OpenMath Content Dictionary: bessel

Canonical URL:
http://www.openmath.org/CDs/bessel.ocd
CD File:
bessel.ocd
CD as XML Encoded OpenMath:
bessel.omcd
Defines:
BesselI, BesselJ, BesselK, BesselY, HankelH1, HankelH2, SphericalBesselI_MinusnPlusHalf, SphericalBesselI_nPlusHalf, SphericalBesselj, SphericalBesselK_nPlusHalf, SphericalBessely, SphericalHankel1, SphericalHankel2
Date:
Version:
(Revision )
Review Date:
Status:
private
Uses CD:
alg1, arith1, calculus1, fns1, relation1

1/1/5000

This content dictionary contains symbols to describe the bessel functions and associated functions.


BesselJ

The BesselJ symbol represents the Bessel Functions of the first kind (of Integer Order). These are the solutions ,$J_{\pm \nu}(z)$, to the differential equation $z^2 \frac{d^2w}{dz^2}+z\frac{dw}{dz} + (z^2-\nu^2)w=0$ which has a branch cut along the negative real axis in the complex $z$ plane. When $\nu$ is integral they are analytic in the entire complex z plane.

They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.1.1] [9.1.20], [9.1.21]

Commented Mathematical property (CMP):
J : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
J : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
J : (symbolic, symbolic) $\rightarrow$ symbolic
Formal Mathematical property (FMP):
<OMOBJ>
        <OMA>
	  <OMS cd="relation1" name="eq"/>
	  <OMA>
	    <OMS cd="arith1" name="plus"/>
	    <OMA>
	      <OMS cd="arith1" name="times"/>
	      <OMA>
	        <OMS cd="arith1" name="power"/>
		<OMV name="z"/>
		<OMI>2</OMI>
	      </OMA>
	      <OMA>
	        <OMA>
	          <OMS cd="calculus1" name="diff"/>
		  <OMBIND>
		    <OMS cd="fns1" name="lambda"/>
		    <OMBVAR>
		      <OMV name="z2"/>
		    </OMBVAR>
		    <OMA>
		      <OMA>
		        <OMS cd="calculus1" name="diff"/>
		        <OMBIND>
		          <OMS cd="fns1" name="lambda"/>
		          <OMBVAR>
		            <OMV name="z3"/>
		          </OMBVAR>
		          <OMA>
		            <OMS cd="bessel" name="BesselJ"/>
			    <OMV name="nu"/>
			    <OMV name="z3"/>
		          </OMA>
		        </OMBIND>
		      </OMA>
		      <OMV name="z2"/>
		    </OMA>
		  </OMBIND>
		</OMA>
		<OMV name="z"/>
	      </OMA>
	    </OMA>
	    <OMA>
	      <OMS cd="arith1" name="times"/>
	      <OMV name="z"/>
	      <OMA>
	        <OMA>
		  <OMS cd="calculus1" name="diff"/>
		  <OMBIND>
		    <OMS cd="fns1" name="lambda"/>
		    <OMBVAR>
		      <OMV name="z2"/>
		    </OMBVAR>
		    <OMA>
		      <OMA>
		        <OMS cd="bessel" name="BesselJ"/>
			<OMV name="nu"/>
			<OMV name="z2"/>
		      </OMA>
		    </OMA>
		  </OMBIND>
		</OMA>
		<OMV name="z"/>
	      </OMA>
	    </OMA>
	    <OMA>
	      <OMS cd="arith1" name="times"/>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
		<OMV name="z"/>
		<OMV name="nu"/>
	      </OMA>
	      <OMV name="w"/>
	    </OMA>
	  </OMA>
	  <OMS cd="alg1" name="zero"/>
	</OMA>
      </OMOBJ>

eq (plus (times (power ( z, 2) , diff (lambda [ z2 ] . (diff (lambda [ z3 ] . (BesselJ ( nu, z3) ) ) ( z2) ) ) ( z) ) , times ( z, diff (lambda [ z2 ] . (BesselJ ( nu, z2) () ) ) ( z) ) , times (minus ( z, nu) , w) ) , zero)

Commented Mathematical property (CMP):
$J_{\nu}(z)=\sum_{k=0}^{\infty} \frac{(-1)^k}{k!\Gamma(\nu+k+1)}(\frac{z}{2})^{\nu+2k}$, $|arg z| /lt \pi$, ascending series
Commented Mathematical property (CMP):
$J_n(z)=\frac{1}{\pi} \int_0^{\pi} cos(n \theta- zsin \theta) d \theta$, integral representation
Commented Mathematical property (CMP):
$J_{-n}(z)=(-1)^nJ_n(z)$, symmetry
Commented Mathematical property (CMP):
$J^{\prime}_0(z)=-J_1(z)$, derivative
Signatures:
sts


[Next: BesselY] [Last: SphericalBesselK_nPlusHalf] [Top]

BesselY

The symbol BesselY represents Bessel Functions of the second kind (of integer order). These are also known as Weber /functions. We denote these by $Y_{\nu}(z)$. $Y_{\nu}(z)$ are the second kind of Bessel functions. They are defined in terms of first kind of Bessel functions as $Y_{\nu}(z)=\frac{J_{\nu}(z)cos(\nu\pi)-J_{-\nu}(z)}{sin(\nu\pi)}$. which has a branch cut along the negative real axis in the complex $z$ plane.

They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.1.2]

Commented Mathematical property (CMP):
Y : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
Y : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
Y : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$Y_{-n}(z)=(-1)^nY_n(z)$, symmetry
Commented Mathematical property (CMP):
$Y^{\prime}_0(z)=-Y_1(z)$, derivative
Signatures:
sts


[Next: HankelH1] [Previous: BesselJ] [Top]

HankelH1

The symbol HankelH1 represents the first variety of Bessel Functions of the third kind (also called the Hankel functions of the first kind). They are denoted as $H_{\nu}^{(1)}(z)$. They are defined by $H_{\nu}^{(1)}(z)=J_{\nu}(z)+iY_{\nu}(z)$. They have a branch cut along the negative real axis in the complex $z$ plane.

They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.1.3]

Function
Commented Mathematical property (CMP):
H^(1) : (real, real) $\rightarrow$ real
Commented Mathematical property (CMP):
H^(1) : (complex, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
H^(1) : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$H_{-\nu}^{(1)}(z)=e^{\nu \pi i} H_{\nu}^{(1)}(z)$
Commented Mathematical property (CMP):
H^{(1)}_\nu(z)=J_\nu(z)+iY_\nu(z)
Signatures:
sts


[Next: HankelH2] [Previous: BesselY] [Top]

HankelH2

The symbol HankelH2 represents the second variety of Bessel Functions of the third kind (also called the Hankel functions of the second kind). They are denoted as $H_{\nu}^{(2)}(z)$. They are defined by $H_{\nu}^{(1)}(z)=J_{\nu}(z)-iY_{\nu}(z)$. They have a branch cut along the negative real axis in the complex $z$ plane.

They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.1.4]

Commented Mathematical property (CMP):
H^(2) : (real, real) $\rightarrow$ real
Commented Mathematical property (CMP):
H^(2) : (complex, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
H^(2) : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$H_{-\nu}^{(2)}(z)=e^{-\nu \pi i} H_{\nu}^{(2)}(z)$
Commented Mathematical property (CMP):
H^{(2)}_\nu(z)=J_\nu(z)-iY_\nu(z)
Signatures:
sts


[Next: BesselI] [Previous: HankelH1] [Top]

BesselI

The symbol BesselI represents the Modified Bessel Functions of the first kind. They are denoted by $I_{\pm\nu}(z)$. The $I_{\pm\nu}(z)$ are the solutions of the differential equation $z^2\frac{d^2w}{d^2z}+z\frac{dw}{dz}-(z^2+\nu^2)w=0$ which has a branch cut along the negative real axis in the complex $z$ plane. When $\nu$ is integral, $I_{\nu}$ is an entire function of z.

They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.6.1]

Function
Commented Mathematical property (CMP):
I : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
I : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
I : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$I_{\nu}(z)=\sum_{k=0}^{\infty} \frac{1}{k!\Gamma(\nu+k+1)} (\frac{z}{2})^{2k+\nu}$, ascending series
Commented Mathematical property (CMP):
$I_{-n}(z)=I_n(z)$
Signatures:
sts


[Next: BesselK] [Previous: HankelH2] [Top]

BesselK

The symbol BesselK represents the Modified Bessel Functions of the second kind. They are denoted by $K_{\nu}(z)$. They are the solutions of the differential equation $z^2\frac{d^2w}{d^2z}+z\frac{dw}{dz}-(z^2+\nu^2)w=0$ which has a branch cut along the negative real axis in the complex $z$ plane.

They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.6.1]

Function
Commented Mathematical property (CMP):
K : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
K : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
K : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$K_{\nu}(z)=\frac{1}{2}\pi \frac{I_{-\nu}(z)-I_{\nu}(z)} {sin(\nu\pi)}$, relation to $I_{\nu}(z)$
Commented Mathematical property (CMP):
$\sqrt{\frac{\pi}{2z}}K_{n+\frac{1}{2}}(z)= \frac{1}{2}\pi(-1)^{n+1} \sqrt{\frac{\pi}{2z}} [I_{n+\frac{1}{2}}(z) -I_{-n-\frac{1}{2}}(z)]$, relation to $I_{\nu}(z)$
Commented Mathematical property (CMP):
$K_{-\nu}(z)=K_{\nu}(z)$
Signatures:
sts


[Next: SphericalBesselj] [Previous: BesselI] [Top]

SphericalBesselj

The symbol SphericalBesselj represents the Spherical Bessel Functions of the first kind. They are denoted by $j_{n}(z)$. They satisfy the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2-n(n+1)]w=0$. They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.1.1]

Commented Mathematical property (CMP):
j : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
j : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
j : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$j_{n}(z)=\sqrt{\frac{1}{2}\pi/z}J_{n+\frac{1}{2}}(z)$
Signatures:
sts


[Next: SphericalBessely] [Previous: BesselK] [Top]

SphericalBessely

The symbol SphericalBessely represents the Spherical Bessel Functions of the second kind. They are denoted by $y_{n}(z)$. They satisfy the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2-n(n+1)]w=0$.

They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.1.1]

Function
Commented Mathematical property (CMP):
y : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
y : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
y : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$y_{n}(z)=\sqrt{\frac{1}{2}\pi/z}Y_{n+\frac{1}{2}}(z)$
Signatures:
sts


[Next: SphericalHankel1] [Previous: SphericalBesselj] [Top]

SphericalHankel1

The symbol SphericalHankel1 represents the Spherical Hankel Functions of the first kind (also called the first variety of the spherical Bessel functions of the third kind). They are denoted by $h_{n}^{(1)}(z)$. They satisfy the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2-n(n+1)]w=0$. They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.1.1]

Function
Commented Mathematical property (CMP):
h^(1) : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
h^(1) : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
h^(1) : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$h_{n}^{(1)}(z)=\sqrt{\frac{1}{2}\pi/z}H_{n+\frac{1}{2}}^{(1)}(z)$
Signatures:
sts


[Next: SphericalHankel2] [Previous: SphericalBessely] [Top]

SphericalHankel2

The symbol SphericalHankel2 represents the Spherical Hankel Functions of the second kind (also called the second variety of the spherical Bessel functions of the third kind). They are denoted by $h_{n}^{(2)}(z)$. They satisfy the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2-n(n+1)]w=0$. They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.1.1]

Commented Mathematical property (CMP):
h^(2) : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
h^(2) : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
h^(2) : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$h_{n}^{(2)}(z)=\sqrt{\frac{1}{2}\pi/z}H_{n+\frac{1}{2}} ^{(2)}(z)$
Signatures:
sts


[Next: SphericalBesselI_nPlusHalf] [Previous: SphericalHankel1] [Top]

SphericalBesselI_nPlusHalf

The symbol SphericalBesselI_nPlusHalf represents the Modified Spherical Bessel Functions of the first kind. They are denoted by $I_{n+\frac{1}{2}}(z)$. They satisfy the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2+n(n+1)]w=0$ where $n \geq 0$. They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.2.2]

Function
Commented Mathematical property (CMP):
I_{n+1/2} : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
I_{n+1/2} : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
I_{n+1/2} : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$\sqrt{\frac{1}{2} \pi/z} I_{n+\frac{1}{2}}(z)= e^{-n \pi i/2}j_n(ze^{\pi i/2})$, (-$\pi \lt argz \leq \pi/2$)
Signatures:
sts


[Next: SphericalBesselI_MinusnPlusHalf] [Previous: SphericalHankel2] [Top]

SphericalBesselI_MinusnPlusHalf

The symbol SphericalBesselI_MinusnPlusHalf represents the Modified Spherical Bessel Functions of the second kind. They are denoted by $I_{-n-\frac{1}{2}}(z)$. They satisfy the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2+n(n+1)]w=0$ where $n \geq 0$. They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.2.3]

Commented Mathematical property (CMP):
I_{-n-1/2} : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
I_{-n-1/2} : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
I_{-n-1/2} : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$\sqrt{\frac{1}{2} \pi/z} I_{-n-\frac{1}{2}}(z)= e^{3(n+1) \pi i/2}y_n(ze^{\pi i/2})$, (-$\pi \lt argz \leq \pi/2$)
Signatures:
sts


[Next: SphericalBesselK_nPlusHalf] [Previous: SphericalBesselI_nPlusHalf] [Top]

SphericalBesselK_nPlusHalf

The symbol SphericalBesselK_nPlusHalf represent the Modified Spherical Bessel Functions of the third kind. They are denoted by $K_{n+\frac{1}{2}}(z)$. They satisfy the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2+n(n+1)]w=0$ where $n \geq 0$. They are defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.2.4]

Commented Mathematical property (CMP):
K_{n+1/2} : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
K_{n+1/2} : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
K_{n+1/2} : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$\sqrt{\frac{1}{2} \pi/z} K_{n+\frac{1}{2}}(z)= \frac{1}{2} \pi (-1)^{n+1}\sqrt{\frac{1}{2} \pi/z}[I_{n+\frac{1}{2}} -I_{-n-\frac{1}{2}}]$
Signatures:
sts


[First: BesselJ] [Previous: SphericalBesselI_MinusnPlusHalf] [Top]