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:

2003-11-30

Version:

0 (Revision 3)

Review Date:

2017-12-31

Status:

experimental

---

  Author: Yasushi Tamura

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="hypergeo2">kummer</csymbol><ci>a</ci><ci>c</ci><ci>z</ci></apply>
  <apply><csymbol cd="hypergeo1">hypergeometric1F1</csymbol><ci>a</ci><ci>c</ci><ci>z</ci></apply>
 </apply>
</math>

Prefix

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

Popcorn

hypergeo2.kummer($a, $c, $z) = hypergeo1.hypergeometric1F1($a, $c, $z)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: besselJ] [Last: airyBi] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="hypergeo2">besselJ</csymbol><ci>v</ci><ci>z</ci></apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="arith1">power</csymbol>
    <apply><csymbol cd="arith1">divide</csymbol><ci>z</ci><cn type="integer">2</cn></apply>
    <ci>v</ci>
   </apply>
   <apply><csymbol cd="arith1">sum</csymbol>
    <apply><csymbol cd="interval1">integer_interval</csymbol><cn type="integer">0</cn><ci>infty</ci></apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>n</ci></bvar>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">divide</csymbol>
       <apply><csymbol cd="arith1">power</csymbol>
        <apply><csymbol cd="arith1">unary_minus</csymbol><cn type="integer">1</cn></apply>
        <ci>n</ci>
       </apply>
       <apply><csymbol cd="arith1">times</csymbol>
        <apply><csymbol cd="integer1">factorial</csymbol><ci>n</ci></apply>
        <apply><csymbol cd="hypergeo0">gamma</csymbol>
         <apply><csymbol cd="arith1">plus</csymbol>
          <apply><csymbol cd="arith1">plus</csymbol><ci>v</ci><ci>n</ci></apply>
          <cn type="integer">1</cn>
         </apply>
        </apply>
       </apply>
      </apply>
      <apply><csymbol cd="arith1">power</csymbol>
       <apply><csymbol cd="arith1">divide</csymbol><ci>z</ci><cn type="integer">2</cn></apply>
       <apply><csymbol cd="arith1">times</csymbol><cn type="integer">2</cn><ci>n</ci></apply>
      </apply>
     </apply>
    </bind>
   </apply>
  </apply>
 </apply>
</math>

Prefix

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) ) ) ) ) ) )

Popcorn

hypergeo2.besselJ($v, $z) = ($z / 2) ^ $v * arith1.sum(interval1.integer_interval(0, $infty), fns1.lambda[$n -> -(1) ^ $n / (integer1.factorial($n) * hypergeo0.gamma($v + $n + 1)) * ($z / 2) ^ (2 * $n)])

Rendered Presentation MathML

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

Example:

OpenMath XML (source)

<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 cd="hypergeo2" name="besselJ"/>
          <OMV name="v"/>
          <OMV name="z"/>
        </OMA>
      </OMA>
    </OMA>
    <OMI> 0 </OMI>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="arith1">plus</csymbol>
   <apply><csymbol cd="arith1">plus</csymbol>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><cn type="integer">2</cn></apply>
     <apply><csymbol cd="calculus1">diff</csymbol>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>z</ci></bvar>
       <apply><csymbol cd="calculus1">diff</csymbol>
        <bind><csymbol cd="fns1">lambda</csymbol>
         <bvar><ci>z</ci></bvar>
         <apply><csymbol cd="hypergeo2">besselJ</csymbol><ci>v</ci><ci>z</ci></apply>
        </bind>
       </apply>
      </bind>
     </apply>
    </apply>
    <apply><csymbol cd="arith1">times</csymbol>
     <ci>z</ci>
     <apply><csymbol cd="calculus1">diff</csymbol>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>z</ci></bvar>
       <apply><csymbol cd="bypergeo2">besselJ</csymbol><ci>v</ci><ci>z</ci></apply>
      </bind>
     </apply>
    </apply>
   </apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <apply><csymbol cd="arith1">minus</csymbol>
     <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><cn type="integer">2</cn></apply>
     <apply><csymbol cd="arith1">power</csymbol><ci>v</ci><cn type="integer">2</cn></apply>
    </apply>
    <apply><csymbol cd="hypergeo2">besselJ</csymbol><ci>v</ci><ci>z</ci></apply>
   </apply>
  </apply>
  <cn type="integer">0</cn>
 </apply>
</math>

Prefix

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 )

Popcorn

$z ^ 2 * calculus1.diff(fns1.lambda[$z -> calculus1.diff(fns1.lambda[$z -> hypergeo2.besselJ($v, $z)])]) + $z * calculus1.diff(fns1.lambda[$z -> bypergeo2.besselJ($v, $z)]) + ($z ^ 2 - $v ^ 2) * hypergeo2.besselJ($v, $z) = 0

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: besselY] [Previous: kummer] [Top]

---

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):

OpenMath XML (source)

<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 cd="hypergeo2" 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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="hypergeo2">besselY</csymbol><ci>v</ci><ci>z</ci></apply>
  <apply><csymbol cd="arith1">divide</csymbol>
   <apply><csymbol cd="arith1">minus</csymbol>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="transc1">cos</csymbol>
      <apply><csymbol cd="arith1">times</csymbol><ci>v</ci><csymbol cd="nums1">pi</csymbol></apply>
     </apply>
     <apply><csymbol cd="hypergeo2">besselJ</csymbol><ci>v</ci><ci>z</ci></apply>
    </apply>
    <apply><csymbol cd="hypergeo2">besselJ</csymbol>
     <apply><csymbol cd="arith1">unary_minus</csymbol><ci>v</ci></apply>
     <ci>z</ci>
    </apply>
   </apply>
   <apply><csymbol cd="transc1">sin</csymbol>
    <apply><csymbol cd="arith1">times</csymbol><ci>v</ci><csymbol cd="nums1">pi</csymbol></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

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

Popcorn

hypergeo2.besselY($v, $z) = (cos($v * nums1.pi) * hypergeo2.besselJ($v, $z) - hypergeo2.besselJ( -($v), $z)) / sin($v * nums1.pi)

Rendered Presentation MathML

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

Example:

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="arith1">plus</csymbol>
   <apply><csymbol cd="arith1">plus</csymbol>
    <apply><csymbol cd="arith1">times</csymbol>
     <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><cn type="integer">2</cn></apply>
     <apply><csymbol cd="calculus1">diff</csymbol>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>z</ci></bvar>
       <apply><csymbol cd="calculus1">diff</csymbol>
        <bind><csymbol cd="fns1">lambda</csymbol>
         <bvar><ci>z</ci></bvar>
         <apply><csymbol cd="hypergeo2">besselY</csymbol><ci>v</ci><ci>z</ci></apply>
        </bind>
       </apply>
      </bind>
     </apply>
    </apply>
    <apply><csymbol cd="arith1">times</csymbol>
     <ci>z</ci>
     <apply><csymbol cd="calculus1">diff</csymbol>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>z</ci></bvar>
       <apply><csymbol cd="hypergeo2">besselY</csymbol><ci>v</ci><ci>z</ci></apply>
      </bind>
     </apply>
    </apply>
   </apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <apply><csymbol cd="arith1">minus</csymbol>
     <apply><csymbol cd="arith1">power</csymbol><ci>z</ci><cn type="integer">2</cn></apply>
     <apply><csymbol cd="arith1">power</csymbol><ci>v</ci><cn type="integer">2</cn></apply>
    </apply>
    <apply><csymbol cd="hypergeo2">besselY</csymbol><ci>v</ci><ci>z</ci></apply>
   </apply>
  </apply>
  <cn type="integer">0</cn>
 </apply>
</math>

Prefix

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 )

Popcorn

$z ^ 2 * calculus1.diff(fns1.lambda[$z -> calculus1.diff(fns1.lambda[$z -> hypergeo2.besselY($v, $z)])]) + $z * calculus1.diff(fns1.lambda[$z -> hypergeo2.besselY($v, $z)]) + ($z ^ 2 - $v ^ 2) * hypergeo2.besselY($v, $z) = 0

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: hankel1] [Previous: besselJ] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="hypergeo2">hankel1</csymbol><ci>v</ci><ci>z</ci></apply>
  <apply><csymbol cd="arith1">plus</csymbol>
   <apply><csymbol cd="hypergeo2">besselJ</csymbol><ci>v</ci><ci>z</ci></apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <csymbol cd="nums1">i</csymbol>
    <apply><csymbol cd="hypergeo2">besselY</csymbol><ci>v</ci><ci>z</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

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

Popcorn

hypergeo2.hankel1($v, $z) = hypergeo2.besselJ($v, $z) + nums1.i * hypergeo2.besselY($v, $z)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: hankel2] [Previous: besselY] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="hypergeo2">hankel1</csymbol><ci>v</ci><ci>z</ci></apply>
  <apply><csymbol cd="arith1">minus</csymbol>
   <apply><csymbol cd="hypergeo2">besselJ</csymbol><ci>v</ci><ci>z</ci></apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <csymbol cd="nums1">i</csymbol>
    <apply><csymbol cd="hypergeo2">besselY</csymbol><ci>v</ci><ci>z</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

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

Popcorn

hypergeo2.hankel1($v, $z) = hypergeo2.besselJ($v, $z) - nums1.i * hypergeo2.besselY($v, $z)

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: airyAi] [Previous: hankel1] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="arith1">minus</csymbol>
   <apply><csymbol cd="calculus1">diff</csymbol>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>z</ci></bvar>
     <apply><csymbol cd="calculus1">diff</csymbol>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>z</ci></bvar>
       <apply><csymbol cd="hypergeo2">airyAi</csymbol><ci>z</ci></apply>
      </bind>
     </apply>
    </bind>
   </apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <ci>z</ci>
    <apply><csymbol cd="hypergeo2">airyAi</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
  <cn type="integer">0</cn>
 </apply>
</math>

Prefix

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

Popcorn

calculus1.diff(fns1.lambda[$z -> calculus1.diff(fns1.lambda[$z -> hypergeo2.airyAi($z)])]) - $z * hypergeo2.airyAi($z) = 0

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: airyBi] [Previous: hankel2] [Top]

---

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):

OpenMath XML (source)

<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>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="arith1">minus</csymbol>
   <apply><csymbol cd="calculus1">diff</csymbol>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>z</ci></bvar>
     <apply><csymbol cd="calculus1">diff</csymbol>
      <bind><csymbol cd="fns1">lambda</csymbol>
       <bvar><ci>z</ci></bvar>
       <apply><csymbol cd="hypergeo2">airyBi</csymbol><ci>z</ci></apply>
      </bind>
     </apply>
    </bind>
   </apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <ci>z</ci>
    <apply><csymbol cd="hypergeo2">airyBi</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
  <cn type="integer">0</cn>
 </apply>
</math>

Prefix

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

Popcorn

calculus1.diff(fns1.lambda[$z -> calculus1.diff(fns1.lambda[$z -> hypergeo2.airyBi($z)])]) - $z * hypergeo2.airyBi($z) = 0

Rendered Presentation MathML

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

Signatures:

sts

---

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