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

Review Date:

2002-09-01

Status:

experimental

Uses CD:

alg1, arith1, expint, fns1, interval1, list1, nums1, transc1, relation1, quant1, logic1

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>
              <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>
     <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) , t [divide (exp ( t) , t) ] . () ) ) )

Popcorn

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

Rendered Presentation MathML

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

Signatures:

sts