Bessel functions#

This module provides symbolic Bessel and Hankel functions, and their spherical versions. 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 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.

• When solving for separable solutions of Laplace’s equation in spherical coordinates, the radial equation has the form:

$x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} + [x^2 - n(n+1)]y = 0.$

The spherical Bessel functions $$j_n$$ and $$y_n$$, are two linearly independent solutions to this equation. They are related to the ordinary Bessel functions $$J_n$$ and $$Y_n$$ by:

$j_n(x) = \sqrt{\frac{\pi}{2x}} J_{n+1/2}(x),$
$y_n(x) = \sqrt{\frac{\pi}{2x}} Y_{n+1/2}(x) = (-1)^{n+1} \sqrt{\frac{\pi}{2x}} J_{-n-1/2}(x).$

EXAMPLES:

Evaluate the Bessel J function symbolically and numerically:

sage: # needs sage.symbolic
sage: bessel_J(0, x)
bessel_J(0, x)
sage: bessel_J(0, 0)
1
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)                                            # needs scipy
2.404825557695773

>>> from sage.all import *
>>> # needs sage.symbolic
>>> bessel_J(Integer(0), x)
bessel_J(0, x)
>>> bessel_J(Integer(0), Integer(0))
1
>>> bessel_J(Integer(0), x).diff(x)
-1/2*bessel_J(1, x) + 1/2*bessel_J(-1, x)
>>> N(bessel_J(Integer(0), Integer(0)), digits=Integer(20))
1.0000000000000000000
>>> find_root(bessel_J(Integer(0),x), Integer(0), Integer(5))                                            # needs scipy
2.404825557695773


Plot the Bessel J function:

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

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(Bessel(Integer(0))(x)).function(x); f                                                    # needs sage.symbolic
x |--> bessel_J(0, x)
>>> plot(f, (x, Integer(1), Integer(10)))                                                       # needs sage.plot sage.symbolic
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)            # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> complex_plot(bessel_Y(Integer(0), x), (-Integer(5), Integer(5)), (-Integer(5), Integer(5)), plot_points=Integer(20))            # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive


Evaluate a combination of Bessel functions:

sage: # needs sage.symbolic
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

>>> from sage.all import *
>>> # needs sage.symbolic
>>> __tmp__=var("x"); f = symbolic_expression(bessel_J(Integer(1), x) - bessel_Y(Integer(0), x)).function(x)
>>> f(pi)
bessel_J(1, pi) - bessel_Y(0, pi)
>>> f(pi).n()
-0.0437509653365599
>>> f(pi).n(digits=Integer(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: # needs sage.symbolic
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))

>>> from sage.all import *
>>> # needs sage.symbolic
>>> y = function('y')(x)
>>> a, b = var('a, b')
>>> diffeq = x**Integer(2)*diff(y,x,x) + x*diff(y,x) + x**Integer(2)*y == Integer(0)
>>> f = desolve(diffeq, y, [Integer(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:

• 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)[source]#

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(typ='K')
bessel_K

sage: # needs sage.symbolic
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(0, typ='I')(x)
bessel_I(0, x)

>>> from sage.all import *
>>> Bessel()
bessel_J
>>> Bessel(typ='K')
bessel_K

>>> # needs sage.symbolic
>>> Bessel(Integer(1))(x)
bessel_J(1, x)
>>> Bessel(Integer(1), 'Y')(x)
bessel_Y(1, x)
>>> Bessel(-Integer(2), 'Y')(x)
bessel_Y(-2, x)
>>> Bessel(Integer(0), typ='I')(x)
bessel_I(0, x)


Evaluation:

sage: f = Bessel(1)
sage: f(3.0)                                                                    # needs mpmath
0.339058958525936

sage: # needs sage.symbolic
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

>>> from sage.all import *
>>> f = Bessel(Integer(1))
>>> f(RealNumber('3.0'))                                                                    # needs mpmath
0.339058958525936

>>> # needs sage.symbolic
>>> f(Integer(3))
bessel_J(1, 3)
>>> f(Integer(3)).n(digits=Integer(50))
0.33905895852593645892551459720647889697308041819801
>>> g = Bessel(typ='J')
>>> g(Integer(1),Integer(3))
bessel_J(1, 3)
>>> g(Integer(2), Integer(3)+I).n()
0.634160370148554 + 0.0253384000032695*I
>>> abs(numerical_integral(Integer(1)/pi*cos(Integer(3)*sin(x)), RealNumber('0.0'), pi)[Integer(0)]
...      - Bessel(Integer(0), 'J')(RealNumber('3.0'))) < RealNumber('1e-15')
True


Symbolic calculus:

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

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(Bessel(Integer(0), 'J')(x)                                                  ).function(x)# needs sage.symbolic
>>> derivative(f, x)                                                          # needs sage.symbolic
x |--> -1/2*bessel_J(1, x) + 1/2*bessel_J(-1, x)
>>> derivative(f, x, x)                                                       # needs sage.symbolic
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)                                                        # needs sage.symbolic
sage: diffeq = x^2*derivative(y,x,x) + x*derivative(y,x) + x^2*y == 0           # needs sage.symbolic
sage: diffeq.test_relation(proof=False)                                         # needs sage.symbolic
True

>>> from sage.all import *
>>> y = bessel_J(Integer(0), x)                                                        # needs sage.symbolic
>>> diffeq = x**Integer(2)*derivative(y,x,x) + x*derivative(y,x) + x**Integer(2)*y == Integer(0)           # needs sage.symbolic
>>> diffeq.test_relation(proof=False)                                         # needs sage.symbolic
True


Conversion to other systems:

sage: # needs sage.symbolic
sage: x,y = var('x,y')
sage: f = Bessel(typ='K')(x,y)
sage: expected = f.derivative(y)
sage: actual = maxima(f).derivative('_SAGE_VAR_y').sage()
sage: bool(actual == expected)
True

>>> from sage.all import *
>>> # needs sage.symbolic
>>> x,y = var('x,y')
>>> f = Bessel(typ='K')(x,y)
>>> expected = f.derivative(y)
>>> actual = maxima(f).derivative('_SAGE_VAR_y').sage()
>>> bool(actual == expected)
True


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

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

>>> from sage.all import *
>>> # needs sage.symbolic
>>> y = function('y')(x)
>>> diffeq = x**Integer(2)*diff(y,x,x) + x*diff(y,x) + x**Integer(2)*y == Integer(0)
>>> f = desolve(diffeq, y, [Integer(1), Integer(1), Integer(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))
>>> f.subs(x=Integer(1)).n()  # numerical verification
1.00000000000000
>>> fp = f.diff(x)
>>> fp.subs(x=Integer(1)).n()
1.00000000000000

>>> f.subs(x=Integer(1)).simplify_full()  # symbolic verification                      # needs sage.symbolic
1
>>> fp = f.diff(x)                                                            # needs sage.symbolic
>>> fp.subs(x=Integer(1)).simplify_full()                                              # needs sage.symbolic
1

>>> plot(f, (x,Integer(0),Integer(5)))                                                          # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive


Plotting:

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

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

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

>>> from sage.all import *
>>> __tmp__=var("x"); f = symbolic_expression(Bessel(Integer(0))(x)).function(x); f                                                    # needs sage.symbolic
x |--> bessel_J(0, x)
>>> plot(f, (x, Integer(1), Integer(10)))                                                       # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive

>>> plot([Bessel(i, 'J') for i in range(Integer(5))], Integer(2), Integer(10))                           # needs sage.plot
Graphics object consisting of 5 graphics primitives

>>> G = Graphics()                                                            # needs sage.plot
>>> G += sum(plot(Bessel(i), Integer(0), Integer(4)*pi, rgbcolor=hue(sin(pi*i/Integer(10))))             # needs sage.plot sage.symbolic
...          for i in range(Integer(5)))
>>> show(G)                                                                   # needs sage.plot


A recreation of Abramowitz and Stegun Figure 9.1:

sage: # needs sage.plot sage.symbolic
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)

>>> from sage.all import *
>>> # needs sage.plot sage.symbolic
>>> G  = plot(Bessel(Integer(0), 'J'), Integer(0), Integer(15), color='black')
>>> G += plot(Bessel(Integer(0), 'Y'), Integer(0), Integer(15), color='black')
>>> G += plot(Bessel(Integer(1), 'J'), Integer(0), Integer(15), color='black', linestyle='dotted')
>>> G += plot(Bessel(Integer(1), 'Y'), Integer(0), Integer(15), color='black', linestyle='dotted')
>>> show(G, ymin=-Integer(1), ymax=Integer(1))

class sage.functions.bessel.Function_Bessel_I[source]#

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(1.0, 1.0)                                                        # needs mpmath
0.565159103992485

sage: # needs sage.symbolic
sage: bessel_I(1, x)
bessel_I(1, x)
sage: n = var('n')
sage: bessel_I(n, x)
bessel_I(n, x)
sage: bessel_I(2, I).n()
-0.114903484931900

>>> from sage.all import *
>>> bessel_I(RealNumber('1.0'), RealNumber('1.0'))                                                        # needs mpmath
0.565159103992485

>>> # needs sage.symbolic
>>> bessel_I(Integer(1), x)
bessel_I(1, x)
>>> n = var('n')
>>> bessel_I(n, x)
bessel_I(n, x)
>>> bessel_I(Integer(2), I).n()
-0.114903484931900


Examples of symbolic manipulation:

sage: # needs sage.symbolic
sage: a = bessel_I(pi, bessel_I(1, I))
sage: N(a, digits=20)
0.00026073272117205890524 - 0.0011528954889080572268*I
sage: f = bessel_I(2, x)
sage: f.diff(x)
1/2*bessel_I(3, x) + 1/2*bessel_I(1, x)

>>> from sage.all import *
>>> # needs sage.symbolic
>>> a = bessel_I(pi, bessel_I(Integer(1), I))
>>> N(a, digits=Integer(20))
0.00026073272117205890524 - 0.0011528954889080572268*I
>>> f = bessel_I(Integer(2), x)
>>> f.diff(x)
1/2*bessel_I(3, x) + 1/2*bessel_I(1, x)


Special identities that bessel_I satisfies:

sage: # needs sage.symbolic
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

>>> from sage.all import *
>>> # needs sage.symbolic
>>> bessel_I(Integer(1)/Integer(2), x)
sqrt(2)*sqrt(1/(pi*x))*sinh(x)
>>> eq = bessel_I(Integer(1)/Integer(2), x) == bessel_I(RealNumber('0.5'), x)
>>> eq.test_relation()
True
>>> bessel_I(-Integer(1)/Integer(2), x)
sqrt(2)*sqrt(1/(pi*x))*cosh(x)
>>> eq = bessel_I(-Integer(1)/Integer(2), x) == bessel_I(-RealNumber('0.5'), x)
>>> eq.test_relation()
True


Examples of asymptotic behavior:

sage: limit(bessel_I(0, x), x=oo)                                               # needs sage.symbolic
+Infinity
sage: limit(bessel_I(0, x), x=0)                                                # needs sage.symbolic
1

>>> from sage.all import *
>>> limit(bessel_I(Integer(0), x), x=oo)                                               # needs sage.symbolic
+Infinity
>>> limit(bessel_I(Integer(0), x), x=Integer(0))                                                # needs sage.symbolic
1


High precision and complex valued inputs:

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

>>> from sage.all import *
>>> bessel_I(Integer(0), Integer(1)).n(Integer(128))                                                     # needs sage.symbolic
1.2660658777520083355982446252147175376
>>> bessel_I(Integer(0), RealField(Integer(200))(Integer(1)))                                            # needs sage.rings.real_mpfr
1.2660658777520083355982446252147175376076703113549622068081
>>> bessel_I(Integer(0), ComplexField(Integer(200))(RealNumber('0.5')+I))                                     # needs sage.symbolic
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')                             # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive
sage: complex_plot(bessel_I(1, x), (-5, 5), (-5, 5), plot_points=20)            # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> plot(bessel_I(Integer(1), x), (x, Integer(0), Integer(5)), color='blue')                             # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive
>>> complex_plot(bessel_I(Integer(1), x), (-Integer(5), Integer(5)), (-Integer(5), Integer(5)), plot_points=Integer(20))            # needs sage.plot sage.symbolic
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).

REFERENCES:

class sage.functions.bessel.Function_Bessel_J[source]#

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)                                                        # needs mpmath
0.440050585744933

sage: # needs sage.symbolic
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)

>>> from sage.all import *
>>> bessel_J(RealNumber('1.0'), RealNumber('1.0'))                                                        # needs mpmath
0.440050585744933

>>> # needs sage.symbolic
>>> bessel_J(Integer(2), I).n(digits=Integer(30))
-0.135747669767038281182852569995
>>> bessel_J(Integer(1), x)
bessel_J(1, x)
>>> n = var('n')
>>> bessel_J(n, x)
bessel_J(n, x)


Examples of symbolic manipulation:

sage: # needs sage.symbolic
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)

>>> from sage.all import *
>>> # needs sage.symbolic
>>> a = bessel_J(pi, bessel_J(Integer(1), I)); a
bessel_J(pi, bessel_J(1, I))
>>> N(a, digits=Integer(20))
0.00059023706363796717363 - 0.0026098820470081958110*I
>>> f = bessel_J(Integer(2), x)
>>> 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)                       # needs sage.symbolic
sage: A[0]  # abs tol 1e-14                                                     # needs sage.symbolic
0.44005058574493355
sage: bessel_J(1.0, 1.0) - A[0] < 1e-15                                         # needs sage.symbolic
True

>>> from sage.all import *
>>> A = numerical_integral(Integer(1)/pi*cos(x - sin(x)), Integer(0), pi)                       # needs sage.symbolic
>>> A[Integer(0)]  # abs tol 1e-14                                                     # needs sage.symbolic
0.44005058574493355
>>> bessel_J(RealNumber('1.0'), RealNumber('1.0')) - A[Integer(0)] < RealNumber('1e-15')                                         # needs sage.symbolic
True


Integration is supported directly and through Maxima:

sage: f = bessel_J(2, x)                                                        # needs sage.symbolic
sage: f.integrate(x)                                                            # needs sage.symbolic
1/24*x^3*hypergeometric((3/2,), (5/2, 3), -1/4*x^2)

>>> from sage.all import *
>>> f = bessel_J(Integer(2), x)                                                        # needs sage.symbolic
>>> f.integrate(x)                                                            # needs sage.symbolic
1/24*x^3*hypergeometric((3/2,), (5/2, 3), -1/4*x^2)


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

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

>>> from sage.all import *
>>> plot(bessel_J(Integer(1),x), (x,Integer(0),Integer(5)), color='blue')                                # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive
>>> complex_plot(bessel_J(Integer(1), x), (-Integer(5), Integer(5)), (-Integer(5), Integer(5)), plot_points=Integer(20))            # needs sage.plot sage.symbolic
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 (Issue #10251):

sage: bessel_J(5, 1.5) in RR                                                    # needs mpmath
True

>>> from sage.all import *
>>> bessel_J(Integer(5), RealNumber('1.5')) in RR                                                    # needs mpmath
True


REFERENCES:

class sage.functions.bessel.Function_Bessel_K[source]#

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(1.0, 1.0)                                                        # needs mpmath
0.601907230197235

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

>>> from sage.all import *
>>> bessel_K(RealNumber('1.0'), RealNumber('1.0'))                                                        # needs mpmath
0.601907230197235

>>> # needs sage.symbolic
>>> bessel_K(Integer(1), x)
bessel_K(1, x)
>>> n = var('n')
>>> bessel_K(n, x)
bessel_K(n, x)
>>> bessel_K(Integer(2), I).n()
-2.59288617549120 + 0.180489972066962*I


Examples of symbolic manipulation:

sage: # needs sage.symbolic
sage: a = bessel_K(pi, bessel_K(1, I)); a
bessel_K(pi, bessel_K(1, I))
sage: N(a, digits=20)
3.8507583115005220156 + 0.068528298579883425456*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)
sqrt(1/2)*sqrt(pi)*e^(-x)/sqrt(x)
sage: bessel_K(1/2, -1)
-I*sqrt(1/2)*sqrt(pi)*e
sage: bessel_K(1/2, 1)
sqrt(1/2)*sqrt(pi)*e^(-1)

>>> from sage.all import *
>>> # needs sage.symbolic
>>> a = bessel_K(pi, bessel_K(Integer(1), I)); a
bessel_K(pi, bessel_K(1, I))
>>> N(a, digits=Integer(20))
3.8507583115005220156 + 0.068528298579883425456*I
>>> f = bessel_K(Integer(2), x)
>>> f.diff(x)
-1/2*bessel_K(3, x) - 1/2*bessel_K(1, x)
>>> bessel_K(Integer(1)/Integer(2), x)
sqrt(1/2)*sqrt(pi)*e^(-x)/sqrt(x)
>>> bessel_K(Integer(1)/Integer(2), -Integer(1))
-I*sqrt(1/2)*sqrt(pi)*e
>>> bessel_K(Integer(1)/Integer(2), Integer(1))
sqrt(1/2)*sqrt(pi)*e^(-1)


Examples of asymptotic behavior:

sage: bessel_K(0, 0.0)                                                          # needs mpmath
+infinity
sage: limit(bessel_K(0, x), x=0)                                                # needs sage.symbolic
+Infinity
sage: limit(bessel_K(0, x), x=oo)                                               # needs sage.symbolic
0

>>> from sage.all import *
>>> bessel_K(Integer(0), RealNumber('0.0'))                                                          # needs mpmath
+infinity
>>> limit(bessel_K(Integer(0), x), x=Integer(0))                                                # needs sage.symbolic
+Infinity
>>> limit(bessel_K(Integer(0), x), x=oo)                                               # needs sage.symbolic
0


High precision and complex valued inputs:

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

>>> from sage.all import *
>>> bessel_K(Integer(0), Integer(1)).n(Integer(128))                                                     # needs sage.symbolic
0.42102443824070833333562737921260903614
>>> bessel_K(Integer(0), RealField(Integer(200))(Integer(1)))                                            # needs sage.rings.real_mpfr
0.42102443824070833333562737921260903613621974822666047229897
>>> bessel_K(Integer(0), ComplexField(Integer(200))(RealNumber('0.5')+I))                                     # needs sage.rings.real_mpfr sage.symbolic
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')                                # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive
sage: complex_plot(bessel_K(1, x), (-5, 5), (-5, 5), plot_points=20)            # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> plot(bessel_K(Integer(1),x), (x,Integer(0),Integer(5)), color='blue')                                # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive
>>> complex_plot(bessel_K(Integer(1), x), (-Integer(5), Integer(5)), (-Integer(5), Integer(5)), plot_points=Integer(20))            # needs sage.plot sage.symbolic
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).

REFERENCES:

class sage.functions.bessel.Function_Bessel_Y[source]#

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)                                                            # needs sage.symbolic
bessel_Y(1, x)
sage: bessel_Y(1.0, 1.0)                                                        # needs mpmath
-0.781212821300289

sage: # needs sage.symbolic
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

>>> from sage.all import *
>>> bessel_Y(Integer(1), x)                                                            # needs sage.symbolic
bessel_Y(1, x)
>>> bessel_Y(RealNumber('1.0'), RealNumber('1.0'))                                                        # needs mpmath
-0.781212821300289

>>> # needs sage.symbolic
>>> n = var('n')
>>> bessel_Y(n, x)
bessel_Y(n, x)
>>> bessel_Y(Integer(2), I).n()
1.03440456978312 - 0.135747669767038*I
>>> bessel_Y(Integer(0), Integer(0)).n()
-infinity
>>> bessel_Y(Integer(0), Integer(1)).n(Integer(128))
0.088256964215676957982926766023515162828


Examples of symbolic manipulation:

sage: # needs sage.symbolic
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)

>>> from sage.all import *
>>> # needs sage.symbolic
>>> a = bessel_Y(pi, bessel_Y(Integer(1), I)); a
bessel_Y(pi, bessel_Y(1, I))
>>> N(a, digits=Integer(20))
4.2059146571791095708 + 21.307914215321993526*I
>>> f = bessel_Y(Integer(2), x)
>>> f.diff(x)
-1/2*bessel_Y(3, x) + 1/2*bessel_Y(1, x)


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

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

>>> from sage.all import *
>>> bessel_Y(Integer(0), Integer(1)).n(Integer(128))                                                     # needs sage.symbolic
0.088256964215676957982926766023515162828
>>> bessel_Y(Integer(0), RealField(Integer(200))(Integer(1)))                                            # needs sage.rings.real_mpfr
0.088256964215676957982926766023515162827817523090675546711044
>>> bessel_Y(Integer(0), ComplexField(Integer(200))(RealNumber('0.5')+I))                                     # needs sage.symbolic
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')                             # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive
sage: complex_plot(bessel_Y(1, x), (-5, 5), (-5, 5), plot_points=20)            # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> plot(bessel_Y(Integer(1), x), (x, Integer(0), Integer(5)), color='blue')                             # needs sage.plot sage.symbolic
Graphics object consisting of 1 graphics primitive
>>> complex_plot(bessel_Y(Integer(1), x), (-Integer(5), Integer(5)), (-Integer(5), Integer(5)), plot_points=Integer(20))            # needs sage.plot sage.symbolic
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).

REFERENCES:

class sage.functions.bessel.Function_Hankel1[source]#

The Hankel function of the first kind

DEFINITION:

$H_\nu^{(1)}(z) = J_{\nu}(z) + iY_{\nu}(z)$

EXAMPLES:

sage: hankel1(3, x)                                                             # needs sage.symbolic
hankel1(3, x)
sage: hankel1(3, 4.)                                                            # needs mpmath
0.430171473875622 - 0.182022115953485*I
sage: latex(hankel1(3, x))                                                      # needs sage.symbolic
H_{3}^{(1)}\left(x\right)
sage: hankel1(3., x).series(x == 2, 10).subs(x=3).n()  # abs tol 1e-12          # needs sage.symbolic
0.309062682819597 - 0.512591541605233*I
sage: hankel1(3, 3.)                                                            # needs mpmath
0.309062722255252 - 0.538541616105032*I

>>> from sage.all import *
>>> hankel1(Integer(3), x)                                                             # needs sage.symbolic
hankel1(3, x)
>>> hankel1(Integer(3), RealNumber('4.'))                                                            # needs mpmath
0.430171473875622 - 0.182022115953485*I
>>> latex(hankel1(Integer(3), x))                                                      # needs sage.symbolic
H_{3}^{(1)}\left(x\right)
>>> hankel1(RealNumber('3.'), x).series(x == Integer(2), Integer(10)).subs(x=Integer(3)).n()  # abs tol 1e-12          # needs sage.symbolic
0.309062682819597 - 0.512591541605233*I
>>> hankel1(Integer(3), RealNumber('3.'))                                                            # needs mpmath
0.309062722255252 - 0.538541616105032*I


REFERENCES:

class sage.functions.bessel.Function_Hankel2[source]#

The Hankel function of the second kind

DEFINITION:

$H_\nu^{(2)}(z) = J_{\nu}(z) - iY_{\nu}(z)$

EXAMPLES:

sage: hankel2(3, x)                                                             # needs sage.symbolic
hankel2(3, x)
sage: hankel2(3, 4.)                                                            # needs mpmath
0.430171473875622 + 0.182022115953485*I
sage: latex(hankel2(3, x))                                                      # needs sage.symbolic
H_{3}^{(2)}\left(x\right)
sage: hankel2(3., x).series(x == 2, 10).subs(x=3).n()  # abs tol 1e-12          # needs sage.symbolic
0.309062682819597 + 0.512591541605234*I
sage: hankel2(3, 3.)                                                            # needs mpmath
0.309062722255252 + 0.538541616105032*I

>>> from sage.all import *
>>> hankel2(Integer(3), x)                                                             # needs sage.symbolic
hankel2(3, x)
>>> hankel2(Integer(3), RealNumber('4.'))                                                            # needs mpmath
0.430171473875622 + 0.182022115953485*I
>>> latex(hankel2(Integer(3), x))                                                      # needs sage.symbolic
H_{3}^{(2)}\left(x\right)
>>> hankel2(RealNumber('3.'), x).series(x == Integer(2), Integer(10)).subs(x=Integer(3)).n()  # abs tol 1e-12          # needs sage.symbolic
0.309062682819597 + 0.512591541605234*I
>>> hankel2(Integer(3), RealNumber('3.'))                                                            # needs mpmath
0.309062722255252 + 0.538541616105032*I


REFERENCES:

class sage.functions.bessel.Function_Struve_H[source]#

The Struve functions, solutions to the non-homogeneous Bessel differential equation:

$x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-\alpha^2)y=\frac{4\bigl(\frac{x}{2}\bigr)^{\alpha+1}}{\sqrt\pi\Gamma(\alpha+\tfrac12)},$
$\mathrm{H}_\alpha(x) = y(x)$

EXAMPLES:

sage: struve_H(-1/2, x)                                                         # needs sage.symbolic
sqrt(2)*sqrt(1/(pi*x))*sin(x)
sage: struve_H(2, x)                                                            # needs sage.symbolic
struve_H(2, x)
sage: struve_H(1/2, pi).n()                                                     # needs sage.symbolic
0.900316316157106

>>> from sage.all import *
>>> struve_H(-Integer(1)/Integer(2), x)                                                         # needs sage.symbolic
sqrt(2)*sqrt(1/(pi*x))*sin(x)
>>> struve_H(Integer(2), x)                                                            # needs sage.symbolic
struve_H(2, x)
>>> struve_H(Integer(1)/Integer(2), pi).n()                                                     # needs sage.symbolic
0.900316316157106


REFERENCES:

class sage.functions.bessel.Function_Struve_L[source]#

The modified Struve functions.

$\mathrm{L}_\alpha(x) = -i\cdot e^{-\frac{i\alpha\pi}{2}}\cdot\mathrm{H}_\alpha(ix)$

EXAMPLES:

sage: struve_L(2, x)                                                            # needs sage.symbolic
struve_L(2, x)
sage: struve_L(1/2, pi).n()                                                     # needs sage.symbolic
4.76805417696286
sage: diff(struve_L(1,x), x)                                                    # needs sage.symbolic
1/3*x/pi - 1/2*struve_L(2, x) + 1/2*struve_L(0, x)

>>> from sage.all import *
>>> struve_L(Integer(2), x)                                                            # needs sage.symbolic
struve_L(2, x)
>>> struve_L(Integer(1)/Integer(2), pi).n()                                                     # needs sage.symbolic
4.76805417696286
>>> diff(struve_L(Integer(1),x), x)                                                    # needs sage.symbolic
1/3*x/pi - 1/2*struve_L(2, x) + 1/2*struve_L(0, x)


REFERENCES:

class sage.functions.bessel.SphericalBesselJ[source]#

The spherical Bessel function of the first kind

DEFINITION:

$j_n(z) = \sqrt{\frac{\pi}{2z}} \,J_{n + \frac{1}{2}}(z)$

EXAMPLES:

sage: spherical_bessel_J(3, 3.)                                                 # needs mpmath
0.152051662030533
sage: spherical_bessel_J(2.,3.)      # rel tol 1e-10                            # needs mpmath
0.2986374970757335

sage: # needs sage.symbolic
sage: spherical_bessel_J(3, x)
spherical_bessel_J(3, x)
sage: spherical_bessel_J(3 + 0.2 * I, 3)
0.150770999183897 - 0.0260662466510632*I
sage: spherical_bessel_J(3, x).series(x == 2, 10).subs(x=3).n()
0.152051648665037
sage: spherical_bessel_J(4, x).simplify()
-((45/x^2 - 105/x^4 - 1)*sin(x) + 5*(21/x^2 - 2)*cos(x)/x)/x
sage: integrate(spherical_bessel_J(1,x)^2,(x,0,oo))
1/6*pi
sage: latex(spherical_bessel_J(4, x))
j_{4}\left(x\right)

>>> from sage.all import *
>>> spherical_bessel_J(Integer(3), RealNumber('3.'))                                                 # needs mpmath
0.152051662030533
>>> spherical_bessel_J(RealNumber('2.'),RealNumber('3.'))      # rel tol 1e-10                            # needs mpmath
0.2986374970757335

>>> # needs sage.symbolic
>>> spherical_bessel_J(Integer(3), x)
spherical_bessel_J(3, x)
>>> spherical_bessel_J(Integer(3) + RealNumber('0.2') * I, Integer(3))
0.150770999183897 - 0.0260662466510632*I
>>> spherical_bessel_J(Integer(3), x).series(x == Integer(2), Integer(10)).subs(x=Integer(3)).n()
0.152051648665037
>>> spherical_bessel_J(Integer(4), x).simplify()
-((45/x^2 - 105/x^4 - 1)*sin(x) + 5*(21/x^2 - 2)*cos(x)/x)/x
>>> integrate(spherical_bessel_J(Integer(1),x)**Integer(2),(x,Integer(0),oo))
1/6*pi
>>> latex(spherical_bessel_J(Integer(4), x))
j_{4}\left(x\right)


REFERENCES:

class sage.functions.bessel.SphericalBesselY[source]#

The spherical Bessel function of the second kind

DEFINITION:

$y_n(z) = \sqrt{\frac{\pi}{2z}} \,Y_{n + \frac{1}{2}}(z)$

EXAMPLES:

sage: # needs sage.symbolic
sage: spherical_bessel_Y(3, x)
spherical_bessel_Y(3, x)
sage: spherical_bessel_Y(3 + 0.2 * I, 3)
-0.505215297588210 - 0.0508835883281404*I
sage: spherical_bessel_Y(-3, x).simplify()
((3/x^2 - 1)*sin(x) - 3*cos(x)/x)/x
sage: spherical_bessel_Y(3 + 2 * I, 5 - 0.2 * I)
-0.270205813266440 - 0.615994702714957*I
sage: integrate(spherical_bessel_Y(0, x), x)
-1/2*Ei(I*x) - 1/2*Ei(-I*x)
sage: integrate(spherical_bessel_Y(1,x)^2,(x,0,oo))
-1/6*pi
sage: latex(spherical_bessel_Y(0, x))
y_{0}\left(x\right)

>>> from sage.all import *
>>> # needs sage.symbolic
>>> spherical_bessel_Y(Integer(3), x)
spherical_bessel_Y(3, x)
>>> spherical_bessel_Y(Integer(3) + RealNumber('0.2') * I, Integer(3))
-0.505215297588210 - 0.0508835883281404*I
>>> spherical_bessel_Y(-Integer(3), x).simplify()
((3/x^2 - 1)*sin(x) - 3*cos(x)/x)/x
>>> spherical_bessel_Y(Integer(3) + Integer(2) * I, Integer(5) - RealNumber('0.2') * I)
-0.270205813266440 - 0.615994702714957*I
>>> integrate(spherical_bessel_Y(Integer(0), x), x)
-1/2*Ei(I*x) - 1/2*Ei(-I*x)
>>> integrate(spherical_bessel_Y(Integer(1),x)**Integer(2),(x,Integer(0),oo))
-1/6*pi
>>> latex(spherical_bessel_Y(Integer(0), x))
y_{0}\left(x\right)


REFERENCES:

class sage.functions.bessel.SphericalHankel1[source]#

The spherical Hankel function of the first kind

DEFINITION:

$h_n^{(1)}(z) = \sqrt{\frac{\pi}{2z}} \,H_{n + \frac{1}{2}}^{(1)}(z)$

EXAMPLES:

sage: # needs sage.symbolic
sage: spherical_hankel1(3, x)
spherical_hankel1(3, x)
sage: spherical_hankel1(3 + 0.2 * I, 3)
0.201654587512037 - 0.531281544239273*I
sage: spherical_hankel1(1, x).simplify()
-(x + I)*e^(I*x)/x^2
sage: spherical_hankel1(3 + 2 * I, 5 - 0.2 * I)
1.25375216869913 - 0.518011435921789*I
sage: integrate(spherical_hankel1(3, x), x)
Ei(I*x) - 6*gamma(-1, -I*x) - 15*gamma(-2, -I*x) - 15*gamma(-3, -I*x)
sage: latex(spherical_hankel1(3, x))
h_{3}^{(1)}\left(x\right)

>>> from sage.all import *
>>> # needs sage.symbolic
>>> spherical_hankel1(Integer(3), x)
spherical_hankel1(3, x)
>>> spherical_hankel1(Integer(3) + RealNumber('0.2') * I, Integer(3))
0.201654587512037 - 0.531281544239273*I
>>> spherical_hankel1(Integer(1), x).simplify()
-(x + I)*e^(I*x)/x^2
>>> spherical_hankel1(Integer(3) + Integer(2) * I, Integer(5) - RealNumber('0.2') * I)
1.25375216869913 - 0.518011435921789*I
>>> integrate(spherical_hankel1(Integer(3), x), x)
Ei(I*x) - 6*gamma(-1, -I*x) - 15*gamma(-2, -I*x) - 15*gamma(-3, -I*x)
>>> latex(spherical_hankel1(Integer(3), x))
h_{3}^{(1)}\left(x\right)


REFERENCES:

class sage.functions.bessel.SphericalHankel2[source]#

The spherical Hankel function of the second kind

DEFINITION:

$h_n^{(2)}(z) = \sqrt{\frac{\pi}{2z}} \,H_{n + \frac{1}{2}}^{(2)}(z)$

EXAMPLES:

sage: # needs sage.symbolic
sage: spherical_hankel2(3, x)
spherical_hankel2(3, x)
sage: spherical_hankel2(3 + 0.2 * I, 3)
0.0998874108557565 + 0.479149050937147*I
sage: spherical_hankel2(1, x).simplify()
-(x - I)*e^(-I*x)/x^2
sage: spherical_hankel2(2,i).simplify()
-e
sage: spherical_hankel2(2,x).simplify()
(-I*x^2 - 3*x + 3*I)*e^(-I*x)/x^3
sage: spherical_hankel2(3 + 2*I, 5 - 0.2*I)
0.0217627632692163 + 0.0224001906110906*I
sage: integrate(spherical_hankel2(3, x), x)
Ei(-I*x) - 6*gamma(-1, I*x) - 15*gamma(-2, I*x) - 15*gamma(-3, I*x)
sage: latex(spherical_hankel2(3, x))
h_{3}^{(2)}\left(x\right)

>>> from sage.all import *
>>> # needs sage.symbolic
>>> spherical_hankel2(Integer(3), x)
spherical_hankel2(3, x)
>>> spherical_hankel2(Integer(3) + RealNumber('0.2') * I, Integer(3))
0.0998874108557565 + 0.479149050937147*I
>>> spherical_hankel2(Integer(1), x).simplify()
-(x - I)*e^(-I*x)/x^2
>>> spherical_hankel2(Integer(2),i).simplify()
-e
>>> spherical_hankel2(Integer(2),x).simplify()
(-I*x^2 - 3*x + 3*I)*e^(-I*x)/x^3
>>> spherical_hankel2(Integer(3) + Integer(2)*I, Integer(5) - RealNumber('0.2')*I)
0.0217627632692163 + 0.0224001906110906*I
>>> integrate(spherical_hankel2(Integer(3), x), x)
Ei(-I*x) - 6*gamma(-1, I*x) - 15*gamma(-2, I*x) - 15*gamma(-3, I*x)
>>> latex(spherical_hankel2(Integer(3), x))
h_{3}^{(2)}\left(x\right)


REFERENCES:

sage.functions.bessel.spherical_bessel_f(F, n, z)[source]#

Numerically evaluate the spherical version, $$f$$, of the Bessel function $$F$$ by computing $$f_n(z) = \sqrt{\frac{1}{2}\pi/z} F_{n + \frac{1}{2}}(z)$$. According to Abramowitz & Stegun, this identity holds for the Bessel functions $$J$$, $$Y$$, $$K$$, $$I$$, $$H^{(1)}$$, and $$H^{(2)}$$.

EXAMPLES:

sage: from sage.functions.bessel import spherical_bessel_f
sage: spherical_bessel_f('besselj', 3, 4)                                       # needs mpmath
mpf('0.22924385795503024')
sage: spherical_bessel_f('hankel1', 3, 4)                                       # needs mpmath
mpc(real='0.22924385795503024', imag='-0.21864196590306359')

>>> from sage.all import *
>>> from sage.functions.bessel import spherical_bessel_f
>>> spherical_bessel_f('besselj', Integer(3), Integer(4))                                       # needs mpmath
mpf('0.22924385795503024')
>>> spherical_bessel_f('hankel1', Integer(3), Integer(4))                                       # needs mpmath
mpc(real='0.22924385795503024', imag='-0.21864196590306359')