OpenMath Content Dictionary: transc1

Canonical URL:
http://www.openmath.org/cd/transc1.ocd
CD File:
transc1.ocd
CD as XML Encoded OpenMath:
transc1.omcd
Defines:
arccos, arccosh, arccot, arccoth, arccsc, arccsch, arcsec, arcsech, arcsin, arcsinh, arctan, arctanh, cos, cosh, cot, coth, csc, csch, exp, ln, log, sec, sech, sin, sinh, tan, tanh
Date:
2001-03-12
Version:
2
Review Date:
2003-04-01
Status:
official
Uses CD:
alg1, arith1, interval1, logic1, nums1, quant1, relation1, set1, setname1, complex1

This CD holds the definitions of many transcendental functions. They are defined as in Abromowitz and Stegun (ninth printing on), with precise reductions to logs in the case of inverse functions.

Note that, if signed zeros are supported, some strict inequalities have to become weak . It is intended to be `compatible' with the MathML elements denoting trancendental functions. Some additional functions are in the CD transc2.


log

This symbol represents a binary log function; the first argument is the base, to which the second argument is log'ed. It is defined in Abramowitz and Stegun, Handbook of Mathematical Functions, section 4.1

Commented Mathematical property (CMP):
a^b = c implies log_a c = b
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="logic1" name="implies"/>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="arith1" name="power"/>
          <OMV name="a"/>
          <OMV name="b"/>
        </OMA>
        <OMV name="c"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="transc1" name="log"/>
          <OMV name="a"/>
          <OMV name="c"/>
        </OMA>
        <OMV name="b"/>
      </OMA>
    </OMA>
  </OMOBJ>

implies (eq (power ( a, b) , c) , eq (log ( a, c) , b) )

Example:
log 100 to base 10 (which is 2).
<OMOBJ>
  <OMA>
    <OMS cd="transc1" name="log"/>
    <OMF dec="10"/>
    <OMF dec="100"/>
  </OMA>
</OMOBJ>

log ( 10 , 100 )

Signatures:
sts


[Next: ln] [Last: arccoth] [Top]

ln

This symbol represents the ln function (natural logarithm) as described in Abramowitz and Stegun, section 4.1. It takes one argument. Note the description in the CMP/FMP of the branch cut. If signed zeros are in use, the inequality needs to be non-strict.

Commented Mathematical property (CMP):
-pi < Im ln x <= pi
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS name="and" cd="logic1"/>
      <OMA>
        <OMS name="lt" cd="relation1"/>
        <OMA>
          <OMS name="unary_minus" cd="arith1"/>
          <OMS name="pi" cd="nums1"/>
        </OMA>
        <OMA>
          <OMS name="imaginary" cd="complex1"/>
          <OMA>
            <OMS name="ln" cd="transc1"/>
            <OMV name="x"/>
          </OMA>
        </OMA>
      </OMA>
      <OMA>
        <OMS name="leq" cd="relation1"/>
        <OMA>
          <OMS name="imaginary" cd="complex1"/>
          <OMA>
            <OMS name="ln" cd="transc1"/>
            <OMV name="x"/>
          </OMA>
        </OMA>
        <OMS name="pi" cd="nums1"/>
      </OMA>
    </OMA>
  </OMOBJ>

and (lt (unary_minus (pi) , imaginary (ln ( x) ) ) , leq (imaginary (ln ( x) ) , pi) )

Example:
ln 1 (which is 0).
<OMOBJ>
  <OMA>
    <OMS cd="transc1" name="ln"/>
    <OMF dec="1"/>
  </OMA>
</OMOBJ>

ln ( 1 )

Signatures:
sts


[Next: exp] [Previous: log] [Top]

exp

This symbol represents the exponentiation function as described in Abramowitz and Stegun, section 4.2. It takes one argument.

Commented Mathematical property (CMP):
for all k if k is an integer then e^(z+2*pi*k*i)=e^z
Formal Mathematical property (FMP):
<OMOBJ>
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="k"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="k"/>
      <OMS cd="setname1" name="Z"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="transc1" name="exp"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMV name="z"/>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMI>2</OMI>
	    <OMS cd="nums1" name="pi"/>
	    <OMV name="k"/>
	    <OMS cd="nums1" name="i"/>
	  </OMA>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="exp"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

forall [ k ] . (implies (in ( k, Z) , eq (exp (plus ( z, times (2, pi, k, i) ) ) , exp ( z) ) ) )

Signatures:
sts


[Next: sin] [Previous: ln] [Top]

sin

This symbol represents the sin function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Commented Mathematical property (CMP):
sin(x) = (exp(ix)-exp(-ix))/2i
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS name="sin" cd="transc1"/>
        <OMV name="x"/>
      </OMA>
      <OMA>
        <OMS name="divide" cd="arith1"/>
        <OMA>
          <OMS name="minus" cd="arith1"/>
          <OMA>
            <OMS name="exp" cd="transc1"/>
            <OMA>
              <OMS name="times" cd="arith1"/>
              <OMS name="i" cd="nums1"/>
              <OMV name="x"/>
            </OMA>
          </OMA>
          <OMA>
            <OMS name="exp" cd="transc1"/>
            <OMA>
              <OMS name="times" cd="arith1"/>
              <OMA>
                <OMS name="unary_minus" cd="arith1"/>
                <OMS name="i" cd="nums1"/>
              </OMA>
              <OMV name="x"/>
            </OMA>
          </OMA>
        </OMA>
        <OMA>
          <OMS name="times" cd="arith1"/>
          <OMI>2</OMI>
          <OMS name="i" cd="nums1"/>
        </OMA>
      </OMA>
    </OMA>
  </OMOBJ>

eq (sin ( x) , divide (minus (exp (times (i, x) ) , exp (times (unary_minus (i) , x) ) ) , times (2, i) ) )

Commented Mathematical property (CMP):
sin(A + B) = sin A cos B + cos A sin B
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="transc1" name="sin"/>
        <OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMV name="A"/>
    	  <OMV name="B"/>
        </OMA>
      </OMA>
      <OMA>
	<OMS cd="arith1" name="plus"/>
	<OMA>
	  <OMS cd="arith1" name="times"/>
	  <OMA>
	    <OMS cd="transc1" name="sin"/>
	    <OMV name="A"/>
	  </OMA>
	  <OMA>
	    <OMS cd="transc1" name="cos"/>
	    <OMV name="B"/>
	  </OMA>
	</OMA>
	<OMA>
	  <OMS cd="arith1" name="times"/>
	  <OMA>
	    <OMS cd="transc1" name="cos"/>
	    <OMV name="A"/>
	  </OMA>
	  <OMA>
	    <OMS cd="transc1" name="sin"/>
	    <OMV name="B"/>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMOBJ>

eq (sin (plus ( A, B) ) , plus (times (sin ( A) , cos ( B) ) , times (cos ( A) , sin ( B) ) ) )

Commented Mathematical property (CMP):
sin A = - sin(-A)
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="transc1" name="sin"/>
	<OMV name="A"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMA>
	  <OMS cd="transc1" name="sin"/>
	  <OMA>
	    <OMS cd="arith1" name="unary_minus"/>
	    <OMV name="A"/>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMOBJ>

eq (sin ( A) , unary_minus (sin (unary_minus ( A) ) ) )

Signatures:
sts


[Next: cos] [Previous: exp] [Top]

cos

This symbol represents the cos function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Commented Mathematical property (CMP):
cos(x) = (exp(ix)+exp(-ix))/2
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS name="cos" cd="transc1"/>
        <OMV name="x"/>
      </OMA>
      <OMA>
        <OMS name="divide" cd="arith1"/>
        <OMA>
          <OMS name="plus" cd="arith1"/>
          <OMA>
            <OMS name="exp" cd="transc1"/>
            <OMA>
              <OMS name="times" cd="arith1"/>
              <OMS name="i" cd="nums1"/>
              <OMV name="x"/>
            </OMA>
          </OMA>
          <OMA>
            <OMS name="exp" cd="transc1"/>
            <OMA>
              <OMS name="times" cd="arith1"/>
              <OMA>
                <OMS name="unary_minus" cd="arith1"/>
                <OMS name="i" cd="nums1"/>
              </OMA>
              <OMV name="x"/>
            </OMA>
          </OMA>
        </OMA>
        <OMI>2</OMI>
      </OMA>
    </OMA>
  </OMOBJ>

eq (cos ( x) , divide (plus (exp (times (i, x) ) , exp (times (unary_minus (i) , x) ) ) , 2) )

Commented Mathematical property (CMP):
cos 2A = cos^2 A - sin^2 A
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="cos"/>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMI> 2 </OMI>
	<OMV name="A"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="minus"/>
      <OMA>
        <OMS cd="arith1" name="power"/>
	<OMA>
	  <OMS cd="transc1" name="cos"/>
	  <OMV name="A"/>
	</OMA>
	<OMI> 2 </OMI>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="power"/>
	<OMA>
	  <OMS cd="transc1" name="sin"/>
	  <OMV name="A"/>
	</OMA>
	<OMI> 2 </OMI>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (cos (times ( 2 , A) ) , minus (power (cos ( A) , 2 ) , power (sin ( A) , 2 ) ) )

Commented Mathematical property (CMP):
cos A = cos(-A)
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="cos"/>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="cos"/>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMV name="A"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (cos ( A) , cos (unary_minus ( A) ) )

Signatures:
sts


[Next: tan] [Previous: sin] [Top]

tan

This symbol represents the tan function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Commented Mathematical property (CMP):
tan A = sin A / cos A
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="tan"/>
    <OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMA>
      <OMS cd="transc1" name="sin"/>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="cos"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (tan ( A) , divide (sin ( A) , cos ( A) ) )

Signatures:
sts


[Next: sec] [Previous: cos] [Top]

sec

This symbol represents the sec function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Commented Mathematical property (CMP):
sec A = 1/cos A
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="sec"/><OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="transc1" name="cos"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (sec ( A) , divide (one, cos ( A) ) )

Signatures:
sts


[Next: csc] [Previous: tan] [Top]

csc

This symbol represents the csc function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Commented Mathematical property (CMP):
csc A = 1/sin A
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="csc"/><OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="transc1" name="sin"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (csc ( A) , divide (one, sin ( A) ) )

Signatures:
sts


[Next: cot] [Previous: sec] [Top]

cot

This symbol represents the cot function as described in Abramowitz and Stegun, section 4.3. It takes one argument.

Commented Mathematical property (CMP):
cot A = 1/tan A
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="cot"/><OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="transc1" name="tan"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (cot ( A) , divide (one, tan ( A) ) )

Signatures:
sts


[Next: sinh] [Previous: csc] [Top]

sinh

This symbol represents the sinh function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Commented Mathematical property (CMP):
sinh A = 1/2 * (e^A - e^(-A))
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="sinh"/>
    <OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMA>
      <OMS cd="nums1" name="rational"/>
      <OMS cd="alg1" name="one"/>
      <OMI> 2 </OMI>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="minus"/>
      <OMA>
        <OMS cd="arith1" name="power"/>
	<OMS cd="nums1" name="e"/>
	<OMV name="A"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="power"/>
	<OMS cd="nums1" name="e"/>
	<OMA>
	  <OMS cd="arith1" name="unary_minus"/>
	  <OMV name="A"/>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (sinh ( A) , times (rational (one, 2 ) , minus (power (e, A) , power (e, unary_minus ( A) ) ) ) )

Signatures:
sts


[Next: cosh] [Previous: cot] [Top]

cosh

This symbol represents the cosh function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Commented Mathematical property (CMP):
cosh A = 1/2 * (e^A + e^(-A))
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="cosh"/>
    <OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMA>
      <OMS cd="nums1" name="rational"/>
      <OMS cd="alg1" name="one"/>
      <OMI> 2 </OMI>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="plus"/>
      <OMA>
        <OMS cd="arith1" name="power"/>
	<OMS cd="nums1" name="e"/>
	<OMV name="A"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="power"/>
	<OMS cd="nums1" name="e"/>
	<OMA>
	  <OMS cd="arith1" name="unary_minus"/>
	  <OMV name="A"/>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (cosh ( A) , times (rational (one, 2 ) , plus (power (e, A) , power (e, unary_minus ( A) ) ) ) )

Signatures:
sts


[Next: tanh] [Previous: sinh] [Top]

tanh

This symbol represents the tanh function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Commented Mathematical property (CMP):
tanh A = sinh A / cosh A
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="tanh"/>
    <OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMA>
      <OMS cd="transc1" name="sinh"/>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="cosh"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (tanh ( A) , divide (sinh ( A) , cosh ( A) ) )

Signatures:
sts


[Next: sech] [Previous: cosh] [Top]

sech

This symbol represents the sech function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Commented Mathematical property (CMP):
sech A = 1/cosh A
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="sech"/><OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="transc1" name="cosh"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (sech ( A) , divide (one, cosh ( A) ) )

Signatures:
sts


[Next: csch] [Previous: tanh] [Top]

csch

This symbol represents the csch function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Commented Mathematical property (CMP):
csch A = 1/sinh A
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="csch"/><OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="transc1" name="sinh"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (csch ( A) , divide (one, sinh ( A) ) )

Signatures:
sts


[Next: coth] [Previous: sech] [Top]

coth

This symbol represents the coth function as described in Abramowitz and Stegun, section 4.5. It takes one argument.

Commented Mathematical property (CMP):
coth A = 1/tanh A
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="coth"/><OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="divide"/>
    <OMS cd="alg1" name="one"/>
    <OMA>
      <OMS cd="transc1" name="tanh"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (coth ( A) , divide (one, tanh ( A) ) )

Signatures:
sts


[Next: arcsin] [Previous: csch] [Top]

arcsin

This symbol represents the arcsin function. This is the inverse of the sin function as described in Abramowitz and Stegun, section 4.4. It takes one argument.

Commented Mathematical property (CMP):
arcsin(z) = -i ln (sqrt(1-z^2)+iz)
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS name="eq" cd="relation1"/>
      <OMA>
        <OMS name="arcsin" cd="transc1"/>
        <OMV name="z"/>
      </OMA>
      <OMA>
        <OMS name="times" cd="arith1"/>
        <OMA>
          <OMS name="unary_minus" cd="arith1"/>
          <OMS name="i" cd="nums1"/>
        </OMA>
        <OMA>
          <OMS name="ln" cd="transc1"/>
          <OMA>
            <OMS name="plus" cd="arith1"/>
            <OMA>
              <OMS name="root" cd="arith1"/>
              <OMA>
                <OMS name="minus" cd="arith1"/>
                <OMS name="one" cd="alg1"/>
                <OMA>
                  <OMS name="power" cd="arith1"/>
                  <OMV name="z"/>
                  <OMI> 2 </OMI>
                </OMA>
              </OMA>
              <OMI> 2 </OMI>
            </OMA>
            <OMA>
              <OMS name="times" cd="arith1"/>
              <OMS name="i" cd="nums1"/>
              <OMV name="z"/>
            </OMA>
          </OMA>
        </OMA>
      </OMA>
    </OMA>
  </OMOBJ>

eq (arcsin ( z) , times (unary_minus (i) , ln (plus (root (minus (one, power ( z, 2 ) ) , 2 ) , times (i, z) ) ) ) )

Commented Mathematical property (CMP):
x in [-(pi/2),(pi/2)] implies arcsin(sin x) = x
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMA>
    <OMS cd="set1" name="in"/>
    <OMV name="x"/>
    <OMA>
      <OMS cd="interval1" name="interval_cc"/>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMA>
	  <OMS cd="arith1" name="divide"/>
	  <OMS cd="nums1" name="pi"/>
	  <OMI> 2 </OMI>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="divide"/>
	<OMS cd="nums1" name="pi"/>
	<OMI> 2 </OMI>
      </OMA>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arcsin"/>
      <OMA>
        <OMS cd="transc1" name="sin"/>
	<OMV name="x"/>
      </OMA>
    </OMA>
    <OMV name="x"/>
  </OMA>
</OMA>
</OMOBJ>

implies (in ( x, interval_cc (unary_minus (divide (pi, 2 ) ) , divide (pi, 2 ) ) ) , eq (arcsin (sin ( x) ) , x) )

Signatures:
sts


[Next: arccos] [Previous: coth] [Top]

arccos

This symbol represents the arccos function. This is the inverse of the cos function as described in Abramowitz and Stegun, section 4.4. It takes one argument.

Commented Mathematical property (CMP):
arccos(z) = -i ln(z+i \sqrt(1-z^2))
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccos"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMS cd="nums1" name="i"/>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMV name="z"/>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMS cd="nums1" name="i"/>
	    <OMA>
	      <OMS cd="arith1" name="root"/>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
		<OMS cd="alg1" name="one"/>
		<OMA>
		  <OMS cd="arith1" name="power"/>
		  <OMV name="z"/>
		  <OMI> 2 </OMI>
		</OMA>
	      </OMA>
	      <OMI> 2 </OMI>
	    </OMA>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (arccos ( z) , times (unary_minus (i) , ln (plus ( z, times (i, root (minus (one, power ( z, 2 ) ) , 2 ) ) ) ) ) )

Commented Mathematical property (CMP):
x in [0,pi] implies arccos(cos x) = x
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMA>
    <OMS cd="set1" name="in"/>
    <OMV name="x"/>
    <OMA>
      <OMS cd="interval1" name="interval_cc"/>
      <OMS cd="alg1" name="zero"/>
      <OMS cd="nums1" name="pi"/>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccos"/>
      <OMA>
        <OMS cd="transc1" name="cos"/>
	<OMV name="x"/>
      </OMA>
    </OMA>
    <OMV name="x"/>
  </OMA>
</OMA>
</OMOBJ>

implies (in ( x, interval_cc (zero, pi) ) , eq (arccos (cos ( x) ) , x) )

Signatures:
sts


[Next: arctan] [Previous: arcsin] [Top]

arctan

This symbol represents the arctan function. This is the inverse of the tan function as described in Abramowitz and Stegun, section 4.4. It takes one argument.

Commented Mathematical property (CMP):
arctan(z) = (ln(1+iz)-ln(1-iz))/2i
Formal Mathematical property (FMP):
<OMOBJ>
    <OMA>
      <OMS name="eq" cd="relation1"/>
      <OMA>
        <OMS name="arctan" cd="transc1"/>
        <OMV name="z"/>
      </OMA>
      <OMA>
        <OMS name="divide" cd="arith1"/>
        <OMA>
          <OMS name="minus" cd="arith1"/>
          <OMA>
            <OMS name="ln" cd="transc1"/>
            <OMA>
              <OMS name="plus" cd="arith1"/>
              <OMS name="one" cd="alg1"/>
              <OMA>
                <OMS name="times" cd="arith1"/>
                <OMS name="i" cd="nums1"/>
                <OMV name="z"/>
              </OMA>
            </OMA>
          </OMA>
          <OMA>
            <OMS name="ln" cd="transc1"/>
            <OMA>
              <OMS name="minus" cd="arith1"/>
              <OMS name="one" cd="alg1"/>
              <OMA>
                <OMS name="times" cd="arith1"/>
                <OMS name="i" cd="nums1"/>
                <OMV name="z"/>
              </OMA>
            </OMA>
          </OMA>
        </OMA>
        <OMA>
          <OMS name="times" cd="arith1"/>
          <OMI> 2 </OMI>
          <OMS name="i" cd="nums1"/>
        </OMA>
      </OMA>
    </OMA>
  </OMOBJ>

eq (arctan ( z) , divide (minus (ln (plus (one, times (i, z) ) ) , ln (minus (one, times (i, z) ) ) ) , times ( 2 , i) ) )

Commented Mathematical property (CMP):
x in (-(pi/2),(pi/2)) implies arctan(tan x) = x
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMA>
    <OMS cd="set1" name="in"/>
    <OMV name="x"/>
    <OMA>
      <OMS cd="interval1" name="interval_oo"/>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMA>
	  <OMS cd="arith1" name="divide"/>
	  <OMS cd="nums1" name="pi"/>
	  <OMI> 2 </OMI>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="divide"/>
	<OMS cd="nums1" name="pi"/>
	<OMI> 2 </OMI>
      </OMA>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arctan"/>
      <OMA>
        <OMS cd="transc1" name="tan"/>
	<OMV name="x"/>
      </OMA>
    </OMA>
    <OMV name="x"/>
  </OMA>
</OMA>
</OMOBJ>

implies (in ( x, interval_oo (unary_minus (divide (pi, 2 ) ) , divide (pi, 2 ) ) ) , eq (arctan (tan ( x) ) , x) )

Signatures:
sts


[Next: arcsec] [Previous: arccos] [Top]

arcsec

This symbol represents the arcsec function as described in Abramowitz and Stegun, section 4.4.

Commented Mathematical property (CMP):
arcsec(z) = -i ln(1/z + i \sqrt(1-1/z^2))
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arcsec"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMS cd="nums1" name="i"/>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMA>
	    <OMS cd="arith1" name="divide"/>
	    <OMS cd="alg1" name="one"/>
	    <OMV name="z"/>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMS cd="nums1" name="i"/>
	    <OMA>
	      <OMS cd="arith1" name="root"/>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
		<OMS cd="alg1" name="one"/>
		<OMA>
		  <OMS cd="arith1" name="divide"/>
		  <OMS cd="alg1" name="one"/>
		  <OMA>
		    <OMS cd="arith1" name="power"/>
		    <OMV name="z"/>
		    <OMI> 2 </OMI>
		  </OMA>
		</OMA>
	      </OMA>
	      <OMI> 2 </OMI>
	    </OMA>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (arcsec ( z) , times (unary_minus (i) , ln (plus (divide (one, z) , times (i, root (minus (one, divide (one, power ( z, 2 ) ) ) , 2 ) ) ) ) ) )

Commented Mathematical property (CMP):
for all z | arcsec z = i * arcsech z
Formal Mathematical property (FMP):
<OMOBJ>
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="z"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arcsec"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMS cd="nums1" name="i"/>
      <OMA>
        <OMS cd="transc1" name="arcsech"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

forall [ z ] . (eq (arcsec ( z) , times (i, arcsech ( z) ) ) )

Signatures:
sts


[Next: arccsc] [Previous: arctan] [Top]

arccsc

This symbol represents the arccsc function as described in Abramowitz and Stegun, section 4.4.

Commented Mathematical property (CMP):
arccsc(z) = -i ln(i/z + \sqrt(1 - 1/z^2))
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccsc"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
	<OMS cd="nums1" name="i"/>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMA>
	    <OMS cd="arith1" name="divide"/>
	    <OMS cd="nums1" name="i"/>
	    <OMV name="z"/>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="root"/>
	    <OMA>
	      <OMS cd="arith1" name="minus"/>
	      <OMS cd="alg1" name="one"/>
	      <OMA>
		<OMS cd="arith1" name="divide"/>
		<OMS cd="alg1" name="one"/>
		<OMA>
		  <OMS cd="arith1" name="power"/>
		  <OMV name="z"/>
		  <OMI> 2 </OMI>
		</OMA>
	      </OMA>
	    </OMA>
	    <OMI> 2 </OMI>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (arccsc ( z) , times (unary_minus (i) , ln (plus (divide (i, z) , root (minus (one, divide (one, power ( z, 2 ) ) ) , 2 ) ) ) ) )

Commented Mathematical property (CMP):
arccsc(z) = i * arccsch(i * z)
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arccsc"/>
    <OMV name="z"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMS cd="nums1" name="i"/>
    <OMA>
      <OMS cd="transc1" name="arccsch"/>
      <OMA>
        <OMS cd="arith1" name="times"/>
        <OMS cd="nums1" name="i"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (arccsc ( z) , times (i, arccsch (times (i, z) ) ) )

Commented Mathematical property (CMP):
arccsc(-z) = - arccsc(z)
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arccsc"/>
    <OMA>
      <OMS cd="arith1" name="unary_minus"/>
      <OMV name="z"/>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="unary_minus"/>
    <OMA>
      <OMS cd="transc1" name="arccsc"/>
      <OMV name="z"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (arccsc (unary_minus ( z) ) , unary_minus (arccsc ( z) ) )

Signatures:
sts


[Next: arccot] [Previous: arcsec] [Top]

arccot

This symbol represents the arccot function as described in Abramowitz and Stegun, section 4.4.

Commented Mathematical property (CMP):
arccot(-z) = - arccot(z)
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arccot"/>
    <OMA>
      <OMS cd="arith1" name="unary_minus"/>
      <OMV name="z"/>
    </OMA>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="unary_minus"/>
    <OMA>
      <OMS cd="transc1" name="arccot"/>
      <OMV name="z"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (arccot (unary_minus ( z) ) , unary_minus (arccot ( z) ) )

Commented Mathematical property (CMP):
for all real x | arccot(x) = 1/(2*i) * ln ((x + i)/(x - i))
Formal Mathematical property (FMP):
<OMOBJ>
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccot"/>
      <OMV name="x"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="arith1" name="divide"/>
	<OMS cd="alg1" name="one"/>
	<OMA>
	  <OMS cd="arith1" name="times"/>
	  <OMI> 2 </OMI>
	  <OMS cd="nums1" name="i"/>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="ln"/>
        <OMA>
          <OMS cd="arith1" name="divide"/>
	  <OMA>
	    <OMS cd="arith1" name="plus"/>
	    <OMV name="x"/>
	    <OMS cd="nums1" name="i"/>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="minus"/>
	    <OMV name="x"/>
	    <OMS cd="nums1" name="i"/>
	  </OMA>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

forall [ x ] . (eq (arccot ( x) , times (divide (one, times ( 2 , i) ) , ln (divide (plus ( x, i) , minus ( x, i) ) ) ) ) )

Signatures:
sts


[Next: arcsinh] [Previous: arccsc] [Top]

arcsinh

This symbol represents the arcsinh function as described in Abramowitz and Stegun, section 4.6.

Commented Mathematical property (CMP):
arcsinh z = ln(z + \sqrt(1+z^2))
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arcsinh"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="ln"/>
      <OMA>
        <OMS cd="arith1" name="plus"/>
	<OMV name="z"/>
	<OMA>
	  <OMS cd="arith1" name="root"/>
	  <OMA>
	    <OMS cd="arith1" name="plus"/>
	    <OMS cd="alg1" name="one"/>
	    <OMA>
	      <OMS cd="arith1" name="power"/>
	      <OMV name="z"/>
	      <OMI> 2 </OMI>
	    </OMA>
	  </OMA>
	  <OMI> 2 </OMI>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (arcsinh ( z) , ln (plus ( z, root (plus (one, power ( z, 2 ) ) , 2 ) ) ) )

Commented Mathematical property (CMP):
arcsinh(z) = - i * arcsin(i * z)
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arcsinh"/>
    <OMV name="z"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMA>
      <OMS cd="arith1" name="unary_minus"/>
      <OMS cd="nums1" name="i"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="arcsin"/>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMS cd="nums1" name="i"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (arcsinh ( z) , times (unary_minus (i) , arcsin (times (i, z) ) ) )

Signatures:
sts


[Next: arccosh] [Previous: arccot] [Top]

arccosh

This symbol represents the arccosh function as described in Abramowitz and Stegun, section 4.6.

Commented Mathematical property (CMP):
arccosh(z) = 2*ln(\sqrt((z+1)/2) + \sqrt((z-1)/2))
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccosh"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMI> 2 </OMI>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMA>
	    <OMS cd="arith1" name="root"/>
	    <OMA>
	      <OMS cd="arith1" name="divide"/>
	      <OMA>
	        <OMS cd="arith1" name="plus"/>
		<OMV name="z"/>
		<OMS cd="alg1" name="one"/>
	      </OMA>
	      <OMI> 2 </OMI>
	    </OMA>
	    <OMI> 2 </OMI>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="root"/>
	    <OMA>
	      <OMS cd="arith1" name="divide"/>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
		<OMV name="z"/>
		<OMS cd="alg1" name="one"/>
	      </OMA>
	      <OMI> 2 </OMI>
	    </OMA>
	    <OMI> 2 </OMI>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (arccosh ( z) , times ( 2 , ln (plus (root (divide (plus ( z, one) , 2 ) , 2 ) , root (divide (minus ( z, one) , 2 ) , 2 ) ) ) ) )

Commented Mathematical property (CMP):
arccosh z = i * (pi - arccos z)
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arccosh"/>
    <OMV name="z"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMS cd="nums1" name="i"/>
    <OMA>
      <OMS cd="arith1" name="minus"/>
      <OMS cd="nums1" name="pi"/>
      <OMA>
	<OMS cd="transc1" name="arccos"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (arccosh ( z) , times (i, minus (pi, arccos ( z) ) ) )

Signatures:
sts


[Next: arctanh] [Previous: arcsinh] [Top]

arctanh

This symbol represents the arctanh function as described in Abramowitz and Stegun, section 4.6.

Commented Mathematical property (CMP):
arctanh(z) = - i * arctan(i * z)
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arctanh"/>
    <OMV name="z"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMA>
      <OMS cd="arith1" name="unary_minus"/>
	<OMS cd="nums1" name="i"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="arctan"/>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMS cd="nums1" name="i"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (arctanh ( z) , times (unary_minus (i) , arctan (times (i, z) ) ) )

Commented Mathematical property (CMP):
for all x where 0 <= x^2 < 1 | arctanh(x) = 1/2 * ln((1 + x)/(1 - x))
Formal Mathematical property (FMP):
<OMOBJ>
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="relation1" name="leq"/>
	<OMS cd="alg1" name="zero"/>
	<OMA>
	  <OMS cd="arith1" name="power"/>
	  <OMV name="x"/>
	  <OMI> 2 </OMI>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="lt"/>
	<OMA>
	  <OMS cd="arith1" name="power"/>
	  <OMV name="x"/>
	  <OMI> 2 </OMI>
	</OMA>
	<OMS cd="alg1" name="one"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="transc1" name="arctanh"/>
	<OMV name="x"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="times"/>
        <OMA>
          <OMS cd="nums1" name="rational"/>
          <OMI> 1 </OMI>
          <OMI> 2 </OMI>
        </OMA>
        <OMA>
          <OMS cd="transc1" name="ln"/>
          <OMA>
            <OMS cd="arith1" name="divide"/>
	    <OMA>
	      <OMS cd="arith1" name="plus"/>
	      <OMV name="x"/>
	      <OMS cd="nums1" name="i"/>
	    </OMA>
	    <OMA>
	      <OMS cd="arith1" name="minus"/>
	      <OMV name="x"/>
	      <OMS cd="nums1" name="i"/>
	    </OMA>
          </OMA>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

forall [ x ] . (implies (and (leq (zero, power ( x, 2 ) ) , lt (power ( x, 2 ) , one) ) , eq (arctanh ( x) , times (rational ( 1 , 2 ) , ln (divide (plus ( x, i) , minus ( x, i) ) ) ) ) ) )

Signatures:
sts


[Next: arcsech] [Previous: arccosh] [Top]

arcsech

This symbol represents the arcsech function as described in Abramowitz and Stegun, section 4.6.

Commented Mathematical property (CMP):
arcsech(z) = 2 ln(\sqrt((1+z)/(2z)) + \sqrt((1-z)/(2z)))
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arcsech"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMI> 2 </OMI>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMA>
	    <OMS cd="arith1" name="root"/>
	    <OMA>
	      <OMS cd="arith1" name="divide"/>
	      <OMA>
	        <OMS cd="arith1" name="plus"/>
		<OMS cd="alg1" name="one"/>
		<OMV name="z"/>
	      </OMA>
	      <OMA>
	        <OMS cd="arith1" name="times"/>
		<OMI> 2 </OMI>
		<OMV name="z"/>
	      </OMA>
	    </OMA>
	    <OMI> 2 </OMI>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="root"/>
	    <OMA>
	      <OMS cd="arith1" name="divide"/>
	      <OMA>
	        <OMS cd="arith1" name="minus"/>
		<OMS cd="alg1" name="one"/>
		<OMV name="z"/>
	      </OMA>
	      <OMA>
	        <OMS cd="arith1" name="times"/>
		<OMI> 2 </OMI>
		<OMV name="z"/>
	      </OMA>
	    </OMA>
	    <OMI> 2 </OMI>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (arcsech ( z) , times ( 2 , ln (plus (root (divide (plus (one, z) , times ( 2 , z) ) , 2 ) , root (divide (minus (one, z) , times ( 2 , z) ) , 2 ) ) ) ) )

Commented Mathematical property (CMP):
for all x in (0..1] | arcsech x = ln(1/x + (1/(x^2) - 1)^(1/2))
Formal Mathematical property (FMP):
<OMOBJ>
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="x"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="x"/>
      <OMA>
        <OMS cd="interval1" name="interval_oc"/>
	<OMI> 0 </OMI> <OMI> 1 </OMI>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="transc1" name="arcsech"/>
	<OMV name="x"/>
      </OMA>
      <OMA>
        <OMS cd="transc1" name="ln"/>
	<OMA>
	  <OMS cd="arith1" name="plus"/>
	  <OMA>
	    <OMS cd="arith1" name="divide"/>
	    <OMS cd="alg1" name="one"/>
	    <OMV name="x"/>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="power"/>
	    <OMA>
	      <OMS cd="arith1" name="minus"/>
	      <OMA>
	        <OMS cd="arith1" name="divide"/>
		<OMS cd="alg1" name="one"/>
		<OMA>
		  <OMS cd="arith1" name="power"/>
		  <OMV name="x"/>
		  <OMI> 2 </OMI>
		</OMA>
	      </OMA>
	      <OMS cd="alg1" name="one"/>
	    </OMA>
	    <OMA>
	      <OMS cd="nums1" name="rational"/>
	      <OMI> 1 </OMI> <OMI> 2 </OMI>
	    </OMA>
	  </OMA>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

forall [ x ] . (implies (in ( x, interval_oc ( 0 , 1 ) ) , eq (arcsech ( x) , ln (plus (divide (one, x) , power (minus (divide (one, power ( x, 2 ) ) , one) , rational ( 1 , 2 ) ) ) ) ) ) )

Signatures:
sts


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arccsch

This symbol represents the arccsch function as described in Abramowitz and Stegun, section 4.6.

Commented Mathematical property (CMP):
arccsch(z) = ln(1/z + \sqrt(1+(1/z)^2))
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccsch"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="transc1" name="ln"/>
      <OMA>
        <OMS cd="arith1" name="plus"/>
	<OMA>
	  <OMS cd="arith1" name="divide"/>
	  <OMS cd="alg1" name="one"/>
	  <OMV name="z"/>
	</OMA>
	<OMA>
	  <OMS cd="arith1" name="root"/>
	  <OMA>
	    <OMS cd="arith1" name="plus"/>
	    <OMS cd="alg1" name="one"/>
	    <OMA>
	      <OMS cd="arith1" name="power"/>
	      <OMA>
	        <OMS cd="arith1" name="divide"/>
		<OMS cd="alg1" name="one"/>
		<OMV name="z"/>
	      </OMA>
	      <OMI> 2 </OMI>
	    </OMA>
	  </OMA>
	  <OMI> 2 </OMI>
	</OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (arccsch ( z) , ln (plus (divide (one, z) , root (plus (one, power (divide (one, z) , 2 ) ) , 2 ) ) ) )

Commented Mathematical property (CMP):
arccsch(z) = i * arccsc(i * z)
Formal Mathematical property (FMP):
<OMOBJ>
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="transc1" name="arccsch"/>
    <OMV name="z"/>
  </OMA>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMS cd="nums1" name="i"/>
    <OMA>
      <OMS cd="transc1" name="arccsc"/>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMS cd="nums1" name="i"/>
	<OMV name="z"/>
      </OMA>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (arccsch ( z) , times (i, arccsc (times (i, z) ) ) )

Signatures:
sts


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arccoth

This symbol represents the arccoth function as described in Abramowitz and Stegun, section 4.6.

Commented Mathematical property (CMP):
arccoth(z) = (ln(-1-z)-ln(1-z))/2
Formal Mathematical property (FMP):
<OMOBJ>
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="transc1" name="arccoth"/>
      <OMV name="z"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="divide"/>
      <OMA>
        <OMS cd="arith1" name="minus"/>
        <OMA>
          <OMS cd="transc1" name="ln"/>
          <OMA>
            <OMS cd="arith1" name="minus"/>
	    <OMA>
	      <OMS cd="arith1" name="unary_minus"/>
	      <OMS cd="alg1" name="one"/>
	    </OMA>
	    <OMV name="z"/>
          </OMA>
        </OMA>
        <OMA>
          <OMS cd="transc1" name="ln"/>
          <OMA>
            <OMS cd="arith1" name="minus"/>
	    <OMS cd="alg1" name="one"/>
	    <OMV name="z"/>
          </OMA>
        </OMA>
      </OMA>
      <OMI> 2 </OMI>
    </OMA>
  </OMA>
</OMOBJ>

eq (arccoth ( z) , divide (minus (ln (minus (unary_minus (one) , z) ) , ln (minus (one, z) ) ) , 2 ) )

Commented Mathematical property (CMP):
for all z | if z is not zero then arccoth(z) = i * arccot(i * z)
Formal Mathematical property (FMP):
<OMOBJ>
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="z"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="relation1" name="neq"/>
      <OMV name="z"/>
      <OMS cd="alg1" name="zero"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="transc1" name="arccoth"/>
	<OMV name="z"/>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="times"/>
	<OMS cd="nums1" name="i"/>
        <OMA>
          <OMS cd="transc1" name="arccot"/>
	  <OMA>
            <OMS cd="arith1" name="times"/>
	    <OMS cd="nums1" name="i"/>
	    <OMV name="z"/>
          </OMA>
        </OMA>
      </OMA> 
    </OMA>
  </OMA>
</OMBIND>
</OMOBJ>

forall [ z ] . (implies (neq ( z, zero) , eq (arccoth ( z) , times (i, arccot (times (i, z) ) ) ) ) )

Signatures:
sts


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