OpenMath Content Dictionary: hypergeo2

Canonical URL:

http://www.openxm.org/...

CD File:

hypergeo2.ocd

CD as XML Encoded OpenMath:

hypergeo2.omcd

Defines:

airyAi, airyBi, besselJ, besselY, hankel1, hankel2, kummer

Date:

2002-08-11, 2003-11-30

Version:

0 (Revision 1)

Review Date:

2003-08-11

Status:

experimental

Uses CD:

arith1, relation1, calculus1, alg1, interval1, nums1, hypergeo0, hypergeo1

This CD defines some famous hypergeometric functions such as Bessel functions and Airy functions. These functions are described in the following books. (1) Handbook of Mathematical Functions, Abramowitz, Stegun (2) Higher transcendental functions. Krieger Publishing Co., Inc., Melbourne, Fla., 1981, Erdlyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G.

kummer

Description:

Kummer's hypergeometric function.

Commented Mathematical property (CMP):

kummer(a,c;z) = hypergeo1.hypergeometric1F1(a,c;z)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeo2" name="kummer"/>
      <OMV name="a"/>
      <OMV name="c"/>
      <OMV name="z"/>
    </OMA>
    <OMA><OMS cd="hypergeo1" name="hypergeometric1F1"/>
      <OMV name="a"/>
      <OMV name="c"/>
      <OMV name="z"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (kummer ( a, c, z) , hypergeometric1F1 ( a, c, z) )

$$\mathrm{kummer}(a,c,z)=\mathrm{hypergeometric1F1}(a,c,z)$$

Signatures:

sts

besselJ

Description:

The Bessel function. This function is one of the famous two solutions of the Bessel differential equation at z=0.

Commented Mathematical property (CMP):

besselJ(v,z) = (\frac{z}{2})^v \sum_{n=0}^{+\infty} \frac{(-1)^n}{n! \Gamma(v+n+1)} (\frac{z}{2})^2n

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeo2" name="besselJ"/>
      <OMV name="v"/>
      <OMV name="z"/>
    </OMA>
    <OMA><OMS cd="arith1" name="times"/>
      <OMA><OMS cd="arith1" name="power"/>
        <OMA><OMS cd="arith1" name="divide"/>
          <OMV name="z"/>
          <OMI> 2 </OMI>
        </OMA>
        <OMV name="v"/>
      </OMA>
      <OMA><OMS cd="arith1" name="sum"/>
        <OMA><OMS cd="interval1" name="integer_interval"/>
          <OMI> 0 </OMI>
          <OMV name="infty"/>
        </OMA>
        <OMBIND>
          <OMS cd="fns1" name="lambda"/>
          <OMBVAR>
            <OMV name="n"/>
          </OMBVAR>
          <OMA><OMS cd="arith1" name="times"/>
            <OMA><OMS cd="arith1" name="divide"/>
              <OMA><OMS cd="arith1" name="power"/>
                <OMA><OMS cd="arith1" name="unary_minus"/>
                  <OMI> 1 </OMI>
                </OMA>
                <OMV name="n"/>
              </OMA>
              <OMA><OMS cd="arith1" name="times"/>
                <OMA><OMS cd="integer1" name="factorial"/>
                  <OMV name="n"/>
                </OMA>
                <OMA><OMS cd="hypergeo0" name="gamma"/>
                  <OMA><OMS cd="arith1" name="plus"/>
                    <OMA><OMS cd="arith1" name="plus"/>
                      <OMV name="v"/>
                      <OMV name="n"/>
                    </OMA>
                    <OMI> 1 </OMI>
                  </OMA>
                </OMA>
              </OMA>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMA><OMS cd="arith1" name="divide"/>
                <OMV name="z"/>
                <OMI> 2 </OMI>
              </OMA>
              <OMA><OMS cd="arith1" name="times"/>
                <OMI> 2 </OMI>
                <OMV name="n"/>
              </OMA>
            </OMA>
          </OMA>
        </OMBIND>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (besselJ ( v, z) , times (power (divide ( z, 2 ) , v) , sum (integer_interval ( 0 , infty) , lambda [ n ] . (times (divide (power (unary_minus ( 1 ) , n) , times (factorial ( n) , gamma (plus (plus ( v, n) , 1 ) ) ) ) , power (divide ( z, 2 ) , times ( 2 , n) ) ) ) ) ) )

$$\mathrm{besselJ}(v,z)={\left(\frac{z}{2}\right)}^v\sum _{n=0}^{\mathrm{infty}}\frac{{-1}^n}{n!\mathrm{gamma}\left(v+n+1\right)}{\left(\frac{z}{2}\right)}^{(2n)}$$

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="arith1" name="plus"/>
      <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><OMS cd="calculus1" name="diff"/>
            <OMBIND>
              <OMS cd="fns1" name="lambda"/>
              <OMBVAR>
                <OMV name="z"/>
              </OMBVAR>
              <OMA><OMS cd="calculus1" name="diff"/>
                <OMBIND>
                  <OMS cd="fns1" name="lambda"/>
                  <OMBVAR>
                    <OMV name="z"/>
                  </OMBVAR>
                  <OMA><OMS cd="hypergeo2" name="besselJ"/>
                    <OMV name="v"/>
                    <OMV name="z"/>
                  </OMA>
                </OMBIND>
              </OMA>
            </OMBIND>
          </OMA>
        </OMA>
        <OMA><OMS cd="arith1" name="times"/>
          <OMV name="z"/>
          <OMA><OMS cd="calculus1" name="diff"/>
            <OMBIND>
              <OMS cd="fns1" name="lambda"/>
              <OMBVAR>
                <OMV name="z"/>
              </OMBVAR>
              <OMA><OMS cd="bypergeo2" name="besselJ"/>
                <OMV name="v"/>
                <OMV name="z"/>
              </OMA>
            </OMBIND>
          </OMA>
        </OMA>
      </OMA>
      <OMA><OMS cd="arith1" name="times"/>
        <OMA><OMS cd="arith1" name="minus"/>
          <OMA><OMS cd="arith1" name="power"/>
            <OMV name="z"/>
            <OMI> 2 </OMI>
          </OMA>
          <OMA><OMS cd="arith1" name="power"/>
            <OMV name="v"/>
            <OMI> 2 </OMI>
          </OMA>
        </OMA>
        <OMA><OMS name="besselJ"/>
          <OMV name="v"/>
          <OMV name="z"/>
        </OMA>
      </OMA>
    </OMA>
    <OMI> 0 </OMI>
  </OMA>
</OMOBJ>

eq (plus (plus (times (power ( z, 2 ) , diff (lambda [ z ] . (diff (lambda [ z ] . (besselJ ( v, z) ) ) ) ) ) , times ( z, diff (lambda [ z ] . (besselJ ( v, z) ) ) ) ) , times (minus (power ( z, 2 ) , power ( v, 2 ) ) , besselJ ( v, z) ) ) , 0 )

$$z^2\frac{d}{dz}\left(\frac{d}{dz}\left(\mathrm{besselJ}(v,z)\right)\right)+z\frac{d}{dz}\left(\mathrm{besselJ}(v,z)\right)+(z^2-v^2)\mathrm{besselJ}(v,z)=0$$

Signatures:

sts

besselY

Description:

The Bessel function. This function is the another one of the famous two solutions of the Bessel differential equation at z=0.

Commented Mathematical property (CMP):

besselY(v,z) = (\cos(v \pi) besselJ(v,z) - besselJ(-v,z))/\sin(v \pi)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeo2" name="besselY"/>
      <OMV name="v"/>
      <OMV name="z"/>
    </OMA>
    <OMA><OMS cd="arith1" name="divide"/>
      <OMA><OMS cd="arith1" name="minus"/>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="transc1" name="cos"/>
            <OMA><OMS cd="arith1" name="times"/>
              <OMV name="v"/>
              <OMS cd="nums1" name="pi"/>
            </OMA>
          </OMA>
          <OMA><OMS cd="hypergeo2" name="besselJ"/>
            <OMV name="v"/>
            <OMV name="z"/>
          </OMA>
        </OMA>
        <OMA><OMS name="besselJ"/>
          <OMA><OMS cd="arith1" name="unary_minus"/>
            <OMV name="v"/>
          </OMA>
          <OMV name="z"/>
        </OMA>
      </OMA>
      <OMA><OMS cd="transc1" name="sin"/>
        <OMA><OMS cd="arith1" name="times"/>
          <OMV name="v"/>
          <OMS cd="nums1" name="pi"/>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (besselY ( v, z) , divide (minus (times (cos (times ( v, pi) ) , besselJ ( v, z) ) , besselJ (unary_minus ( v) , z) ) , sin (times ( v, pi) ) ) )

$$\mathrm{besselY}(v,z)=\frac{\cos \left(v\pi \right)\mathrm{besselJ}(v,z)-\mathrm{besselJ}(-v,z)}{\sin \left(v\pi \right)}$$

Example:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="arith1" name="plus"/>
      <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><OMS cd="calculus1" name="diff"/>
            <OMBIND>
              <OMS cd="fns1" name="lambda"/>
              <OMBVAR>
                <OMV name="z"/>
              </OMBVAR>
              <OMA><OMS cd="calculus1" name="diff"/>
                <OMBIND>
                  <OMS cd="fns1" name="lambda"/>
                  <OMBVAR>
                    <OMV name="z"/>
                  </OMBVAR>
                  <OMA><OMS cd="hypergeo2" name="besselY"/>
                    <OMV name="v"/>
                    <OMV name="z"/>
                  </OMA>
                </OMBIND>
              </OMA>
            </OMBIND>
          </OMA>
        </OMA>
        <OMA><OMS cd="arith1" name="times"/>
          <OMV name="z"/>
          <OMA><OMS cd="calculus1" name="diff"/>
            <OMBIND>
              <OMS cd="fns1" name="lambda"/>
              <OMBVAR>
                <OMV name="z"/>
              </OMBVAR>
              <OMA><OMS cd="hypergeo2" name="besselY"/>
                <OMV name="v"/>
                <OMV name="z"/>
              </OMA>
            </OMBIND>
          </OMA>
        </OMA>
      </OMA>
      <OMA><OMS cd="arith1" name="times"/>
        <OMA><OMS cd="arith1" name="minus"/>
          <OMA><OMS cd="arith1" name="power"/>
            <OMV name="z"/>
            <OMI> 2 </OMI>
          </OMA>
          <OMA><OMS cd="arith1" name="power"/>
            <OMV name="v"/>
            <OMI> 2 </OMI>
          </OMA>
        </OMA>
        <OMA><OMS cd="hypergeo2" name="besselY"/>
          <OMV name="v"/>
          <OMV name="z"/>
        </OMA>
      </OMA>
    </OMA>
    <OMI> 0 </OMI>
  </OMA>
</OMOBJ>

eq (plus (plus (times (power ( z, 2 ) , diff (lambda [ z ] . (diff (lambda [ z ] . (besselY ( v, z) ) ) ) ) ) , times ( z, diff (lambda [ z ] . (besselY ( v, z) ) ) ) ) , times (minus (power ( z, 2 ) , power ( v, 2 ) ) , besselY ( v, z) ) ) , 0 )

$$z^2\frac{d}{dz}\left(\frac{d}{dz}\left(\mathrm{besselY}(v,z)\right)\right)+z\frac{d}{dz}\left(\mathrm{besselY}(v,z)\right)+(z^2-v^2)\mathrm{besselY}(v,z)=0$$

Signatures:

sts

hankel1

Description:

The first Hankel function. This function is one of the famous two solutions of the Bessel differential equation at z=\infty.

Commented Mathematical property (CMP):

hankel1(v,z) = besselJ(v,z) + i BesselY(v,z)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeo2" name="hankel1"/>
      <OMV name="v"/>
      <OMV name="z"/>
    </OMA>
    <OMA><OMS cd="arith1" name="plus"/>
      <OMA><OMS cd="hypergeo2" name="besselJ"/>
        <OMV name="v"/>
        <OMV name="z"/>
      </OMA>
      <OMA><OMS cd="arith1" name="times"/>
        <OMS cd="nums1" name="i"/>
        <OMA><OMS cd="hypergeo2" name="besselY"/>
          <OMV name="v"/>
          <OMV name="z"/>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (hankel1 ( v, z) , plus (besselJ ( v, z) , times (i, besselY ( v, z) ) ) )

$$\mathrm{hankel1}(v,z)=\mathrm{besselJ}(v,z)+i\mathrm{besselY}(v,z)$$

Signatures:

sts

hankel2

Description:

The second Hankel function. This function is the another one of the famous two solutions of the Bessel differential equation at z=\infty.

Commented Mathematical property (CMP):

hankel2(v,z) = besselJ(v,z) - i BesselY(v,z)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeo2" name="hankel1"/>
      <OMV name="v"/>
      <OMV name="z"/>
    </OMA>
    <OMA><OMS cd="arith1" name="minus"/>
      <OMA><OMS cd="hypergeo2" name="besselJ"/>
        <OMV name="v"/>
        <OMV name="z"/>
      </OMA>
      <OMA><OMS cd="arith1" name="times"/>
        <OMS cd="nums1" name="i"/>
        <OMA><OMS cd="hypergeo2" name="besselY"/>
          <OMV name="v"/>
          <OMV name="z"/>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (hankel1 ( v, z) , minus (besselJ ( v, z) , times (i, besselY ( v, z) ) ) )

$$\mathrm{hankel1}(v,z)=\mathrm{besselJ}(v,z)-i\mathrm{besselY}(v,z)$$

Signatures:

sts

airyAi

Description:

The first Airy function. This function is one of the famous two solutions of the Airy differential equation, and converges to 0 when z->\infty

Commented Mathematical property (CMP):

(\frac{d^2}{dz^2} - z) airyAi(z) = 0

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="arith1" name="minus"/>
      <OMA><OMS cd="calculus1" name="diff"/>
        <OMBIND>
          <OMS cd="fns1" name="lambda"/>
          <OMBVAR>
            <OMV name="z"/>
          </OMBVAR>
          <OMA><OMS cd="calculus1" name="diff"/>
            <OMBIND>
              <OMS cd="fns1" name="lambda"/>
              <OMBVAR>
                <OMV name="z"/>
              </OMBVAR>
              <OMA><OMS cd="hypergeo2" name="airyAi"/>
                <OMV name="z"/>
              </OMA>
            </OMBIND>
          </OMA>
        </OMBIND>
      </OMA>
      <OMA><OMS cd="arith1" name="times"/>
        <OMV name="z"/>
        <OMA><OMS cd="hypergeo2" name="airyAi"/>
          <OMV name="z"/>
        </OMA>
      </OMA>
    </OMA>
    <OMI> 0 </OMI>
  </OMA>
</OMOBJ>

eq (minus (diff (lambda [ z ] . (diff (lambda [ z ] . (airyAi ( z) ) ) ) ) , times ( z, airyAi ( z) ) ) , 0 )

$$\frac{d}{dz}\left(\frac{d}{dz}\left(\mathrm{airyAi}\left(z\right)\right)\right)-z\mathrm{airyAi}\left(z\right)=0$$

Signatures:

sts

airyBi

Description:

The second Airy function. This function is the another one of the famous two solutions of the Airy differential equation, and diverges when z->\infty

Commented Mathematical property (CMP):

(\frac{d^2}{dz^2} - z) airyBi(z) = 0

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="arith1" name="minus"/>
      <OMA><OMS cd="calculus1" name="diff"/>
        <OMBIND>
          <OMS cd="fns1" name="lambda"/>
          <OMBVAR>
            <OMV name="z"/>
          </OMBVAR>
          <OMA><OMS cd="calculus1" name="diff"/>
            <OMBIND>
              <OMS cd="fns1" name="lambda"/>
              <OMBVAR>
                <OMV name="z"/>
              </OMBVAR>
              <OMA><OMS cd="hypergeo2" name="airyBi"/>
                <OMV name="z"/>
              </OMA>
            </OMBIND>
          </OMA>
        </OMBIND>
      </OMA>
      <OMA><OMS cd="arith1" name="times"/>
        <OMV name="z"/>
        <OMA><OMS cd="hypergeo2" name="airyBi"/>
          <OMV name="z"/>
        </OMA>
      </OMA>
    </OMA>
    <OMI> 0 </OMI>
  </OMA>
</OMOBJ>

eq (minus (diff (lambda [ z ] . (diff (lambda [ z ] . (airyBi ( z) ) ) ) ) , times ( z, airyBi ( z) ) ) , 0 )

$$\frac{d}{dz}\left(\frac{d}{dz}\left(\mathrm{airyBi}\left(z\right)\right)\right)-z\mathrm{airyBi}\left(z\right)=0$$

Signatures:

sts