OpenMath Content Dictionary: elementryInts

Canonical URL:
http://www.openmath.org/CDs/elementryInts.ocd
CD File:
elementryInts.ocd
CD as XML Encoded OpenMath:
elementryInts.omcd
Defines:
expIntegralE, expIntegralEi, logIntegral
Date:
Version:
(Revision )
Review Date:
Status:
private
Uses CD:
alg1, arith1, calculus1, complex1, fns1, integer1, interval1, logic1, nums1, relation1, setname1, sts, transc1

1/1/2002

This content dictionary contains symbols to describe the exponential integral function and related functions. They are defined as in Abromowitz and Stegun (ninth printing on).


expIntegralE

The expIntegralE symbol represents the Exponential Integral function. It takes two arguments, the first argument is the order of the function, the second is the point about which it is being taken.

The exponential integral function is defined by $E_n(z)=\int_{1}^{\infty} \frac{e^{-zt}}{t^n}dt$ where $(n=0, 1, 2, \dots;\ \Re{z}>0)$.

It has a branch cut along the negative real axis in the complex $z$ plane.

It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [5.1.4]

Commented Mathematical property (CMP):
$\Re{z}>0 /Rightarrow E_n(z)=\int_{1}^{\infty} \frac{e^{-zt}}{t^n}dt$
Formal Mathematical property (FMP):
<OMOBJ>
	<OMA>
	  <OMS cd="logic1" name="implies"/>
	  <OMA>
	    <OMS cd="relation1" name="gt"/>
	    <OMA>
	      <OMS cd="complex1" name="real"/>
              <OMV name="z"/>
	    </OMA>
	    <OMS cd="alg1" name="zero"/>
	  </OMA>
	  <OMA>
	    <OMS cd="relation1" name="eq"/>
	    <OMA>
	      <OMS cd="elementryInts" name="expIntegralE"/>
              <OMV name="n"/>
              <OMV name="z"/>
	    </OMA>
	    <OMA>
	      <OMS cd="calculus1" name="defint"/>
	      <OMA>
		<OMS cd="interval1" name="interval_co"/>
		<OMS cd="alg1" name="one"/>
		<OMS cd="nums1" name="infinity"/>
	      </OMA>
              <OMBIND>
		<OMS cd="fns1" name="lambda"/>
		<OMBVAR>
		  <OMV name="t"/>
		</OMBVAR>
	        <OMA>
		  <OMS cd="arith1" name="divide"/>
		  <OMA>
		    <OMS cd="arith1" name="power"/>
		    <OMS cd="nums1" name="e"/>
		    <OMA>
		      <OMS cd="arith1" name="unary_minus"/>
		      <OMA>
			<OMS cd="arith1" name="times"/>
			<OMV name="z"/>
			<OMV name="t"/>
		      </OMA>
		    </OMA>
		  </OMA>
		  <OMA>
		    <OMS cd="arith1" name="power"/>
		    <OMV name="t"/>
		    <OMV name="n"/>
		  </OMA>
		</OMA>
              </OMBIND>
	    </OMA>
	  </OMA>
	</OMA>
      </OMOBJ>

implies (gt (real ( z) , zero) , eq (expIntegralE ( n, z) , defint (interval_co (one, infinity) , lambda [ t ] . (divide (power (e, unary_minus (times ( z, t) ) ) , power ( t, n) ) ) ) ) )

Commented Mathematical property (CMP):
n \in Z, x \in R implies expIntegralE(n,x) \in R
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA><OMS cd="logic1" name="and"/>
        <OMA><OMS cd="set1" name="in"/>
          <OMV name="n"/>
          <OMS cd="setname1" name="Z"/>
        </OMA>
        <OMA><OMS cd="set1" name="in"/>
          <OMV name="x"/>
          <OMS cd="setname1" name="R"/>
        </OMA>
      </OMA>
      <OMA><OMS cd="set1" name="in"/>
        <OMA><OMS cd="elementryInts" name="expIntegralE"/>
          <OMV name="n"/>
          <OMV name="x"/>
        </OMA>
        <OMS cd="setname1" name="R"/>
      </OMA>
    </OMA>
  </OMOBJ>

implies (and (in ( n, Z) , in ( x, R) ) , in (expIntegralE ( n, x) , R) )

Commented Mathematical property (CMP):
n \in Z, x \in C implies expIntegralE(n,x) \in C
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA><OMS cd="logic1" name="and"/>
        <OMA><OMS cd="set1" name="in"/>
          <OMV name="n"/>
          <OMS cd="setname1" name="Z"/>
        </OMA>
        <OMA><OMS cd="set1" name="in"/>
          <OMV name="z"/>
          <OMS cd="setname1" name="C"/>
        </OMA>
      </OMA>
      <OMA><OMS cd="set1" name="in"/>
        <OMA><OMS cd="elementryInts" name="expIntegralE"/>
          <OMV name="n"/>
          <OMV name="z"/>
        </OMA>
        <OMS cd="setname1" name="C"/>
      </OMA>
    </OMA>
  </OMOBJ>

implies (and (in ( n, Z) , in ( z, C) ) , in (expIntegralE ( n, z) , C) )

Commented Mathematical property (CMP):
exists(n,x,y) s.t. expIntegralE(n,x)=y
Formal Mathematical property (FMP):
<OMOBJ>
    <OMBIND>
      <OMS cd="quant1" name="exists"/>
      <OMBVAR>
        <OMV name="n"/>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMBVAR>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="elementryInts" name="expIntegralE"/>
          <OMV name="n"/>
          <OMV name="x"/>
        </OMA>
        <OMV name="y"/>
      </OMA>
    </OMBIND>
  </OMOBJ>

exists [ n x y ] . (eq (expIntegralE ( n, x) , y) )

Commented Mathematical property (CMP):
$E_n(\bar{z})=\overline{E_n(z)}$, symmetry
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
	<OMS cd="elementryInts" name="expIntegralE"/>
	<OMV name="n"/>
	<OMA>
	  <OMS cd="complex1" name="conjugate"/>
	  <OMV name="z"/>
	</OMA>
      </OMA>
      <OMA>
	<OMS cd="complex1" name="conjugate"/>
	<OMA>
	  <OMS cd="elementryInts" name="expIntegralE"/>
	  <OMV name="n"/>
	  <OMV name="z"/>
	</OMA>
      </OMA>
    </OMA>
  </OMOBJ>

eq (expIntegralE ( n, conjugate ( z) ) , conjugate (expIntegralE ( n, z) ) )

Commented Mathematical property (CMP):
$E_{n+1}(z)=\frac{1}{n}[e^{-z}-zE_n(z)]$, $n=1, 2, \dots$, recurrence relation
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="elementryInts" name="expIntegralE"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMV name="n"/>
	  <OMS cd="alg1" name="one"/>
	</OMA>
	<OMV name="z"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMA>
	  <OMS cd="arith1" name="divide"/>
	  <OMS cd="alg1" name="one"/>
	  <OMV name="n"/>
	</OMA>
	<OMA>
	  <OMS cd="arith1" name="minus"/>
	  <OMA>
	    <OMS cd="transc1" name="exp"/>
	    <OMA>
	      <OMS cd="arith1" name="unary_minus"/>
	      <OMV name="z"/>
	    </OMA>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMV name="z"/>
	    <OMA>
	      <OMS cd="elementryInts" name="expIntegralE"/>
	      <OMV name="n"/>
	      <OMV name="z"/>
	    </OMA>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMOBJ>

eq (expIntegralE (plus ( n, one) , z) , times (divide (one, n) , minus (exp (unary_minus ( z) ) , times ( z, expIntegralE ( n, z) ) ) ) )

Commented Mathematical property (CMP):
$\frac{dE_n(z)}{dz}=-E_{n-1}(z)$, $n=1, 2, \dots$, derivative
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="calculus1" name="diff"/>
      <OMBIND>
        <OMBVAR>
	  <OMV name="z"/>
	</OMBVAR>
	<OMA>
	  <OMS cd="elementryInts" name="expIntegralE"/>
	  <OMV name="n"/>
	  <OMV name="z"/>
	</OMA>
      </OMBIND>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="unary_minus"/>
      <OMS cd="elementryInts" name="expIntegralE"/>
      <OMA>
        <OMS cd="arith1" name="minus"/>
	<OMV name="n"/>
	<OMS cd="alg1" name="one"/>
      </OMA>
      <OMV name="z"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (diff ( z [expIntegralE ( n, z) ] . () ) , unary_minus (expIntegralE, minus ( n, one) , z) )

Commented Mathematical property (CMP):
$E_n(0)=\frac{1}{n-1}$, $n>1$, special value
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="relation1" name="gt"/>
      <MOV name="n"/>
      <OMS cd="alg1" name="one"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="elementryInts" name="expIntegralE"/>
        <OMV name="n"/>
        <OMS cd="alg1" name="zero"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="divide"/>
	<OMS cd="alg1" name="one"/>
	<OMA>
          <OMS cd="arith1" name="minus"/>
	  <OMV name="n"/>
	  <OMS cd="alg1" name="one"/>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

implies (gt (, one) , eq (expIntegralE ( n, zero) , divide (one, minus ( n, one) ) ) )

Commented Mathematical property (CMP):
$E_0(z)=\frac{e^z}{z}$, special value
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="elementryInts" name="expIntegralE"/>
      <OMS cd="alg1" name="zero"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMA>
        <OMS cd="transc1" name="exp"/>
	<OMV name="z"/>
      </OMA>
      <OMV name="z"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (expIntegralE (zero, z) , divide (exp ( z) , z) )

Signatures:
sts


[Next: expIntegralEi] [Last: logIntegral] [Top]

expIntegralEi

This symbol represents the exponential integral function, it takes one argument. the exponential integral function is normally denoted by Ei(z).

The exponential integral function is defined by $Ei(z)=-\int_{-z}^{\infty} \frac{e^{-t}}{t}dt$. where the principal value is taken. It has a branch cut along the negative real axis in the complex $z$ plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [5.1.2]

Commented Mathematical property (CMP):
x in Reals implies Ei(x) in Reals
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA><OMS cd="set1" name="in"/>
        <OMV name="x"/>
        <OMS cd="setname1" name="R"/>
      </OMA>
      <OMA><OMS cd="set1" name="in"/>
        <OMA><OMS cd="elementryInts" name="expIntegralEi"/>
          <OMV name="x"/>
        </OMA>
        <OMS cd="setname1" name="R"/>
      </OMA>
    </OMA>
  </OMOBJ>

implies (in ( x, R) , in (expIntegralEi ( x) , R) )

Commented Mathematical property (CMP):
z Complex implies Ei(z) is Complex
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA><OMS cd="set1" name="in"/>
        <OMV name="z"/>
        <OMS cd="setname1" name="C"/>
      </OMA>
      <OMA><OMS cd="set1" name="in"/>
        <OMA><OMS cd="elementryInts" name="expIntegralEi"/>
          <OMV name="z"/>
        </OMA>
        <OMS cd="setname1" name="C"/>
      </OMA>
    </OMA>
  </OMOBJ>

implies (in ( z, C) , in (expIntegralEi ( z) , C) )

Commented Mathematical property (CMP):
exists(x,y) s.t. expIntegralEi(x)=y
Formal Mathematical property (FMP):
<OMOBJ>
    <OMBIND>
      <OMS cd="quant1" name="exists"/>
      <OMBVAR>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMBVAR>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="elementryInts" name="expIntegralEi"/>
          <OMV name="x"/>
        </OMA>
        <OMV name="y"/>
      </OMA>
    </OMBIND>
  </OMOBJ>

exists [ x y ] . (eq (expIntegralEi ( x) , y) )

Commented Mathematical property (CMP):
$Ei(z)=-\int_{-z}^{\infty} \frac{e^{-t}}{t}dt$
Formal Mathematical property (FMP):
<OMOBJ>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="elementryInts" name="expIntegralEi"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="unary_minus"/>
      <OMA>
        <OMS cd="calculus1" name="defint"/>
	<OMA>
	  <OMS cd="interval1" name="interval_co"/>
	  <OMA>
	    <OMS cd="arith1" name="unary_minus"/>
	    <OMV name="z"/>
	  </OMA>
	  <OMS cd="nums1" name="infinity"/>
	</OMA>
	<OMBIND>
	  <OMS cd="fns1" name="lambda"/>
	  <OMBVAR>
	    <OMV name="t"/>
	  </OMBVAR>
	  <OMA>
	    <OMS cd="arith1" name="divide"/>
	    <OMA>
	      <OMS cd="transc1" name="exp"/>
	      <OMA>
	        <OMS cd="arith1" name="unary_minus"/>
	        <OMV name="t"/>
	      </OMA>
	    </OMA>
	    <OMV name="t"/>
	  </OMA>
	</OMBIND>
      </OMA>
    </OMA>
  </OMOBJ>

eq expIntegralEi ( z) unary_minus (defint (interval_co (unary_minus ( z) , infinity) , lambda [ t ] . (divide (exp (unary_minus ( t) ) , t) ) ) )

Commented Mathematical property (CMP):
$Ei(z)=\gamma+ln(-z)+\sum_{n=1}^{\infty} \frac{z^n}{n!n}$, series expansion
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="elementryInts" name="expIntegralEi"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="plus"/>
      <OMS cd="nums1" name="gamma"/>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="unary_minus"/>
	  <OMV name="z"/>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="sum"/>
        <OMA>
          <OMS cd="interval1" name="integer_interval"/>
	  <OMS cd="alg1" name="one"/>
	  <OMS cd="nums1" name="infinity"/>
        </OMA>
        <OMBIND>
          <OMS cd="fns1" name="lambda"/>
          <OMBVAR>
            <OMV name="x"/>
          </OMBVAR>
          <OMA>
	    <OMS cd="arith1" name="divide"/>
	    <OMA>
	      <OMS cd="arith1" name="power"/>
	      <OMV name="z"/>
	      <OMV name="n"/>
	    </OMA>
	    <OMA>
	      <OMS cd="arith1" name="times"/>
	      <OMA>
	        <OMS cd="integer1" name="factorial"/>
		<OMV name="n"/>
	      </OMA>
	      <OMV name="n"/>
	    </OMA>
          </OMA>
        </OMBIND>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (expIntegralEi ( z) , plus (gamma, ln (unary_minus ( z) ) , sum (integer_interval (one, infinity) , lambda [ x ] . (divide (power ( z, n) , times (factorial ( n) , n) ) ) ) ) )

Signatures:
sts


[Next: logIntegral] [Previous: expIntegralE] [Top]

logIntegral

This symbol represents the unary function which calculates the logarithmic integral.

The logarithmic integral function is defined by $li(z)=\int_{0}^{z} \frac{dt}{lnt}$. where the principal value is taken. It has a branch cut along the negative real axis in the complex $z$ plane. It is defined in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [5.1.3]

Commented Mathematical property (CMP):
x in Reals implies li(x) in Reals
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA><OMS cd="set1" name="in"/>
        <OMV name="x"/>
        <OMS cd="setname1" name="R"/>
      </OMA>
      <OMA><OMS cd="set1" name="in"/>
        <OMA><OMS cd="elementryInts" name="logIntegral"/>
          <OMV name="x"/>
        </OMA>
        <OMS cd="setname1" name="R"/>
      </OMA>
    </OMA>
  </OMOBJ>

implies (in ( x, R) , in (logIntegral ( x) , R) )

Commented Mathematical property (CMP):
z Complex implies li(z) is Complex
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA><OMS cd="set1" name="in"/>
        <OMV name="z"/>
        <OMS cd="setname1" name="C"/>
      </OMA>
      <OMA><OMS cd="set1" name="in"/>
        <OMA><OMS cd="elementryInts" name="logIntegral"/>
          <OMV name="z"/>
        </OMA>
        <OMS cd="setname1" name="C"/>
      </OMA>
    </OMA>
  </OMOBJ>

implies (in ( z, C) , in (logIntegral ( z) , C) )

Commented Mathematical property (CMP):
exists(x,y) s.t. logIntegral(x)=y
Formal Mathematical property (FMP):
<OMOBJ>
    <OMBIND>
      <OMS cd="quant1" name="exists"/>
      <OMBVAR>
        <OMV name="x"/>
        <OMV name="y"/>
      </OMBVAR>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="elementryInts" name="logIntegral"/>
          <OMV name="x"/>
        </OMA>
        <OMV name="y"/>
      </OMA>
    </OMBIND>
  </OMOBJ>

exists [ x y ] . (eq (logIntegral ( x) , y) )

Commented Mathematical property (CMP):
$li(z)=\int_{0}^{z} \frac{dt}{lnt}$,
Formal Mathematical property (FMP):
<OMOBJ>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="elementryInts" name="logIntegral"/>
      <OMV name="z"/>
    </OMA>
      <OMA>
        <OMS cd="calculus1" name="defint"/>
	<OMA>
	  <OMS cd="interval1" name="interval_co"/>
	  <OMS cd="alg1" name="zero"/>
	  <OMV name="z"/>
	</OMA>
	<OMBIND>
	  <OMS cd="fns1" name="lambda"/>
	  <OMBVAR>
	    <OMV name="t"/>
	  </OMBVAR>
	  <OMA>
	    <OMS cd="arith1" name="divide"/>
	    <OMS cd="alg1" name="one"/>
	    <OMA>
	      <OMS cd="transc1" name="ln"/>
	      <OMV name="t"/>
	    </OMA>
	  </OMA>
	</OMBIND>
      </OMA>
  </OMOBJ>

eq logIntegral ( z) defint (interval_co (zero, z) , lambda [ t ] . (divide (one, ln ( t) ) ) )

Commented Mathematical property (CMP):
$li(z)=Ei(lnz)$
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="elementryInts" name="logIntegral"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
	<OMS cd="elementryInts" name="expIntegralEi"/>
	<OMA>
	  <OMS cd="transc1" name="ln"/>
	  <OMV name="z"/>
	</OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (logIntegral ( z) , expIntegralEi (ln ( z) ) )

Signatures:
sts


[First: expIntegralE] [Previous: expIntegralEi] [Top]