OpenMath Content Dictionary: cauchypv

Canonical URL:

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

CD File:

cauchypv.ocd

CD as XML Encoded OpenMath:

cauchypv.omcd

Defines:

defint

Date:

2001-09-23

Version:

1 (Revision 1)

Review Date:

2017-12-31

Status:

experimental

---

     This document is distributed in the hope that it will be useful, 
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     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

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     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
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          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
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       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
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     If you have questions about this license please contact the OpenMath
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  Author: James Davenport

This CD extends calculus1 (compatible with the calculus operations in Content MathML) with a Cauchy principal value definite integral.

---

defint

Description:

This symbol is used to represent definite (Cauchy principal value) integration of unary functions. It takes two arguments; the first being the range (e.g. a set) of integration, and the second the function.

Commented Mathematical property (CMP):

for all a,b | integral from a to b = -integral from b to a

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMBIND>
    <OMS cd="quant1" name="forall"/>
    <OMBVAR>
      <OMV name="a"/>
      <OMV name="b"/>
    </OMBVAR>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="cauchypv" name="defint"/>
	<OMA>
	  <OMS cd="interval1" name="interval"/>
	  <OMV name="a"/>
	  <OMV name="b"/>
	</OMA>
	<OMBIND>
	  <OMS cd="fns1" name="lambda"/>
	  <OMBVAR>
	    <OMV name="x"/>
	  </OMBVAR>
	  <OMA>
	    <OMV name="f"/>
	    <OMV name="x"/>
	  </OMA>
	</OMBIND>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
        <OMA>
          <OMS cd="cauchypv" name="defint"/>
	  <OMA>
	    <OMS cd="interval1" name="interval"/>
	    <OMV name="b"/>
	    <OMV name="a"/>
	  </OMA>
	  <OMBIND>
	    <OMS cd="fns1" name="lambda"/>
	    <OMBVAR>
	      <OMV name="x"/>
	    </OMBVAR>
	    <OMA>
	      <OMV name="f"/>
	      <OMV name="x"/>
	    </OMA>
	  </OMBIND>
        </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>a</ci></bvar>
  <bvar><ci>b</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="cauchypv">defint</csymbol>
    <apply><csymbol cd="interval1">interval</csymbol><ci>a</ci><ci>b</ci></apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>x</ci></bvar>
     <apply><ci>f</ci><ci>x</ci></apply>
    </bind>
   </apply>
   <apply><csymbol cd="arith1">unary_minus</csymbol>
    <apply><csymbol cd="cauchypv">defint</csymbol>
     <apply><csymbol cd="interval1">interval</csymbol><ci>b</ci><ci>a</ci></apply>
     <bind><csymbol cd="fns1">lambda</csymbol>
      <bvar><ci>x</ci></bvar>
      <apply><ci>f</ci><ci>x</ci></apply>
     </bind>
    </apply>
   </apply>
  </apply>
 </bind>
</math>

Prefix

forall [ a b ] . (eq (defint (interval ( a, b) , lambda [ x ] . ( f ( x) ) ) , unary_minus (defint (interval ( b, a) , lambda [ x ] . ( f ( x) ) ) ) ) )

Popcorn

quant1.forall[$a, $b -> cauchypv.defint(interval1.interval($a, $b), fns1.lambda[$x -> $f($x)]) = -(cauchypv.defint(interval1.interval($b, $a), fns1.lambda[$x -> $f($x)]))]

Rendered Presentation MathML

$$\forall \hspace{0.17em}a,b.\mathrm{defint}((a,b),\lambda \hspace{0.17em}x.f\left(x\right))=-\mathrm{defint}((b,a),\lambda \hspace{0.17em}x.f\left(x\right))$$

Example:

An example: the definition of the exponential integral. $$Ei(x)=\int_{-\infty}^x\frac{e^t}t dt\qquad(x>0)$$

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
      <OMA>
        <OMS name="implies" cd="logic1"/>
        <OMA>
          <OMS name="gt" cd="relation1"/>
          <OMV name="x"/>
          <OMS name="zero" cd="alg1"/>
        </OMA>
        <OMA>
          <OMS name="eq" cd="relation1"/>
          <OMA>
            <OMS name="Ei" cd="expint"/>
            <OMV name="x"/>
          </OMA>
          <OMA>
            <OMS name="defint" cd="cauchypv"/>
            <OMA>
              <OMS name="interval" cd="interval1"/>
              <OMA>
                <OMS name="unary_minus" cd="arith1"/>
                <OMS name="infinity" cd="nums1"/>
              </OMA>
              <OMV name="x"/>
            </OMA>
            <OMBIND>
	    <OMV name="missing"/>
              <OMBVAR>
                <OMV name="t"/>
              </OMBVAR>
              <OMA>
                <OMS name="divide" cd="arith1"/>
                <OMA>
                  <OMS name="exp" cd="transc1"/>
                  <OMV name="t"/>
                </OMA>
                <OMV name="t"/>
              </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="relation1">gt</csymbol><ci>x</ci><csymbol cd="alg1">zero</csymbol></apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="expint">Ei</csymbol><ci>x</ci></apply>
   <apply><csymbol cd="cauchypv">defint</csymbol>
    <apply><csymbol cd="interval1">interval</csymbol>
     <apply><csymbol cd="arith1">unary_minus</csymbol><csymbol cd="nums1">infinity</csymbol></apply>
     <ci>x</ci>
    </apply>
    <bind>
     <ci>missing</ci>
     <bvar><ci>t</ci></bvar>
     <apply><csymbol cd="arith1">divide</csymbol>
      <apply><csymbol cd="transc1">exp</csymbol><ci>t</ci></apply>
      <ci>t</ci>
     </apply>
    </bind>
   </apply>
  </apply>
 </apply>
</math>

Prefix

implies (gt ( x, zero) , eq (Ei ( x) , defint (interval (unary_minus (infinity) , x) , missing [ t ] . (divide (exp ( t) , t) ) ) ) )

Popcorn

$x > alg1.zero ==> expint.Ei($x) = cauchypv.defint(interval1.interval( -(nums1.infinity), $x), $missing[$t -> exp($t) / $t])

Rendered Presentation MathML

$$x>0\Rightarrow \mathrm{Ei}\left(x\right)=\mathrm{defint}((-\infty ,x),\mathrm{missing}\hspace{0.17em}t.\frac{\exp \left(t\right)}{t})$$

Signatures:

sts

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