OpenMath Content Dictionary: gen_hyperbolic1

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

Review Date:

2005-01-01

Status:

experimental

Uses CD:

alg1, arith1, fns1, integer1, interval1, logic1, nums1, quant1, relation1, set1, setname1, transc1

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):

xml prefix mathml

<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>

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) ) ) ) ) ) )

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

Commented Mathematical property (CMP):

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

Formal Mathematical property (FMP):

xml prefix mathml

<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>

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

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

Commented Mathematical property (CMP):

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

Formal Mathematical property (FMP):

xml prefix mathml

<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>

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

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

Commented Mathematical property (CMP):

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

Formal Mathematical property (FMP):

xml prefix mathml

<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>

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

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

Commented Mathematical property (CMP):

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

Formal Mathematical property (FMP):

xml prefix mathml

<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>

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

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

Commented Mathematical property (CMP):

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

Formal Mathematical property (FMP):

xml prefix mathml

<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>

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

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

Signatures:

sts