Texas Instruments TI-Nspire CX CAS Reference Guide

TI-Nspire™ CAS
Reference Guide
This guidebook applies to TI-Nspire™ software version 4.5. To obtain the latest version of the documentation, go to education.ti.com/go/download.

Important Information

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License
Please see the complete license installed in C:\ProgramFiles\TIEducation\<TI-Nspire™
Product Name>\license.
© 2006 - 2017 Texas Instruments Incorporated
ii
Contents
Important Information
Expression Templates 1
Alphabetical Listing 8
A B C D E F G I L M N O P Q R S T U V W X Z
17 20 45 57 68 77 87
96 112 120 129 131 140 143 157 182 197 198 199 201 202
ii
8
iii
Symbols 210
Empty (Void) Elements 236
Shortcuts for Entering Math Expressions 238
EOS™ (Equation Operating System) Hierarchy 240
Constants and Values 242
Error Codes and Messages 243
Warning Codes and Messages 251
Support and Service 253
Texas Instruments Support and Service Service and Warranty Information
253 253
Index 254
iv

Expression Templates

Expression templates give you an easy way to enter math expressions in standard mathematical notation. When you insert a template, it appears on the entry line with small blocks at positions where you can enter elements. A cursor shows which element you can enter.
Position the cursor on each element, and type a value or expression for the element.
Fraction template
Note: See also / (divide), page 212.
Exponent template
Note: Type the first value, press l, and
then type the exponent. To return the cursor to the baseline, press right arrow (¢).
Note: See also ^ (power), page 213.
Square root template
Note: See also () (square root), page
223.
/p keys
Example:
l key
Example:
/q keys
Example:
Nth root template
Note: See also root(), page 154.
/l keys
Example:
Expression T emplates 1
Nth root template
/l keys
e exponent template
Natural exponential e raised to a power
Note: See also e^(), page 57.
Log template
Calculates log to a specified base. For a default of base 10, omit the base.
Note: See also log(), page 107.
Piecewise template (2-piece)
Lets you create expressions and conditions for a two-piece piecewise function. To add a piece, click in the template and repeat the template.
Note: See also piecewise(), page 133.
u keys
Example:
/s key
Example:
Catalog >
Example:
2 Expression Templates
Piecewise template (N-piece)
Lets you create expressions and conditions for an N-piece piecewise function. Prompts for N.
Note: See also piecewise(), page 133.
Catalog >
Example:
See the example for Piecewisetemplate (2­piece).
System of 2 equations template
Creates a system of two equations. To add a row to an existing system, click in the template and repeat the template.
Note: See also system(), page 182.
System of N equations template
Lets you create a system of N equations. Prompts for N.
Note: See also system(), page 182.
Absolute value template
Note: See also abs(), page 8.
Catalog >
Example:
Catalog >
Example:
See the example for Systemof equations template (2-equation).
Catalog >
Example:
Expression T emplates 3
Absolute value template
Catalog >
dd°mm’ss.ss’’ template
Lets you enter angles in dd°mmss.ss’’ format, where dd is the number of decimal degrees, mm is the number of minutes, and
ss.ss is the number of seconds.
Matrix template (2 x 2)
Creates a 2 x 2 matrix.
Matrix template (1 x 2)
.
Matrix template (2 x 1)
Catalog >
Example:
Catalog >
Example:
Catalog >
Example:
Catalog >
Example:
Matrix template (m x n)
The template appears after you are prompted to specify the number of rows and columns.
4 Expression Templates
Catalog >
Example:
Matrix template (m x n)
Note: If you create a matrix with a large
number of rows and columns, it may take a few moments to appear.
Catalog >
Sum template (Σ)
Note: See also Σ() (s umSeq), page 224.
Product template (Π)
Note: See also Π() (prodSeq), page 223.
First derivative template
The first derivative template can also be used to calculate first derivative at a point.
Note: See also d() (derivative), page 221.
Catalog >
Example:
Catalog >
Example:
Catalog >
Example:
Expression T emplates 5
Second derivative template
The second derivative template can also be used to calculate second derivative at a point.
Note: See also d() (derivative), page 221.
Catalog >
Example:
Nth derivative template
The nth derivative template can be used to calculate the nth derivative.
Note: See also d() (derivative), page 221.
Definite integral template
Note: See also() integral(), page 221.
Indefinite integral template
Note: See also () integral(), page 221.
Catalog >
Example:
Catalog >
Example:
Catalog >
Example:
Limit template
6 Expression Templates
Catalog >
Example:
Limit template
Use or () for left hand limit. Use + for right hand limit.
Note: See also limit(), page 6.
Catalog >
Expression T emplates 7

Alphabetical Listing

Items whose names are not alphabetic (such as +, !, and >) are listed at the end of this section, page 210. Unless otherwise specified, all examples in this section were performed in the default reset mode, and all variables are assumed to be undefined.
A
abs()
abs(Expr1) expression
abs(List1) list abs(Matrix1) matrix
Returns the absolute value of the argument.
Note: See also Absolute value template,
page 3.
If the argument is a complex number, returns the number’s modulus.
Note: All undefined variables are treated as
real variables.
amortTbl()
amortTbl(NPmt,N,I,PV, [Pmt], [FV], [PpY], [CpY], [PmtAt], [roundValue])
matrix
Amortization function that returns a matrix as an amortization table for a set of TVM arguments.
NPmt is the number of payments to be
included in the table. The table starts with the first payment.
N, I, PV, Pmt, FV, PpY, CpY, and PmtAt
are described in the table of TVM arguments, page 195.
If you omit Pmt, it defaults to
Pmt=tvmPmt
(N,I,PV,FV,PpY,CpY,PmtAt).
If you omit FV, it defaults to FV=0.
The defaults for PpY, CpY, and PmtAt are the same as for the TVM functions.
Catalog >
Catalog >
8 Alphabetical Listing
amortTbl()
roundValue specifies the number of
decimal places for rounding. Default=2.
The columns in the result matrix are in this order: Payment number, amount paid to interest, amount paid to principal, and balance.
The balance displayed in row n is the balance after payment n.
You can use the output matrix as input for the other amortization functions ΣInt() and
ΣPrn(), page 225, and bal(), page 17.
Catalog >
and
BooleanExpr1 and BooleanExpr2 Boolean expression
BooleanList1 and BooleanList2 Boolean list
BooleanMatrix1 and BooleanMatrix2 Boolean matrix
Returns true or false or a simplified form of the original entry.
Integer1 andInteger2 integer
Compares two real integers bit-by-bit using an and operation. Internally, both integers are converted to signed, 64-bit binary numbers. When corresponding bits are compared, the result is 1 if both bits are 1; otherwise, the result is 0. The returned value represents the bit results, and is displayed according to the Base mode.
You can enter the integers in any number base. For a binary or hexadecimal entry, you must use the 0b or 0h prefix, respectively. Without a prefix, integers are treated as decimal (base10).
Catalog >
In Hex basemode:
Important: Zero, not the letter O.
In Binbasemode:
In Dec base mode:
Note: A binary entry canhave upto 64 digits
(not countingthe0b prefix). A hexadecimal entry canhave up to 16 digits.
Alphabetical Listing 9
angle()
angle(Expr1) expression
Returns the angle of the argument, interpreting the argument as a complex number.
Note: All undefined variables are treated as
real variables.
angle(List1) list angle(Matrix1) matrix
Returns a list or matrix of angles of the elements in List1 or Matrix1, interpreting each element as a complex number that represents a two-dimensional rectangular coordinate point.
Catalog >
In Degree angle mode:
In Gradianangle mode:
In Radian angle mode:
ANOVA
Catalog >
ANOVA List1,List2[,List3,...,List20][,Flag]
Performs a one-way analysis of variance for comparing the means of two to 20 populations. A summary of results is stored in the stat.results variable. (page 177)
Flag=0 for Data, Flag=1 for Stats
Output variable Description
stat.F Valueof the F statistic
stat.PVal Smallest level of significance at which thenull hypothesis can be rejected
stat.df Degrees of freedom of the groups
stat.SS Sum of squaresof thegroups
stat.MS Mean squares for the groups
10 A lphabetical Listing
Output variable Description
stat.dfError Degrees of freedom of the errors
stat.SSError Sum of squaresof theerrors
stat.MSError Mean square for the errors
stat.sp Pooled standard deviation
stat.xbarlist Mean of the inputof the lists
stat.CLowerList 95%confidence intervals for the mean of each inputlist
stat.CUpperList 95%confidenceintervals for the mean of each inputlist
ANOVA2way
Catalog >
ANOVA2way List1,List2[,List3,,List10]
[,levRow]
Computes a two-way analysis of variance for comparing the means of two to 10 populations. A summary of results is stored in the stat.results variable. (See page 177.)
LevRow=0 for Block
LevRow=2,3,...,Len-1, for Two Factor,
where Len=length(List1)=length(List2) = … = length(List10) and Len/LevRow Î {2,3,…}
Outputs: Block Design
Output variable Description
stat.F Fstatistic of the columnfactor
stat.PVal Smallest level of significance at which thenull hypothesis can be rejected
stat.df Degrees of freedom of the columnfactor
stat.SS Sum of squaresof thecolumn factor
stat.MS Mean squares for columnfactor
stat.FBlock F statisticfor factor
stat.PValBlock Leastprobability atwhichthe nullhypothesiscanbe rejected
stat.dfBlock Degrees of freedom for factor
stat.SSBlock Sumof squares for factor
Alphabetical Listing 11
Output variable Description
stat.MSBlock Mean squares for factor
stat.dfError Degrees of freedom of the errors
stat.SSError Sum of squaresof theerrors
stat.MSError Mean squares for the errors
stat.s Standard deviation of the error
COLUMN FACTOR Outputs
Output variable Description
stat.Fcol F statistico f the columnfactor
stat.PValCol Probability valueof the columnfactor
stat.dfCol Degrees of freedom of the columnfactor
stat.SSCol Sum of squaresof thecolumn factor
stat.MSCol Mean squares for columnfactor
ROW FACTOR Outputs
Output variable Description
stat.FRow F statisticof the row factor
stat.PValRow Probability valueof the row factor
stat.dfRow Degrees of freedom of the ro w factor
stat.SSRow Sum of squaresof the row factor
stat.MSRow Mean squares for row factor
INTERACTION Outputs
Output variable Description
stat.FInteract F statisticof the interaction
stat.PValInteract Probability valueof the interaction
stat.dfInteract Degrees of freedom of the interaction
stat.SSInteract Sum of squaresof theinteraction
stat.MSInteract Mean squares for interaction
ERROR Outputs
12 A lphabetical Listing
Output variable Description
stat.dfError Degrees of freedom of the errors
stat.SSError Sum of squaresof theerrors
stat.MSError Mean squares for the errors
s Standard deviation of the error
Ans
Ans value
Returns the result of the most recently evaluated expression.
approx()
approx(Expr1) expression
Returns the evaluation of the argument as an expression containing decimal values, when possible, regardless of the current
Auto or Approximate mode.
This is equivalent to entering the argument and pressing .
approx(List1) list approx(Matrix1) matrix
Returns a list or matrix where each element has been evaluated to a decimal value, when possible.
approxFraction()
ExprapproxFraction([Tol]) expression
/v keys
Catalog >
Catalog >
ListapproxFraction([Tol]) ⇒ list
MatrixapproxFraction([Tol]) ⇒ matrix
Returns the input as a fraction, using a tolerance of Tol. If Tol is omitted, a tolerance of 5.E-14 is used.
Alphabetical Listing 13
approxFraction()
Note: You can insert this function from the
computer keyboard by typing
@>approxFraction(...).
Catalog >
approxRational()
approxRational(Expr[, Tol]) expression
approxRational(List[, Tol]) list
approxRational(Matrix[, Tol]) matrix
Returns the argument as a fraction using a tolerance of Tol. If Tol is omitted, a tolerance of 5.E-14 is used.
arccos()
arccosh()
arccot()
arccoth()
Catalog >
See cos¹(), page 31.
See cosh¹(), page 32.
See cot¹(), page 33.
See coth¹(), page 34.
arccsc()
arccsch()
14 A lphabetical Listing
See csc¹(), page 37.
See csch¹(), page 37.
arcLen()
arcLen(Expr1,Var,Start,End)
expression
Returns the arc length of Expr1 from
Start to End with respect to variable Var.
Arc length is calculated as an integral assuming a function mode definition.
arcLen(List1,Var,Start,End) list
Returns a list of the arc lengths of each element of List1 from Start to End with respect to Var.
Catalog >
arcsec()
arcsech()
arcsin()
arcsinh()
arctan()
arctanh()
augment()
augment(List1, List2) list
See sec¹(), page 158.
See sech¹(), page 158.
See sin¹(), page 168.
See sinh¹(), page 169.
See tan¹(), page 183.
See tanh¹(), page 185.
Catalog >
Alphabetical Listing 15
augment()
Returns a new list that is List2 appended to the end of List1.
augment(Matrix1, Matrix2) matrix
Returns a new matrix that is Matrix2 appended to Matrix1. When the “,” character is used, the matrices must have equal row dimensions, and Matrix2 is appended to Matrix1 as new columns. Does not alter Matrix1 or Matrix2.
Catalog >
avgRC()
avgRC(Expr1, Var [=Value] [, Step])
expression
avgRC(Expr1, Var [=Value] [, List1])
list
avgRC(List1, Var [=Value] [, Step])
list
avgRC(Matrix1, Var [=Value] [, Step])
matrix
Returns the forward-difference quotient (average rate of change).
Expr1 can be a user-defined function name
(see Func).
When Value is specified, it overrides any prior variable assignment or any current “|” substitution for the variable.
Step is the step value. If Step is omitted, it
defaults to 0.001.
Note that the similar function centralDiff() uses the central-difference quotient.
Catalog >
16 A lphabetical Listing
B
bal()
bal(NPmt,N,I,PV ,[Pmt], [FV], [PpY], [CpY], [PmtAt], [roundValue]) value
bal(NPmt,amortTable) value
Amortization function that calculates schedule balance after a specified payment.
N, I, PV, Pmt, FV, PpY, CpY, and PmtAt
are described in the table of TVM arguments, page 195.
NPmt specifies the payment number after
which you want the data calculated.
N, I, PV, Pmt, FV, PpY, CpY, and PmtAt
are described in the table of TVM arguments, page 195.
If you omit Pmt, it defaults to
Pmt=tvmPmt
(N,I,PV,FV,PpY,CpY,PmtAt).
If you omit FV, it defaults to FV=0.
The defaults for PpY, CpY, and PmtAt are the same as for the TVM functions.
roundValue specifies the number of
decimal places for rounding. Default=2.
bal(NPmt,amortTable) calculates the
balance after payment number NPmt, based on amortization table amortTable. The amortTable argument must be a matrix in the form described under
amortTbl(), page 8. Note: See also ΣInt() and ΣPrn(), page 225.
Catalog >
Base2
Integer1 Base2 integer
Note: You can insert this operator from the
computer keyboard by typing @>Base2.
Catalog >
Alphabetical Listing 17
Base2
Converts Integer1 to a binary number. Binary or hexadecimal numbers always have a 0b or 0h prefix, respectively. Use a zero, not the letter O, followed by b or h.
0b binaryNumber 0h hexadecimalNumber
A binary number can have up to 64 digits. A hexadecimal number can have up to 16.
Without a prefix, Integer1 is treated as decimal (base10). The result is displayed in binary, regardless of the Base mode.
Negative numbers are displayed in “two's complement” form. For example,
1is displayed as
0hFFFFFFFFFFFFFFFFin Hex base mode 0b111...111 (641’s)in Binary base mode
63
2
is displayed as 0h8000000000000000in Hex base mode 0b100...000 (63 zeros)in Binary base mode
If you enter a decimal integer that is outside the range of a signed, 64-bit binary form, a symmetric modulo operation is used to bring the value into the appropriate range. Consider the following examples of values outside the range.
263becomes ⁻263and is displayed as 0h8000000000000000in Hex base mode 0b100...000 (63 zeros)in Binary base mode
264becomes 0 and is displayed as 0h0in Hex base mode 0b0in Binary base mode
263− 1 becomes 263− 1 and is displayed
as 0h7FFFFFFFFFFFFFFFin Hex base mode 0b111...111 (641’s)in Binary base mode
Catalog >
Base10
Integer1 Base10 integer
18 A lphabetical Listing
Catalog >
Base10
Note: You can insert this operator from the
computer keyboard by typing @>Base10.
Converts Integer1 to a decimal (base10) number. A binary or hexadecimal entry must always have a 0b or 0h prefix, respectively.
0b binaryNumber 0h hexadecimalNumber
Zero, not the letter O, followed by b or h.
A binary number can have up to 64 digits. A hexadecimal number can have up to 16.
Without a prefix, Integer1 is treated as decimal. The result is displayed in decimal, regardless of the Base mode.
Catalog >
Base16
Integer1 Base16 integer
Note: You can insert this operator from the
computer keyboard by typing @>Base16.
Converts Integer1 to a hexadecimal number. Binary or hexadecimal numbers always have a 0b or 0h prefix, respectively.
0b binaryNumber 0h hexadecimalNumber
Zero, not the letter O, followed by b or h.
A binary number can have up to 64 digits. A hexadecimal number can have up to 16.
Without a prefix, Integer1 is treated as decimal (base10). The result is displayed in hexadecimal, regardless of the Base mode.
If you enter a decimal integer that is too large for a signed, 64-bit binary form, a symmetric modulo operation is used to bring the value into the appropriate range. For more information, see Base2, page
17.
Catalog >
Alphabetical Listing 19
binomCdf()
binomCdf(n,p) list
binomCdf(n,p,lowBound,upBound)
number if lowBound and upBound are
numbers, list if lowBound and upBound are lists
binomCdf(n,p,upBound)for P(0≤X≤upBound)
number if upBound is a number, list if upBound is a list
Computes a cumulative probability for the discrete binomial distribution with n number of trials and probability p of success on each trial.
For P(X upBound), set lowBound=0
Catalog >
binomPdf()
binomPdf(n,p) list
binomPdf(n,p,XVal) number if XVal is a
number, list if XVal is a list
Computes a probability for the discrete binomial distribution with n number of trials and probability p of success on each trial.
C
ceiling(Expr1) integer
Returns the nearest integer that is the argument.
The argument can be a real or a complex number.
Note: See also floor().
ceiling(List1) list ceiling(Matrix1) matrix
Returns a list or matrix of the ceiling of each element.
Catalog >
Catalog >
20 A lphabetical Listing
centralDiff()
centralDiff(Expr1,Var [=Value][,Step])
expression
centralDiff(Expr1,Var [,Step])|Var=Value
expression
centralDiff(Expr1,Var [=Value][,List])
list
centralDiff(List1,Var [=Value][,Step])
list
centralDiff(Matrix1,Var [=Value][,Step])
matrix
Returns the numerical derivative using the central difference quotient formula.
When Value is specified, it overrides any prior variable assignment or any current “|” substitution for the variable.
Step is the step value. If Step is omitted, it
defaults to 0.001.
When using List1 or Matrix1, the operation gets mapped across the values in the list or across the matrix elements.
Note: See also avgRC() and d().
Catalog >
cFactor()
cFactor(Expr1[,Var]) expression cFactor(List1[,Var]) list cFactor(Matrix1[,Var]) matrix
cFactor(Expr1) returns Expr1 factored with
respect to all of its variables over a common denominator.
Expr1 is factored as much as possible
toward linear rational factors even if this introduces new non-real numbers. This alternative is appropriate if you want factorization with respect to more than one variable.
Catalog >
Alphabetical Listing 21
cFactor()
cFactor(Expr1,Var) returns Expr1 factored
with respect to variable Var.
Expr1 is factored as much as possible
toward factors that are linear in Var, with perhaps non-real constants, even if it introduces irrational constants or subexpressions that are irrational in other variables.
The factors and their terms are sorted with
Var as the main variable. Similar powers of Var are collected in each factor. Include Var if factorization is needed with respect
to only that variable and you are willing to accept irrational expressions in any other variables to increase factorization with respect to Var. There might be some incidental factoring with respect to other variables.
For the Auto setting of the Auto or
Approximate mode, including Var also
permits approximation with floating-point coefficients where irrational coefficients cannot be explicitly expressed concisely in terms of the built-in functions. Even when there is only one variable, including Var might yield more complete factorization.
Note: See also factor().
Catalog >
To see the entire result, press £ and then use¡and¢ to move thecursor.
char()
char(Integer) character
Returns a character string containing the character numbered Integer from the handheld character set. The valid range for
Integer is 0–65535.
22 A lphabetical Listing
Catalog >
charPoly()
charPoly(squareMatrix,Var)
polynomial expression
charPoly(squareMatrix,Expr)
polynomial expression
charPoly(squareMatrix1,Matrix2)
polynomial expression
Returns the characteristic polynomial of
squareMatrix. The characteristic
polynomial of n×n matrix A, denoted by p (λ), is the polynomial defined by
p
(λ) = det(λ•I−A)
A
where I denotes the n×n identity matrix.
squareMatrix1 and squareMatrix2 must
have the equal dimensions.
Catalog >
A
2
χ
2way
2
χ
2way obsMatrix
Catalog >
chi22way obsMatrix
Computes a χ2test for association on the two-way table of counts in the observed matrix obsMatrix. A summary of results is stored in the stat.results variable. (page
177)
For information on the effect of empty elements in a matrix, see “Empty (Void) Elements,” page 236.
Output variable Description
2
stat.χ
stat.PVal Smallest level of significance at which thenull hypothesis can be rejected
stat.df Degrees of freedom for the chi square statistics
stat.ExpMat Matrix of expectedelemental counttable, assuming nullhypothesis
stat.CompMat Matrix of elementalchi square statisticcontributions
Chi square stat: sum (observed - expected)2/expected
Alphabetical Listing 23
2
χ
Cdf()
2
χ
Cdf(lowBound,upBound,df) number if
lowBound and upBound are numbers, list if lowBound and upBound are lists
chi2Cdf(lowBound,upBound,df) number
if lowBound and upBound are numbers, list if lowBound and upBound are lists
Computes the χ2distribution probability between lowBound and upBound for the specified degrees of freedom df.
For P(X upBound), set lowBound = 0.
For information on the effect of empty elements in a list, see “Empty (Void) Elements,” page 236.
Catalog >
2
χ
GOF
2
χ
GOF obsList,expList,df
Catalog >
chi2GOF obsList,expList,df
Performs a test to confirm that sample data is from a population that conforms to a specified distribution. obsList is a list of counts and must contain integers. A summary of results is stored in the
stat.results variable. (See page 177.)
For information on the effect of empty elements in a list, see “Empty (Void) Elements,” page 236.
Output variable Description
2
stat.χ
stat.PVal Smallest level of significance at which thenull hypothesis can be rejected
stat.df Degrees of freedom for the chi square statistics
stat.CompList Elemental chisquare statistic contributions
2
χ
Pdf()
2
χ
Pdf(XVal,df) number if XVal is a
Chi square stat: sum((observed - expected)2/expected
Catalog >
number, list if XVal is a list
24 A lphabetical Listing
2
χ
Pdf()
chi2Pdf(XVal,df) number if XVal is a
number, list if XVal is a list
Computes the probability density function (pdf) for the χ2distribution at a specified
XVal value for the specified degrees of
freedom df.
For information on the effect of empty elements in a list, see “Empty (Void) Elements,” page 236.
Catalog >
ClearAZ
ClearAZ
Clears all single-character variables in the current problem space.
If one or more of the variables are locked, this command displays an error message and deletes only the unlocked variables. See
unLock, page 197.
ClrErr
ClrErr
Clears the error status and sets system variable errCode to zero.
The Else clause of the Try...Else...EndTry block should use ClrErr or PassErr. If the error is to be processed or ignored, use
ClrErr. If what to do with the error is not
known, use PassErr to send it to the next error handler. If there are no more pending
Try...Else...EndTry error handlers, the error
dialog box will be displayed as normal.
Note: See also PassErr, page 132, and Try,
page 191.
Note for entering the example: For
instructions on entering multi-line program and function definitions, refer to the Calculator section of your product guidebook.
Catalog >
Catalog >
For an example of ClrErr, See Example 2 under the Try command, page 191.
Alphabetical Listing 25
colAugment()
colAugment(Matrix1, Matrix2) matrix
Returns a new matrix that is Matrix2 appended to Matrix1. The matrices must have equal column dimensions, and
Matrix2 is appended to Matrix1 as new
rows. Does not alter Matrix1 or Matrix2.
Catalog >
colDim()
colDim(Matrix) expression
Returns the number of columns contained in Matrix.
Note: See also rowDim().
colNorm()
colNorm(Matrix) expression
Returns the maximum of the sums of the absolute values of the elements in the columns in Matrix.
Note: Undefined matrix elements are not
allowed. See also rowNorm().
comDenom()
comDenom(Expr1[,Var]) expression comDenom(List1[,Var]) list comDenom(Matrix1[,Var]) matrix
comDenom(Expr1) returns a reduced ratio
of a fully expanded numerator over a fully expanded denominator.
Catalog >
Catalog >
Catalog >
26 A lphabetical Listing
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