OpenMath Content Dictionary: expint

Canonical URL:

http://www.openmath.org/CDs/expint.ocd

CD File:

expint.ocd

CD as XML Encoded OpenMath:

expint.omcd

Defines:

E, Ei, li

Date:

2002-01-17

Version:

1

Review Date:

2003-04-01

Status:

experimental

Uses CD:

alg1, arith1, calculus1, cauchypv, complex1, fns1, interval1, logic1, nums1, relation1, transc1

This content dictionary contains symbols to describe the Exponential integral and associated functions.

Ei

Description:

The symbol Ei defines the basic exponential integral as in Abramovitz & Stegun equation 5.1.2. This is a Cauchy principal value integral:

$$Ei(x)=\int_{-\infty}^x\frac{e^t}t dt\qquad(x>0)$$

which is then extended by analytic continuation (this latter is not currently represented in the FMPs) to the complex plane slit along the negative real axis

Formal Mathematical property (FMP):

xml prefix mathml

<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>
              <OMS name="lambda" cd="fns1"/>
              <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>

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

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

Signatures:

sts

li

Description:

The symbol li defines the basic logarithmic integral as in Abramovitz & Stegun equation 5.1.2. This is a Cauchy principal value integral:

$$li(x)=\int_0^x\frac1{\ln t}t dt\qquad(x>1)$$

which is then extended by analytic continuation (this latter is not currently represented in the FMPs) to the complex plane slit along the negative real axis

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
      <OMA>
        <OMS name="implies" cd="logic1"/>
        <OMA>
          <OMS name="gt" cd="relation1"/>
          <OMV name="x"/>
          <OMS name="one" cd="alg1"/>
        </OMA>
        <OMA>
          <OMS name="eq" cd="relation1"/>
          <OMA>
            <OMS name="li" cd="expint"/>
            <OMV name="x"/>
          </OMA>
          <OMA>
            <OMS name="defint" cd="cauchypv"/>
            <OMA>
              <OMS name="interval" cd="interval1"/>
              <OMS name="zero" cd="nums1"/>
              <OMV name="x"/>
            </OMA>
            <OMBIND>
              <OMS name="lambda" cd="fns1"/>
              <OMBVAR>
                <OMV name="t"/>
              </OMBVAR>
              <OMA>
                <OMS name="divide" cd="arith1"/>
                <OMS name="one" cd="alg1"/>
                <OMA>
                  <OMS name="ln" cd="transc1"/>
                  <OMV name="t"/>
                </OMA>
              </OMA>
            </OMBIND>
          </OMA>
        </OMA>
      </OMA>
    </OMOBJ>

implies (gt ( x, one) , eq (li ( x) , defint (interval (zero, x) , lambda [ t ] . (divide (one, ln ( t) ) ) ) ) )

$$x>1\Rightarrow \mathrm{li}\left(x\right)=\mathrm{defint}((\mathrm{zero},x),\lambda \hspace{0.17em}t.\frac{1}{\ln \left(t\right)})$$

Signatures:

sts

E

Description:

The symbol E defines the generalised exponential integral as in Abramovitz & Stegun equation 5.1.4. This is an ordinary integral:

$$E_n(z)=\int_1^{-\infty}\frac{e^{-zt}}{t^n} dt\qquad(\Re z>0)$$

which is then extended by analytic continuation (this latter is not currently represented in the FMPs) to the complex plane slit along the negative real axis. Note that OpenMath's definition is curried, i.e. E(n) is a function.

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
      <OMA>
        <OMS name="implies" cd="logic1"/>
        <OMA>
          <OMS name="gt" cd="relation1"/>
          <OMA>
            <OMS name="real" cd="complex1"/>
            <OMV name="z"/>
          </OMA>
          <OMS name="zero" cd="alg1"/>
        </OMA>
        <OMA>
          <OMS name="eq" cd="relation1"/>
          <OMA>
            <OMA>
              <OMS name="E" cd="expint"/>
              <OMV name="n"/>
            </OMA>
            <OMV name="z"/>
          </OMA>
          <OMA>
            <OMS name="defint" cd="calculus1"/>
            <OMA>
              <OMS name="interval" cd="interval1"/>
              <OMS name="one" cd="alg1"/>
              <OMS name="infinity" cd="nums1"/>
            </OMA>
            <OMBIND>
              <OMS name="lambda" cd="fns1"/>
              <OMBVAR>
                <OMV name="t"/>
              </OMBVAR>
              <OMA>
                <OMS name="divide" cd="arith1"/>
                <OMA>
                  <OMS name="exp" cd="transc1"/>
                  <OMA>
                    <OMS name="times" cd="arith1"/>
                    <OMI> -1 </OMI>
                    <OMV name="z"/>
                    <OMV name="t"/>
                  </OMA>
                </OMA>
                <OMA>
                  <OMS name="power" cd="arith1"/>
                  <OMV name="t"/>
                  <OMV name="n"/>
                </OMA>
              </OMA>
            </OMBIND>
          </OMA>
        </OMA>
      </OMA>
    </OMOBJ>

implies (gt (real ( z) , zero) , eq (E ( n) ( z) , defint (interval (one, infinity) , lambda [ t ] . (divide (exp (times ( -1 , z, t) ) , power ( t, n) ) ) ) ) )

$$\mathrm{real}\left(z\right)>0\Rightarrow \left(E\left(n\right)\right)\left(z\right)=\underset{1}{\overset{\infty }{\int }}\frac{\exp \left(-1zt\right)}{t^n}dt$$

Signatures:

sts