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:
for an arbitrary complex number (the order).
In this module, denotes the unique solution of Bessel’s equation
which is non-singular at
. This function is known as the Bessel
Function of the First Kind. This function also arises as a special case
of the hypergeometric function
:
The second linearly independent solution to Bessel’s equation (which is
singular at ) is denoted by
and is called the Bessel
Function of the Second Kind:
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:
The Bessel K Function, or the Modified Bessel Function of the Second Kind, is defined by:
We should note here that the above formulas for Bessel Y and K functions
should be understood as limits when is an integer.
It follows from Bessel’s differential equation that the derivative of
with respect to
is:
Another important formulation of the two linearly independent
solutions to Bessel’s equation are the Hankel functions
and
,
defined by:
where is the imaginary unit (and
and
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) 2.404825557695773Plot 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 primitiveVisualize 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 primitiveEvaluate 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) -0.043750965336559909054985168023342675387737118378169Symbolically solve a second order differential equation with initial conditions
and
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:
- Abramowitz and Stegun: Handbook of Mathematical Functions, http://www.math.sfu.ca/~cbm/aands/
- http://en.wikipedia.org/wiki/Bessel_function
- mpmath Library Bessel Functions
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 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 and
, 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)
Bases: sage.symbolic.function.BuiltinFunction
The Bessel I function, or the Modified Bessel Function of the First Kind.
DEFINITION:
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
Bases: sage.symbolic.function.BuiltinFunction
The Bessel J Function, denoted by bessel_J(, x) or
.
As a Taylor series about
it is equal to:
The parameter 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
when
is an integer or greater than zero and it diverges as
for negative non-integer values of
.
For integer orders there is an integral representation:
This function also arises as a special case of the hypergeometric
function :
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 :
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
Bases: sage.symbolic.function.BuiltinFunction
The Bessel K function, or the modified Bessel function of the second kind.
DEFINITION:
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
Bases: sage.symbolic.function.BuiltinFunction
The Bessel Y functions, also known as the Bessel functions of the second kind, Weber functions, or Neumann functions.
is a holomorphic function of
on the complex plane,
cut along the negative real axis. It is singular at
. When
is fixed,
is an entire function of the order
.
DEFINITION:
Its derivative with respect to is:
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
sage: bessel_Y(0, 1).n(128)
0.088256964215676957982926766023515162828
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).
TESTS:
Check whether the return value is real whenever the argument is real (trac ticket #10251):
sage: bessel_Y(5, 1.5) in RR
True
Coercion works correctly (see trac ticket #17130):
sage: r = bessel_Y(RealField(200)(1), 1.0); r
-0.781212821300289
sage: parent(r)
Real Field with 53 bits of precision
sage: r = bessel_Y(RealField(200)(1), 1); r
-0.78121282130028871654715000004796482054990639071644460784383
sage: parent(r)
Real Field with 200 bits of precision