OpenMath Content Dictionary: trigHypInts

Canonical URL:
http://www.openmath.org/CDs/trigHypInts.ocd
CD File:
trigHypInts.ocd
CD as XML Encoded OpenMath:
trigHypInts.omcd
Defines:
CoshIntegral, CosIntegral, SinhIntegral, SinIntegral
Date:
Version:
(Revision )
Review Date:
Status:
private
Uses CD:
alg1, arith1, calculus1, complex1, fns1, interval1, logic1, nums1, relation1, transc1

1/1/5000

This content dictionary contains symbols which describe the functions which perform the sine, cosine, hyperbolic sine and hyperbolic cosine integrals.


SinIntegral

The symbol SinIntegral represents the sine integral, it is denoted by $Si(z)$. The sine integral is defined by : $Si(z)=\int_{0}^{z} \frac{sint}{t}dt$.

It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [5.2.1]

Commented Mathematical property (CMP):
Si : real $\rightarrow$ real
Commented Mathematical property (CMP):
Si : complex $\rightarrow$ complex
Commented Mathematical property (CMP):
Si : symbolic $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$Si(z)=\int_0^z \frac{sin(t)}{t}dt
Formal Mathematical property (FMP):
<OMOBJ>
        <OMA>
	  <OMS cd="relation1" name="eq"/>
	  <OMA>
	    <OMS cd="trigHypInts" name="SinIntegral"/>
	    <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"/>
		<OMA>
		  <OMS cd="transc1" name="sin"/>
		  <OMV name="t"/>
		</OMA>
		<OMV name="t"/>
	      </OMA>
	    </OMBIND>
          </OMA>
	</OMA>
      </OMOBJ>

eq (SinIntegral ( z) , defint (interval_co (zero, z) , lambda [ t ] . (divide (sin ( t) , t) ) ) )

Commented Mathematical property (CMP):
$Si(z)=\sum_{n=0}^{\infty} \frac{(-1)^{n}z^{2n+1}}{(2n+1)(2n+1)!}$, series expansion
Commented Mathematical property (CMP):
$Si(-z)=-Si(z)$, symmetry
Commented Mathematical property (CMP):
$Si(\bar{z})=\overline{Si(z)}$, symmetry
Signatures:
sts


[Next: CosIntegral] [Last: CoshIntegral] [Top]

CosIntegral

The symbol CosIntegral represents the cosine integral, it is denoted by $Ci(z)$. The cosine integral is defined by $Ci(z)=-\int_z^{\infty} \frac{cost}{t} dt$ where $|argz|< \pi$.

It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions,[5.2.27]

Commented Mathematical property (CMP):
Ci : real $\rightarrow$ real
Commented Mathematical property (CMP):
Ci : complex $\rightarrow$ complex
Commented Mathematical property (CMP):
Ci : symbolic $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$Ci(z) = -\int_z^{\infty} \frac{cos t}{t} dt
Formal Mathematical property (FMP):
<OMOBJ>
        <OMA>
	  <OMS cd="relation1" name="eq"/>
	  <OMA>
	    <OMS cd="trigHypInts" name="CosIntegral"/>
	    <OMV name="z"/>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="unary_minus"/>
	    <OMA>
	      <OMS cd="calculus1" name="defint"/>
	      <OMA>
	        <OMS cd="interval1" name="interval_co"/>
		<OMV name="z"/>
		<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="cos"/>
		    <OMV name="t"/>
		  </OMA>
		  <OMV name="t"/>
		</OMA>
	      </OMBIND>
            </OMA>
	  </OMA>
	</OMA>
      </OMOBJ>

eq (CosIntegral ( z) , unary_minus (defint (interval_co ( z, infinity) , lambda [ t ] . (divide (cos ( t) , t) ) ) ) )

Commented Mathematical property (CMP):
$Ci(z)=\gamma + lnz + \sum_{n=1}^{\infty} \frac{(-1)^{n}z^{2n}}{2n(2n)!}$, series expansion
Commented Mathematical property (CMP):
symmetry relation: $Ci(\bar{z})=\overline{Ci(z)}$, symmetry
Commented Mathematical property (CMP):
$Ci(-z)=Ci(z)-i \pi$, ($0 P argz P \frac{\pi}{2}$), symmetry
Signatures:
sts


[Next: SinhIntegral] [Previous: SinIntegral] [Top]

SinhIntegral

The symbol SinhIntegral represents the hyperbolic sine integral, it is denoted by The hyperbolic sine integral is defined by $Shi(z)=\int_{0}^{z} \frac{sinh(t)}{t} dt$.

It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions,[5.2.3]

Commented Mathematical property (CMP):
Shi : real $\rightarrow$ real
Commented Mathematical property (CMP):
Shi : complex $\rightarrow$ complex
Commented Mathematical property (CMP):
Shi : symbolic $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$Shi(z)=\int_{0}^{z} \frac{sinht}{t} dt$
Formal Mathematical property (FMP):
<OMOBJ>
        <OMA>
	  <OMS cd="relation1" name="eq"/>
	  <OMA>
	    <OMS cd="trigHypInts" name="SinhIntegral"/>
	    <OMV name="z"/>
	  </OMA>
	  <OMA>
	    <OMS cd="calculus1" name="defint"/>
	    <OMA>
	      <OMS cd="interval1" name="interval"/>
	      <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"/>
		<OMA>
		  <OMS cd="transc1" name="sinh"/>
		  <OMV name="t"/>
		</OMA>
		<OMV name="t"/>
	      </OMA>
	    </OMBIND>
	  </OMA>
	</OMA>
      </OMOBJ>

eq (SinhIntegral ( z) , defint (interval (zero, z) , lambda [ t ] . (divide (sinh ( t) , t) ) ) )

Commented Mathematical property (CMP):
$Shi(z)=\sum_{n=0}^{\infty}\frac{z^{2n+1}} {(2n+1)(2n+1)!}$, series expansion
Signatures:
sts


[Next: CoshIntegral] [Previous: CosIntegral] [Top]

CoshIntegral

The symbol CoshIntegral represents the hyperbolic cosine integral, it is denoted by $Chi(z)$. The hyperbolic cosine integral is defined by $Chi(z)=\gamma + lnz + \int_{0}^{z} \frac{cosht-1}{t} dt$ where $\gamma=.5772156649 \dots$ is an Euler's constant and $|argz|P \pi$. It has a branch cut along the negative real axis in the complex $z$ plane.

It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions,[5.2.4]

Commented Mathematical property (CMP):
Chi : real $\rightarrow$ real
Commented Mathematical property (CMP):
Chi : complex $\rightarrow$ complex
Commented Mathematical property (CMP):
Chi : symbolic $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$Chi(z)=\gamma + lnz + \int_{0}^{z} \frac{cosht-1}{t} dt~~~~ (|arg(z)| < /pi$
Formal Mathematical property (FMP):
<OMOBJ>
        <OMA>
          <OMS cd="logic1" name="implies"/>
	  <OMA>
	    <OMS cd="relation1" name="lt"/>
	    <OMA>
	      <OMS cd="arith1" name="abs"/>
	      <OMA>
	        <OMS cd="complex1" name="argument"/>
		<OMV name="z"/>
	      </OMA>
	    </OMA>
	    <OMS cd="nums1" name="pi"/>
	  </OMA>
	  <OMA>
	    <OMS cd="relation1" name="eq"/>
	    <OMA>
	      <OMS cd="trigHypInts" name="CosIntegral"/>
	      <OMV name="z"/>
	    </OMA>
	    <OMA>
	      <OMS cd="arith1" name="plus"/>
	      <OMS cd="nums1" name="gamma"/>
	      <OMA>
	        <OMS cd="transc1" name="ln"/>
	        <OMV name="z"/>
	      </OMA>
	      <OMA>
	        <OMS cd="calculus1" name="defint"/>
		<OMA>
		  <OMS cd="interval1" name="interval"/>
		  <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"/>
		    <OMA>
		      <OMS cd="arith1" name="minus"/>
		      <OMA>
		        <OMS cd="transc1" name="cosh"/>
			<OMV name="t"/>
		      </OMA>
		      <OMS cd="alg1" name="one"/>
		    </OMA>
		    <OMV name="t"/>
		  </OMA>
                </OMBIND>
	      </OMA>
	    </OMA>
	  </OMA>
	</OMA>
      </OMOBJ>

implies (lt (abs (argument ( z) ) , pi) , eq (CosIntegral ( z) , plus (gamma, ln ( z) , defint (interval (zero, z) , lambda [ t ] . (divide (minus (cosh ( t) , one) , t) ) ) ) ) )

Commented Mathematical property (CMP):
$Chi(z)=\gamma + lnz + \sum_{n=1}^{\infty} \frac{z^{2n}}{2n(2n)!}$, series expansion
Signatures:
sts


[First: SinIntegral] [Previous: SinhIntegral] [Top]