eq.test_relation()
True


Examples of asymptotic behavior:

sage: limit(bessel_I(0, x), x=oo)
+Infinity
sage: limit(bessel_I(0, x), x=0)
1


High precision and complex valued inputs:

sage: bessel_I(0, 1).n(128)
1.2660658777520083355982446252147175376
sage: bessel_I(0, RealField(200)(1))
1.2660658777520083355982446252147175376076703113549622068081
sage: bessel_I(0, ComplexField(200)(0.5+I))
0.80644357583493619472428518415019222845373366024179916785502 + 0.22686958987911161141397453401487525043310874687430711021434*I


Visualization (set plot_points to a higher value to get more detail):

sage: plot(bessel_I(1,x), (x,0,5), color='blue')
Graphics object consisting of 1 graphics primitive
sage: complex_plot(bessel_I(1, x), (-5, 5), (-5, 5), plot_points=20)
Graphics object consisting of 1 graphics primitive


ALGORITHM:

Numerical evaluation is handled by the mpmath library. Symbolics are handled by a combination of Maxima and Sage (Ginac/Pynac).

TESTS:

sage: N(bessel_I(1,1),500)
0.565159103992485027207696027609863307328899621621092009480294489479255640964371134092664997766814410064677886055526302676857637684917179812041131208121


Check whether the return value is real whenever the argument is real (trac ticket #10251):

sage: bessel_I(5, 1.5) in RR
True

class sage.functions.bessel.Function_Bessel_J

The Bessel J Function, denoted by bessel_J($$\nu$$, x) or $$J_\nu(x)$$. As a Taylor series about $$x=0$$ it is equal to:

$J_\nu(x) = \sum_{k=0}^\infty \frac{(-1)^k}{k! \Gamma(k+\nu+1)} \left(\frac{x}{2}\right)^{2k+\nu}$

The parameter $$\nu$$ is called the order and may be any real or complex number; however, integer and half-integer values are most common. It is defined for all complex numbers $$x$$ when $$\nu$$ is an integer or greater than zero and it diverges as $$x \to 0$$ for negative non-integer values of $$\nu$$.

For integer orders $$\nu = n$$ there is an integral representation:

$J_n(x) = \frac{1}{\pi} \int_0^\pi \cos(n t - x \sin(t)) \; dt$

This function also arises as a special case of the hypergeometric function $${}_0F_1$$:

$J_\nu(x) = \frac{x^n}{2^\nu \Gamma(\nu + 1)} {}_0F_1\left(\nu + 1, -\frac{x^2}{4}\right).$

EXAMPLES:

sage: bessel_J(1.0, 1.0)
0.440050585744933
sage: bessel_J(2, I).n(digits=30)
-0.135747669767038281182852569995

sage: bessel_J(1, x)
bessel_J(1, x)
sage: n = var('n')
sage: bessel_J(n, x)
bessel_J(n, x)


Examples of symbolic manipulation:

sage: a = bessel_J(pi, bessel_J(1, I)); a
bessel_J(pi, bessel_J(1, I))
sage: N(a, digits=20)
0.00059023706363796717363 - 0.0026098820470081958110*I

sage: f = bessel_J(2, x)
sage: f.diff(x)
-1/2*bessel_J(3, x) + 1/2*bessel_J(1, x)


Comparison to a well-known integral representation of $$J_1(1)$$:

sage: A = numerical_integral(1/pi*cos(x - sin(x)), 0, pi)
sage: A[0]  # abs tol 1e-14
0.44005058574493355
sage: bessel_J(1.0, 1.0) - A[0] < 1e-15
True


Integration is supported directly and through Maxima:

sage: f = bessel_J(2, x)
sage: f.integrate(x)
1/24*x^3*hypergeometric((3/2,), (5/2, 3), -1/4*x^2)
sage: m = maxima(bessel_J(2, x))
sage: m.integrate(x)
hypergeometric([3/2],[5/2,3],-_SAGE_VAR_x^2/4)*_SAGE_VAR_x^3/24


Visualization (set plot_points to a higher value to get more detail):

sage: plot(bessel_J(1,x), (x,0,5), color='blue')
Graphics object consisting of 1 graphics primitive
sage: complex_plot(bessel_J(1, x), (-5, 5), (-5, 5), plot_points=20)
Graphics object consisting of 1 graphics primitive


ALGORITHM:

Numerical evaluation is handled by the mpmath library. Symbolics are handled by a combination of Maxima and Sage (Ginac/Pynac).

Check whether the return value is real whenever the argument is real (trac ticket #10251):

sage: bessel_J(5, 1.5) in RR
True

class sage.functions.bessel.Function_Bessel_K

The Bessel K function, or the modified Bessel function of the second kind.

DEFINITION:

$K_\nu(x) = \frac{\pi}{2} \frac{I_{-\nu}(x)-I_\nu(x)}{\sin(\nu \pi)}$

EXAMPLES:

sage: bessel_K(1, x)
bessel_K(1, x)
sage: bessel_K(1.0, 1.0)
0.601907230197235
sage: n = var('n')
sage: bessel_K(n, x)
bessel_K(n, x)
sage: bessel_K(2, I).n()
-2.59288617549120 + 0.180489972066962*I


Examples of symbolic manipulation:

sage: a = bessel_K(pi, bessel_K(1, I)); a
bessel_K(pi, bessel_K(1, I))
sage: N(a, digits=20)
3.8507583115005220157 + 0.068528298579883425792*I

sage: f = bessel_K(2, x)
sage: f.diff(x)
-1/2*bessel_K(3, x) - 1/2*bessel_K(1, x)

sage: bessel_K(1/2, x)
bessel_K(1/2, x)
sage: bessel_K(1/2, -1)
bessel_K(1/2, -1)
sage: bessel_K(1/2, 1)
sqrt(1/2)*sqrt(pi)*e^(-1)


Examples of asymptotic behavior:

sage: bessel_K(0, 0.0)
+infinity
sage: limit(bessel_K(0, x), x=0)
+Infinity
sage: limit(bessel_K(0, x), x=oo)
0


High precision and complex valued inputs:

sage: bessel_K(0, 1).n(128)
0.42102443824070833333562737921260903614
sage: bessel_K(0, RealField(200)(1))
0.42102443824070833333562737921260903613621974822666047229897
sage: bessel_K(0, ComplexField(200)(0.5+I))
0.058365979093103864080375311643360048144715516692187818271179 - 0.67645499731334483535184142196073004335768129348518210260256*I


Visualization (set plot_points to a higher value to get more detail):

sage: plot(bessel_K(1,x), (x,0,5), color='blue')
Graphics object consisting of 1 graphics primitive
sage: complex_plot(bessel_K(1, x), (-5, 5), (-5, 5), plot_points=20)
Graphics object consisting of 1 graphics primitive


ALGORITHM:

Numerical evaluation is handled by the mpmath library. Symbolics are handled by a combination of Maxima and Sage (Ginac/Pynac).

TESTS:

Verify that trac ticket #3426 is fixed:

The Bessel K function can be evaluated numerically at complex orders:

sage: bessel_K(10 * I, 10).n()
9.82415743819925e-8


For a fixed imaginary order and increasing, real, second component the value of Bessel K is exponentially decaying:

sage: for x in [10, 20, 50, 100, 200]: print bessel_K(5*I, x).n()
5.27812176514912e-6
3.11005908421801e-10
2.66182488515423e-23 - 8.59622057747552e-58*I
4.11189776828337e-45 - 1.01494840019482e-80*I
1.15159692553603e-88 - 6.75787862113718e-125*I


Check whether the return value is real whenever the argument is real (trac ticket #10251):

sage: bessel_K(5, 1.5) in RR
True

class sage.functions.bessel.Function_Bessel_Y

The Bessel Y functions, also known as the Bessel functions of the second kind, Weber functions, or Neumann functions.

$$Y_\nu(z)$$ is a holomorphic function of $$z$$ on the complex plane, cut along the negative real axis. It is singular at $$z = 0$$. When $$z$$ is fixed, $$Y_\nu(z)$$ is an entire function of the order $$\nu$$.

DEFINITION:

$Y_n(z) = \frac{J_\nu(z) \cos(\nu z) - J_{-\nu}(z)}{\sin(\nu z)}$

Its derivative with respect to $$z$$ is:

$\frac{d}{dz} Y_n(z) = \frac{1}{z^n} \left(z^n Y_{n-1}(z) - n z^{n-1} Y_n(z) \right)$

EXAMPLES:

sage: bessel_Y(1, x)
bessel_Y(1, x)
sage: bessel_Y(1.0, 1.0)
-0.781212821300289
sage: n = var('n')
sage: bessel_Y(n, x)
bessel_Y(n, x)
sage: bessel_Y(2, I).n()
1.03440456978312 - 0.135747669767038*I
sage: bessel_Y(0, 0).n()
-infinity


Examples of symbolic manipulation:

sage: a = bessel_Y(pi, bessel_Y(1, I)); a
bessel_Y(pi, bessel_Y(1, I))
sage: N(a, digits=20)
4.2059146571791095708 + 21.307914215321993526*I

sage: f = bessel_Y(2, x)
sage: f.diff(x)
-1/2*bessel_Y(3, x) + 1/2*bessel_Y(1, x)


High precision and complex valued inputs (see trac ticket #4230):

sage: bessel_Y(0, 1).n(128)
0.088256964215676957982926766023515162828
sage: bessel_Y(0, RealField(200)(1))
0.088256964215676957982926766023515162827817523090675546711044
sage: bessel_Y(0, ComplexField(200)(0.5+I))
0.077763160184438051408593468823822434235010300228009867784073 + 1.0142336049916069152644677682828326441579314239591288411739*I


Visualization (set plot_points to a higher value to get more detail):

sage: plot(bessel_Y(1,x), (x,0,5), color='blue')
Graphics object consisting of 1 graphics primitive
sage: complex_plot(bessel_Y(1, x), (-5, 5), (-5, 5), plot_points=20)
Graphics object consisting of 1 graphics primitive


ALGORITHM:

Numerical evaluation is handled by the mpmath library. Symbolics are handled by a combination of Maxima and Sage (Ginac/Pynac).

Check whether the return value is real whenever the argument is real (trac ticket #10251):

sage: bessel_Y(5, 1.5) in RR
True


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