Canonical URL:

http://www.openmath.org/cd/gen_hyperbolic1.ocd

CD File:

gen_hyperbolic1.ocd

CD as XML Encoded OpenMath:

gen_hyperbolic1.omcd

Defines:

generalised_hyperbolic

Date:

2002-11-11

Version:

0 (Revision 1)

Review Date:

2017-12-31

Status:

experimental

     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

  Author: Bill Naylor

This CD contains a symbol to represent the generalised hyperbolic function, and facts relating it to other functions.

generalised_hyperbolic

Description:

This symbol represents the generalised hyperbolic function as recorded by Riccati. It is intended to be applied in the curried form, that is, the symbol should be applied to three arguments in order to return a function which should be applied to one argument. The generalised hyperbolic function may be defined as an infinite sum as in the first CMP/FMP .

Commented Mathematical property (CMP):

for complex \alpha, integral n and r an integer between 0 and r (inclusive) (F^\alpha_{n,r})(x) = \Sigma^\infty_{k=0}{\frac{\alpha^k}{(nk+r)!}x^{nk+r}}

Formal Mathematical property (FMP):

OpenMath XML (source)

    <OMOBJ xmlns="http://www.openmath.org/OpenMath">
      <OMA>
        <OMS cd="logic1" name="implies"/>
        <OMA>
          <OMS cd="logic1" name="and"/>
          <OMA>
            <OMS cd="set1" name="in"/>
            <OMV name="alpha"/>
            <OMS cd="setname1" name="C"/>
          </OMA>
          <OMA>
            <OMS cd="set1" name="in"/>
            <OMV name="n"/>
            <OMS cd="setname1" name="Z"/>
          </OMA>
          <OMA>
            <OMS cd="set1" name="in"/>
            <OMV name="r"/>
            <OMA>
              <OMS cd="interval1" name="integer_interval"/>
              <OMI> 0 </OMI>
              <OMA>
                <OMS cd="arith1" name="minus"/>
                <OMV name="n"/><OMI> 1 </OMI>
              </OMA>
            </OMA>
          </OMA>
        </OMA>
        <OMA>
          <OMS cd="relation1" name="eq"/>
          <OMA>
            <OMA>
              <OMS cd="gen_hyperbolic1" name="generalised_hyperbolic"/>
              <OMV name="alpha"/>
              <OMV name="n"/>
              <OMV name="r"/>
            </OMA>
            <OMV name="x"/>
          </OMA>
          <OMA>
            <OMS cd="arith1" name="sum"/>
            <OMA>
              <OMS cd="interval1" name="integer_interval"/>
              <OMS cd="alg1" name="zero"/>
              <OMS cd="nums1" name="infinity"/>
            </OMA>
            <OMBIND>
              <OMS cd="fns1" name="lambda"/>
              <OMBVAR>
                <OMV name="k"/>
              </OMBVAR>
              <OMA>
                <OMS cd="arith1" name="times"/>
                <OMA>
                  <OMS cd="arith1" name="divide"/>
                  <OMA>
                    <OMS cd="arith1" name="power"/>
                    <OMV name="alpha"/>
                    <OMV name="k"/>
                  </OMA>
                  <OMA>
                    <OMS cd="integer1" name="factorial"/>
                    <OMA>
                      <OMS cd="arith1" name="plus"/>
                      <OMA>
                        <OMS cd="arith1" name="times"/>
                        <OMV name="n"/>
                        <OMV name="k"/>
                      </OMA>
                      <OMV name="r"/>
                    </OMA>
                  </OMA>
                </OMA>
                <OMA>
                  <OMS cd="arith1" name="power"/>
                  <OMV name="x"/>
                  <OMA>
                    <OMS cd="arith1" name="plus"/>
                    <OMA>
                      <OMS cd="arith1" name="times"/>
                      <OMV name="n"/>
                      <OMV name="k"/>
                    </OMA>
                    <OMV name="r"/>
                  </OMA>
                </OMA>
              </OMA>
            </OMBIND>
          </OMA>
        </OMA>
      </OMA>
    </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="set1">in</csymbol><ci>alpha</ci><csymbol cd="setname1">C</csymbol></apply>
   <apply><csymbol cd="set1">in</csymbol><ci>n</ci><csymbol cd="setname1">Z</csymbol></apply>
   <apply><csymbol cd="set1">in</csymbol>
    <ci>r</ci>
    <apply><csymbol cd="interval1">integer_interval</csymbol>
     <cn type="integer">0</cn>
     <apply><csymbol cd="arith1">minus</csymbol><ci>n</ci><cn type="integer">1</cn></apply>
    </apply>
   </apply>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply>
    <apply><csymbol cd="gen_hyperbolic1">generalised_hyperbolic</csymbol><ci>alpha</ci><ci>n</ci><ci>r</ci></apply>
    <ci>x</ci>
   </apply>
   <apply><csymbol cd="arith1">sum</csymbol>
    <apply><csymbol cd="interval1">integer_interval</csymbol>
     <csymbol cd="alg1">zero</csymbol>
     <csymbol cd="nums1">infinity</csymbol>
    </apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>k</ci></bvar>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">divide</csymbol>
       <apply><csymbol cd="arith1">power</csymbol><ci>alpha</ci><ci>k</ci></apply>
       <apply><csymbol cd="integer1">factorial</csymbol>
        <apply><csymbol cd="arith1">plus</csymbol>
         <apply><csymbol cd="arith1">times</csymbol><ci>n</ci><ci>k</ci></apply>
         <ci>r</ci>
        </apply>
       </apply>
      </apply>
      <apply><csymbol cd="arith1">power</csymbol>
       <ci>x</ci>
       <apply><csymbol cd="arith1">plus</csymbol>
        <apply><csymbol cd="arith1">times</csymbol><ci>n</ci><ci>k</ci></apply>
        <ci>r</ci>
       </apply>
      </apply>
     </apply>
    </bind>
   </apply>
  </apply>
 </apply>
</math>

Prefix

implies (and (in ( alpha, C) , in ( n, Z) , in ( r, integer_interval ( 0 , minus ( n, 1 ) ) ) ) , eq (generalised_hyperbolic ( alpha, n, r) ( x) , sum (integer_interval (zero, infinity) , lambda [ k ] . (times (divide (power ( alpha, k) , factorial (plus (times ( n, k) , r) ) ) , power ( x, plus (times ( n, k) , r) ) ) ) ) ) )

Popcorn

set1.in($alpha, setname1.C) and set1.in($n, setname1.Z) and set1.in($r, interval1.integer_interval(0, $n - 1)) ==> gen_hyperbolic1.generalised_hyperbolic($alpha, $n, $r)($x) = arith1.sum(interval1.integer_interval(alg1.zero, nums1.infinity), fns1.lambda[$k -> $alpha ^ $k / integer1.factorial($n * $k + $r) * $x ^ ($n * $k + $r)])

Rendered Presentation MathML

$$\mathrm{alpha}\in \mathbb{C}\wedge n\in \mathbb{Z}\wedge r\in [0,n-1]\Rightarrow \left(\mathrm{generalised\_hyperbolic}(\mathrm{alpha},n,r)\right)\left(x\right)=\sum _{k=0}^{\infty }\frac{{\mathrm{alpha}}^k}{(nk+r)!}{x}^{(nk+r)}$$

Commented Mathematical property (CMP):

for all z \in C F^1_{1,0} (z) = e^z

Formal Mathematical property (FMP):

OpenMath XML (source)

    <OMOBJ xmlns="http://www.openmath.org/OpenMath">
      <OMBIND>
        <OMS cd="quant1" name="forall"/>
        <OMBVAR>
          <OMV name="z"/>
        </OMBVAR>
        <OMA>
          <OMS cd="logic1" name="implies"/>
          <OMA>
            <OMS cd="set1" name="in"/>
            <OMV name="z"/>
            <OMS cd="setname1" name="C"/>
          </OMA>
          <OMA>
            <OMS cd="relation1" name="eq"/>
            <OMA>
              <OMA>
                <OMS cd="gen_hyperbolic1" name="generalised_hyperbolic"/>
                <OMI> 1 </OMI><OMI> 1 </OMI><OMI> 0 </OMI>
              </OMA>
              <OMV name="z"/>
            </OMA>
            <OMA>
              <OMS cd="transc1" name="exp"/>
              <OMV name="z"/>
            </OMA>
          </OMA>
        </OMA>
      </OMBIND>
    </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>z</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="set1">in</csymbol><ci>z</ci><csymbol cd="setname1">C</csymbol></apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply>
     <apply><csymbol cd="gen_hyperbolic1">generalised_hyperbolic</csymbol>
      <cn type="integer">1</cn>
      <cn type="integer">1</cn>
      <cn type="integer">0</cn>
     </apply>
     <ci>z</ci>
    </apply>
    <apply><csymbol cd="transc1">exp</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ z ] . (implies (in ( z, C) , eq (generalised_hyperbolic ( 1 , 1 , 0 ) ( z) , exp ( z) ) ) )

Popcorn

quant1.forall[$z -> set1.in($z, setname1.C) ==> gen_hyperbolic1.generalised_hyperbolic(1, 1, 0)($z) = exp($z)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}z.z\in \mathbb{C}\Rightarrow \left(\mathrm{generalised\_hyperbolic}(1,1,0)\right)\left(z\right)=\exp \left(z\right)$$

Commented Mathematical property (CMP):

for all z \in C F^{-1}_{2,-1} (z) = sin(z)

Formal Mathematical property (FMP):

OpenMath XML (source)

    <OMOBJ xmlns="http://www.openmath.org/OpenMath">
      <OMBIND>
        <OMS cd="quant1" name="forall"/>
        <OMBVAR>
          <OMV name="z"/>
        </OMBVAR>
        <OMA>
          <OMS cd="logic1" name="implies"/>
          <OMA>
            <OMS cd="set1" name="in"/>
            <OMV name="z"/>
            <OMS cd="setname1" name="C"/>
          </OMA>
          <OMA>
            <OMS cd="relation1" name="eq"/>
            <OMA>
              <OMA>
                <OMS cd="gen_hyperbolic1" name="generalised_hyperbolic"/>
                <OMI> -1 </OMI><OMI> 2 </OMI><OMI> -1 </OMI>
              </OMA>
              <OMV name="z"/>
            </OMA>
            <OMA>
              <OMS cd="transc1" name="sin"/>
              <OMV name="z"/>
            </OMA>
          </OMA>
        </OMA>
      </OMBIND>
    </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>z</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="set1">in</csymbol><ci>z</ci><csymbol cd="setname1">C</csymbol></apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply>
     <apply><csymbol cd="gen_hyperbolic1">generalised_hyperbolic</csymbol>
      <cn type="integer">-1</cn>
      <cn type="integer">2</cn>
      <cn type="integer">-1</cn>
     </apply>
     <ci>z</ci>
    </apply>
    <apply><csymbol cd="transc1">sin</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ z ] . (implies (in ( z, C) , eq (generalised_hyperbolic ( -1 , 2 , -1 ) ( z) , sin ( z) ) ) )

Popcorn

quant1.forall[$z -> set1.in($z, setname1.C) ==> gen_hyperbolic1.generalised_hyperbolic(-1, 2, -1)($z) = sin($z)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}z.z\in \mathbb{C}\Rightarrow \left(\mathrm{generalised\_hyperbolic}(-1,2,-1)\right)\left(z\right)=\sin \left(z\right)$$

Commented Mathematical property (CMP):

for all z \in C F^{-1}_{2,0} (z) = cos(z)

Formal Mathematical property (FMP):

OpenMath XML (source)

    <OMOBJ xmlns="http://www.openmath.org/OpenMath">
      <OMBIND>
        <OMS cd="quant1" name="forall"/>
        <OMBVAR>
          <OMV name="z"/>
        </OMBVAR>
        <OMA>
          <OMS cd="logic1" name="implies"/>
          <OMA>
            <OMS cd="set1" name="in"/>
            <OMV name="z"/>
            <OMS cd="setname1" name="C"/>
          </OMA>
          <OMA>
            <OMS cd="relation1" name="eq"/>
            <OMA>
              <OMA>
                <OMS cd="gen_hyperbolic1" name="generalised_hyperbolic"/>
                <OMI> -1 </OMI><OMI> 2 </OMI><OMI> 0 </OMI>
              </OMA>
              <OMV name="z"/>
            </OMA>
            <OMA>
              <OMS cd="transc1" name="cos"/>
              <OMV name="z"/>
            </OMA>
          </OMA>
        </OMA>
      </OMBIND>
    </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>z</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="set1">in</csymbol><ci>z</ci><csymbol cd="setname1">C</csymbol></apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply>
     <apply><csymbol cd="gen_hyperbolic1">generalised_hyperbolic</csymbol>
      <cn type="integer">-1</cn>
      <cn type="integer">2</cn>
      <cn type="integer">0</cn>
     </apply>
     <ci>z</ci>
    </apply>
    <apply><csymbol cd="transc1">cos</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ z ] . (implies (in ( z, C) , eq (generalised_hyperbolic ( -1 , 2 , 0 ) ( z) , cos ( z) ) ) )

Popcorn

quant1.forall[$z -> set1.in($z, setname1.C) ==> gen_hyperbolic1.generalised_hyperbolic(-1, 2, 0)($z) = cos($z)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}z.z\in \mathbb{C}\Rightarrow \left(\mathrm{generalised\_hyperbolic}(-1,2,0)\right)\left(z\right)=\cos \left(z\right)$$

Commented Mathematical property (CMP):

for all z \in C F^{1}_{2,1} (z) = sinh(z)

Formal Mathematical property (FMP):

OpenMath XML (source)

    <OMOBJ xmlns="http://www.openmath.org/OpenMath">
      <OMBIND>
        <OMS cd="quant1" name="forall"/>
        <OMBVAR>
          <OMV name="z"/>
        </OMBVAR>
        <OMA>
          <OMS cd="logic1" name="implies"/>
          <OMA>
            <OMS cd="set1" name="in"/>
            <OMV name="z"/>
            <OMS cd="setname1" name="C"/>
          </OMA>
          <OMA>
            <OMS cd="relation1" name="eq"/>
            <OMA>
              <OMA>
                <OMS cd="gen_hyperbolic1" name="generalised_hyperbolic"/>
                <OMI> 1 </OMI><OMI> 2 </OMI><OMI> 1 </OMI>
              </OMA>
              <OMV name="z"/>
            </OMA>
            <OMA>
              <OMS cd="transc1" name="sinh"/>
              <OMV name="z"/>
            </OMA>
          </OMA>
        </OMA>
      </OMBIND>
    </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>z</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="set1">in</csymbol><ci>z</ci><csymbol cd="setname1">C</csymbol></apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply>
     <apply><csymbol cd="gen_hyperbolic1">generalised_hyperbolic</csymbol>
      <cn type="integer">1</cn>
      <cn type="integer">2</cn>
      <cn type="integer">1</cn>
     </apply>
     <ci>z</ci>
    </apply>
    <apply><csymbol cd="transc1">sinh</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ z ] . (implies (in ( z, C) , eq (generalised_hyperbolic ( 1 , 2 , 1 ) ( z) , sinh ( z) ) ) )

Popcorn

quant1.forall[$z -> set1.in($z, setname1.C) ==> gen_hyperbolic1.generalised_hyperbolic(1, 2, 1)($z) = sinh($z)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}z.z\in \mathbb{C}\Rightarrow \left(\mathrm{generalised\_hyperbolic}(1,2,1)\right)\left(z\right)=\sinh \left(z\right)$$

Commented Mathematical property (CMP):

for all z \in C F^{1}_{2,0} (z) = cosh(z)

Formal Mathematical property (FMP):

OpenMath XML (source)

    <OMOBJ xmlns="http://www.openmath.org/OpenMath">
      <OMBIND>
        <OMS cd="quant1" name="forall"/>
        <OMBVAR>
          <OMV name="z"/>
        </OMBVAR>
        <OMA>
          <OMS cd="logic1" name="implies"/>
          <OMA>
            <OMS cd="set1" name="in"/>
            <OMV name="z"/>
            <OMS cd="setname1" name="C"/>
          </OMA>
          <OMA>
            <OMS cd="relation1" name="eq"/>
            <OMA>
              <OMA>
                <OMS cd="gen_hyperbolic1" name="generalised_hyperbolic"/>
                <OMI> 1 </OMI><OMI> 2 </OMI><OMI> 0 </OMI>
              </OMA>
              <OMV name="z"/>
            </OMA>
            <OMA>
              <OMS cd="transc1" name="cosh"/>
              <OMV name="z"/>
            </OMA>
          </OMA>
        </OMA>
      </OMBIND>
    </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>z</ci></bvar>
  <apply><csymbol cd="logic1">implies</csymbol>
   <apply><csymbol cd="set1">in</csymbol><ci>z</ci><csymbol cd="setname1">C</csymbol></apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply>
     <apply><csymbol cd="gen_hyperbolic1">generalised_hyperbolic</csymbol>
      <cn type="integer">1</cn>
      <cn type="integer">2</cn>
      <cn type="integer">0</cn>
     </apply>
     <ci>z</ci>
    </apply>
    <apply><csymbol cd="transc1">cosh</csymbol><ci>z</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ z ] . (implies (in ( z, C) , eq (generalised_hyperbolic ( 1 , 2 , 0 ) ( z) , cosh ( z) ) ) )

Popcorn

quant1.forall[$z -> set1.in($z, setname1.C) ==> gen_hyperbolic1.generalised_hyperbolic(1, 2, 0)($z) = cosh($z)]

Rendered Presentation MathML

$$\forall \hspace{0.17em}z.z\in \mathbb{C}\Rightarrow \left(\mathrm{generalised\_hyperbolic}(1,2,0)\right)\left(z\right)=\cosh \left(z\right)$$

Signatures:

sts