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_J(n, x) – The Bessel J function

• bessel_Y(n, x) – The Bessel Y function

• bessel_I(n, x) – The Bessel I function

• bessel_K(n, x) – The Bessel K function

• Bessel(...) – A factory function for producing Bessel functions of various kinds and orders

• hankel1(nu, z) – The Hankel function of the first kind

• hankel2(nu, z) – The Hankel function of the second kind

• struve_H(nu, z) – The Struve function

• struve_L(nu, z) – The modified Struve function

• spherical_bessel_J(n, z) – The Spherical Bessel J function

• spherical_bessel_Y(n, z) – The Spherical Bessel J function

• spherical_hankel1(n, z) – The Spherical Hankel function of the first kind

• spherical_hankel2(n, z) – The Spherical Hankel function of the second kind

• 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: 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)
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:

• Some of the documentation here has been adapted from David Joyner’s original documentation of Sage’s special functions module (2006).

REFERENCES:

• [AS-Bessel]

• [AS-Spherical]

• [AS-Struve]

• [DLMF-Bessel]

• [DLMF-Struve]

• [WP-Bessel]

• [WP-Struve]

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

Bases: sage.symbolic.function.BuiltinFunction

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)
0.00026073272117205890524 - 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)

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).

REFERENCES:

• [AS-Bessel]

• [DLMF-Bessel]

• [WP-Bessel]

class sage.functions.bessel.Function_Bessel_J

Bases: sage.symbolic.function.BuiltinFunction

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

REFERENCES:

• [AS-Bessel]

• [DLMF-Bessel]

• [AS-Bessel]

class sage.functions.bessel.Function_Bessel_K

Bases: sage.symbolic.function.BuiltinFunction

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)
3.8507583115005220156 + 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)

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).

REFERENCES:

• [AS-Bessel]

• [DLMF-Bessel]

• [WP-Bessel]

class sage.functions.bessel.Function_Bessel_Y

Bases: sage.symbolic.function.BuiltinFunction

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
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).

REFERENCES:

• [AS-Bessel]

• [DLMF-Bessel]

• [WP-Bessel]

class sage.functions.bessel.Function_Hankel1

Bases: sage.symbolic.function.BuiltinFunction

The Hankel function of the first kind

DEFINITION:

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

EXAMPLES:

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

REFERENCES:

• [AS-Bessel] see 9.1.6

class sage.functions.bessel.Function_Hankel2

Bases: sage.symbolic.function.BuiltinFunction

The Hankel function of the second kind

DEFINITION:

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

EXAMPLES:

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

REFERENCES:

• [AS-Bessel] see 9.1.6

class sage.functions.bessel.Function_Struve_H

Bases: sage.symbolic.function.BuiltinFunction

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

REFERENCES:

• [AS-Struve]

• [DLMF-Struve]

• [WP-Struve]

class sage.functions.bessel.Function_Struve_L

Bases: sage.symbolic.function.BuiltinFunction

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

REFERENCES:

• [AS-Struve]

• [DLMF-Struve]

• [WP-Struve]

class sage.functions.bessel.SphericalBesselJ

Bases: sage.symbolic.function.BuiltinFunction

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

REFERENCES:

• [AS-Spherical]

• [DLMF-Bessel]

• [WP-Bessel]

class sage.functions.bessel.SphericalBesselY

Bases: sage.symbolic.function.BuiltinFunction

The spherical Bessel function of the second kind

DEFINITION:

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

EXAMPLES:

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)

REFERENCES:

• [AS-Spherical]

• [DLMF-Bessel]

• [WP-Bessel]

class sage.functions.bessel.SphericalHankel1

Bases: sage.symbolic.function.BuiltinFunction

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: 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)

REFERENCES:

• [AS-Spherical]

• [DLMF-Bessel]

• [WP-Bessel]

class sage.functions.bessel.SphericalHankel2

Bases: sage.symbolic.function.BuiltinFunction

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: 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)

REFERENCES:

• [AS-Spherical]

• [DLMF-Bessel]

• [WP-Bessel]

sage.functions.bessel.spherical_bessel_f(F, n, z)

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)
mpf('0.22924385795503024')
sage: spherical_bessel_f('hankel1', 3, 4)
mpc(real='0.22924385795503024', imag='-0.21864196590306359')