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# 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:

> - 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

`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)
      0.00026073272117205890528 - 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`

  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

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)
      3.8507583115005220157 + 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`

  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

  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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