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: # 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 2.404825557695773Plot 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 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) # needs sage.plot sage.symbolic Graphics object consisting of 1 graphics primitiveEvaluate 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) -0.043750965336559909054985168023342675387737118378169Symbolically 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))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)#
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 functionBessel()– order is unspecified, type is'J'
where
ordercan be any integer andTmust 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)
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
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)
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
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
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
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
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)
- class sage.functions.bessel.Function_Bessel_I#
Bases:
BuiltinFunctionThe 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
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)
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
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
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
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
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#
Bases:
BuiltinFunctionThe 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)
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)
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
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)
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
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 (github issue #10251):
sage: bessel_J(5, 1.5) in RR # needs mpmath True
REFERENCES:
- class sage.functions.bessel.Function_Bessel_K#
Bases:
BuiltinFunctionThe 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
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)
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
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
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
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#
Bases:
BuiltinFunctionThe 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
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)
High precision and complex valued inputs (see github 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
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
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#
Bases:
BuiltinFunctionThe 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
REFERENCES:
[AS-Bessel] see 9.1.6
- class sage.functions.bessel.Function_Hankel2#
Bases:
BuiltinFunctionThe 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
REFERENCES:
[AS-Bessel] see 9.1.6
- class sage.functions.bessel.Function_Struve_H#
Bases:
BuiltinFunctionThe 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
REFERENCES:
- class sage.functions.bessel.Function_Struve_L#
Bases:
BuiltinFunctionThe 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)
REFERENCES:
- class sage.functions.bessel.SphericalBesselJ#
Bases:
BuiltinFunctionThe 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)
REFERENCES:
- class sage.functions.bessel.SphericalBesselY#
Bases:
BuiltinFunctionThe 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)
REFERENCES:
- class sage.functions.bessel.SphericalHankel1#
Bases:
BuiltinFunctionThe 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)
REFERENCES:
- class sage.functions.bessel.SphericalHankel2#
Bases:
BuiltinFunctionThe 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)
REFERENCES:
- 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) # needs mpmath mpf('0.22924385795503024') sage: spherical_bessel_f('hankel1', 3, 4) # needs mpmath mpc(real='0.22924385795503024', imag='-0.21864196590306359')