Calculus¶
In this chapter, some facilities for doing calculus are described. These include functions implementing differentiation, integration, calculating limits etc.

D
(variable[, n=1]) expression¶ derivative
 Param variable
variable
 Param expression
expression to take derivatives of
 Param n
order
 Returns
n
th derivative ofexpression
with respect tovariable

D
(variable) expression derivative
 Param variable
variable
 Param list
a list of variables
 Param expression
expression to take derivatives of
 Param n
order of derivative
 Returns
derivative of
expression
with respect tovariable
This function calculates the derivative of the expression {expr} with respect to the variable {var} and returns it. If the third calling format is used, the {n}th derivative is determined. Yacas knows how to differentiate standard functions such as {Ln} and {Sin}. The {D} operator is threaded in both {var} and {expr}. This means that if either of them is a list, the function is applied to each entry in the list. The results are collected in another list which is returned. If both {var} and {expr} are a list, their lengths should be equal. In this case, the first entry in the list {expr} is differentiated with respect to the first entry in the list {var}, the second entry in {expr} is differentiated with respect to the second entry in {var}, and so on. The {D} operator returns the original function if \(n=0\), a common mathematical idiom that simplifies many formulae.
 Example
In> D(x)Sin(x*y) Out> y*Cos(x*y); In> D({x,y,z})Sin(x*y) Out> {y*Cos(x*y),x*Cos(x*y),0}; In> D(x,2)Sin(x*y) Out> Sin(x*y)*y^2; In> D(x){Sin(x),Cos(x)} Out> {Cos(x),Sin(x)};
See also
Integrate()
,Taylor()
,Diverge()
,Curl()

Curl
(vector, basis)¶ curl of a vector field
 Param vector
vector field to take the curl of
 Param basis
list of variables forming the basis
This function takes the curl of the vector field
vector
with respect to the variablesbasis
. The curl is defined in the usual way,Curl(f,x) = {D(x[2]) f[3]  D(x[3]) f[2], D(x[3]) f[1]  D(x[1])f[3], D(x[1]) f[2]  D(x[2]) f[1]}
. Bothvector
andbasis
should be lists of length 3.

Diverge
(vector, basis)¶ divergence of a vector field
 Param vector
vector field to calculate the divergence of
 Param basis
list of variables forming the basis
This function calculates the divergence of the vector field
vector
with respect to the variablesbasis
. The divergence is defined asDiverge(f,x) = D(x[1]) f[1] + ... + D(x[n]) f[n]
, wheren
is the length of the listsvector
andbasis
. These lists should have equal length.

HessianMatrix
(function, var)¶ create the Hessian matrix
 Param function
a function in \(n\) variables
 Param var
an \(n\)dimensional vector of variables
The function
HessianMatrix()
calculates the Hessian matrix of a vector. If \(f(x)\) is a function of an \(n\)dimensional vector \(x\), then the \((i,j)\)th element of the Hessian matrix of the function \(f(x)\) is defined as :math:` Deriv(x[i]) Deriv(x[j]) f(x)`. If the second order mixed partials are continuous, then the Hessian matrix is symmetric (a standard theorem of calculus). The Hessian matrix is used in the second derivative test to discern if a critical point is a local maximum, a local minimum or a saddle point. Example
In> HessianMatrix(3*x^22*x*y+y^28*y, {x,y} ) Out> {{6,2},{2,2}}; In> PrettyForm(%) / \  ( 6 ) ( 2 )     ( 2 ) ( 2 )  \ /

JacobianMatrix
(functions, variables)¶ calculate the Jacobian matrix of \(n\) functions in \(n\) variables
 Param functions
an \(n\)dimensional vector of functions
 Param variables
an \(n\)dimensional vector of variables
The function {JacobianMatrix} calculates the Jacobian matrix of n functions in n variables. The \((i,j)\)th element of the Jacobian matrix is defined as the derivative of \(i\)th function with respect to the \(j\)th variable.
 Example
In> JacobianMatrix( {Sin(x),Cos(y)}, {x,y} ); Out> {{Cos(x),0},{0,Sin(y)}}; In> PrettyForm(%) / \  ( Cos( x ) ) ( 0 )     ( 0 ) ( ( Sin( y ) ) )  \ /

Integrate
(var) expr¶ 
Integrate
(var, x1, x2) expr integral
 Param expr
expression to integrate
 Param var
atom, variable to integrate over
 Param x1
first point of definite integration
 Param x2
second point of definite integration
This function integrates the expression expr with respect to the variable var. In the case of definite integral, the integration is carried out from \(var=x1\) to \(var=x2\)”. Some simple integration rules have currently been implemented. Polynomials, some quotients of polynomials, trigonometric functions and their inverses, hyperbolic functions and their inverses, {Exp}, and {Ln}, and products of these functions with polynomials can be integrated.
 Example
In> Integrate(x,a,b) Cos(x) Out> Sin(b)Sin(a); In> Integrate(x) Cos(x) Out> Sin(x);
See also

Limit
(var, val) expr¶ limit of an expression
 Param var
variable
 Param val
number or
Infinity
 Param dir
direction (
Left
orRight
) Param expr
an expression
This command tries to determine the value that the expression “expr” converges to when the variable “var” approaches “val”. One may use {Infinity} or {Infinity} for “val”. The result of {Limit} may be one of the symbols {Undefined} (meaning that the limit does not exist), {Infinity}, or {Infinity}. The second calling sequence is used for unidirectional limits. If one gives “dir” the value {Left}, the limit is taken as “var” approaches “val” from the positive infinity; and {Right} will take the limit from the negative infinity.
 Example
In> Limit(x,0) Sin(x)/x Out> 1; In> Limit(x,0) (Sin(x)Tan(x))/(x^3) Out> 1/2; In> Limit(x,0) 1/x Out> Undefined; In> Limit(x,0,Left) 1/x Out> Infinity; In> Limit(x,0,Right) 1/x Out> Infinity;

Add
(val1, val2, ...)¶ 
Add
(list) find sum of a list of values
 Param val1 val2
expressions
 Param list
list of expressions to add
This function adds all its arguments and returns their sum. It accepts any number of arguments. The arguments can be also passed as a list.
 Example
In> Add(1,4,9); Out> 14; In> Add(1 .. 10); Out> 55;

Multiply
(val1, val2, ...)¶ 
Multiply
(list) product of a list of values
 Param val1 val2
expressions
 Param list
list of expressions to add
Multiply all arguments and returns their product. It accepts any number of arguments. The arguments can be also passed as a list.
 Example
In> Multiply(2,3,4); Out> 24 In> Multiply(1 .. 10) Out> 3628800

Sum
(var, from, to, body)¶ find sum of a sequence
 Param var
variable to iterate over
 Param from
integer value to iterate from
 Param to
integer value to iterate up to
 Param body
expression to evaluate for each iteration
The command finds the sum of the sequence generated by an iterative formula. The expression “body” is evaluated while the variable “var” ranges over all integers from “from” up to “to”, and the sum of all the results is returned. Obviously, “to” should be greater than or equal to “from”. Warning: {Sum} does not evaluate its arguments {var} and {body} until the actual loop is run.
 Example
In> Sum(i, 1, 3, i^2); Out> 14;
See also

Factorize
(list)¶ product of a list of values
 Param list
list of values to multiply
 Param var
variable to iterate over
 Param from
integer value to iterate from
 Param to
integer value to iterate up to
 Param body
expression to evaluate for each iteration
The first form of the {Factorize} command simply multiplies all the entries in “list” and returns their product. If the second calling sequence is used, the expression “body” is evaluated while the variable “var” ranges over all integers from “from” up to “to”, and the product of all the results is returned. Obviously, “to” should be greater than or equal to “from”.
 Example
In> Factorize({1,2,3,4}); Out> 24; In> Factorize(i, 1, 4, i); Out> 24;

Taylor(var, at, order) expr
univariate Taylor series expansion
 Param var
variable
 Param at
point to get Taylor series around
 Param order
order of approximation
 Param expr
expression to get Taylor series for
This function returns the Taylor series expansion of the expression “expr” with respect to the variable “var” around “at” up to order “order”. This is a polynomial which agrees with “expr” at the point “var = at”, and furthermore the first “order” derivatives of the polynomial at this point agree with “expr”. Taylor expansions around removable singularities are correctly handled by taking the limit as “var” approaches “at”.
 Example
In> PrettyForm(Taylor(x,0,9) Sin(x)) 3 5 7 9 x x x x x   +    +  6 120 5040 362880 Out> True;
See also
D()
,InverseTaylor()
,ReversePoly()
,BigOh()

InverseTaylor(var, at, order) expr
Taylor expansion of inverse
 Param var
variable
 Param at
point to get inverse Taylor series around
 Param order
order of approximation
 Param expr
expression to get inverse Taylor series for
This function builds the Taylor series expansion of the inverse of the expression “expr” with respect to the variable “var” around “at” up to order “order”. It uses the function {ReversePoly} to perform the task.
 Example
In> PrettyPrinter'Set("PrettyForm") True In> exp1 := Taylor(x,0,7) Sin(x) 3 5 7 x x x x   +    6 120 5040 In> exp2 := InverseTaylor(x,0,7) ArcSin(x) 5 7 3 x x x      + x 120 5040 6 In> Simplify(exp1exp2) 0
See also
ReversePoly()
,Taylor()
,BigOh()

ReversePoly
(f, g, var, newvar, degree)¶ solve \(h(f(x)) = g(x) + O(x^n)\) for \(h\)
 Param f
function of
var
 Param g
function of
var
 Param var
a variable
 Param newvar
a new variable to express the result in
 Param degree
the degree of the required solution
This function returns a polynomial in “newvar”, say “h(newvar)”, with the property that “h(f(var))” equals “g(var)” up to order “degree”. The degree of the result will be at most “degree1”. The only requirement is that the first derivative of “f” should not be zero. This function is used to determine the Taylor series expansion of the inverse of a function “f”: if we take “g(var)=var”, then “h(f(var))=var” (up to order “degree”), so “h” will be the inverse of “f”.
 Example
In> f(x):=Eval(Expand((1+x)^4)) Out> True; In> g(x) := x^2 Out> True; In> h(y):=Eval(ReversePoly(f(x),g(x),x,y,8)) Out> True; In> BigOh(h(f(x)),x,8) Out> x^2; In> h(x) Out> (2695*(x1)^7)/131072+(791*(x1)^6)/32768 +(119*(x1)^5)/4096+(37*(x1)^4)/1024+(3*(x1)^3)/64+(x1)^2/16;
See also
InverseTaylor()
,Taylor()
,BigOh()

BigOh
(poly, var, degree)¶ drop all terms of a certain order in a polynomial
 Param poly
a univariate polynomial
 Param var
a free variable
 Param degree
positive integer
This function drops all terms of order “degree” or higher in “poly”, which is a polynomial in the variable “var”.
 Example
In> BigOh(1+x+x^2+x^3,x,2) Out> x+1;
See also
Taylor()
,InverseTaylor()

LagrangeInterpolant
(xlist, ylist, var)¶ polynomial interpolation
 Param xlist
list of argument values
 Param ylist
list of function values
 Param var
free variable for resulting polynomial
This function returns a polynomial in the variable “var” which interpolates the points “(xlist, ylist)”. Specifically, the value of the resulting polynomial at “xlist[1]” is “ylist[1]”, the value at “xlist[2]” is “ylist[2]”, etc. The degree of the polynomial is not greater than the length of “xlist”. The lists “xlist” and “ylist” should be of equal length. Furthermore, the entries of “xlist” should be all distinct to ensure that there is one and only one solution. This routine uses the Lagrange interpolant formula to build up the polynomial.
 Example
In> f := LagrangeInterpolant({0,1,2}, \ {0,1,1}, x); Out> (x*(x1))/2x*(x2); In> Eval(Subst(x,0) f); Out> 0; In> Eval(Subst(x,1) f); Out> 1; In> Eval(Subst(x,2) f); Out> 1; In> PrettyPrinter'Set("PrettyForm"); True In> LagrangeInterpolant({x1,x2,x3}, {y1,y2,y3}, x) y1 * ( x  x2 ) * ( x  x3 )  ( x1  x2 ) * ( x1  x3 ) y2 * ( x  x1 ) * ( x  x3 ) +  ( x2  x1 ) * ( x2  x3 ) y3 * ( x  x1 ) * ( x  x2 ) +  ( x3  x1 ) * ( x3  x2 )
See also

n
!
¶ factorial
 Param m
integer
 Param n
integer, halfinteger, or list
 Param a}, {b
numbers
The factorial function {n!} calculates the factorial of integer or halfinteger numbers. For nonnegative integers, \(n! := n*(n1)*(n2)*...*1\). The factorial of halfintegers is defined via Euler’s Gamma function, \(z! := Gamma(z+1)\). If \(n=0\) the function returns \(1\). The “double factorial” function {n!!} calculates \(n*(n2)*(n4)*...\). This product terminates either with \(1\) or with \(2\) depending on whether \(n\) is odd or even. If \(n=0\) the function returns \(1\). The “partial factorial” function {a * b} calculates the product :math:`a*(a+1)*…` which is terminated at the least integer not greater than :math:`b`. The arguments :math:`a` and :math:`b` do not have to be integers; for integer arguments, {a * b} = \(b! / (a1)!\). This function is sometimes a lot faster than evaluating the two factorials, especially if \(a\) and \(b\) are close together. If \(a>b\) the function returns \(1\). The {Subfactorial} function can be interpreted as the number of permutations of {m} objects in which no object appears in its natural place, also called “derangements.” The factorial functions are threaded, meaning that if the argument {n} is a list, the function will be applied to each element of the list. Note: For reasons of Yacas syntax, the factorial sign {!} cannot precede other nonletter symbols such as {+} or {*}. Therefore, you should enter a space after {!} in expressions such as {x! +1}. The factorial functions terminate and print an error message if the arguments are too large (currently the limit is \(n < 65535\)) because exact factorials of such large numbers are computationally expensive and most probably not useful. One can call {Internal’LnGammaNum()} to evaluate logarithms of such factorials to desired precision.
 Example
In> 5! Out> 120; In> 1 * 2 * 3 * 4 * 5 Out> 120; In> (1/2)! Out> Sqrt(Pi)/2; In> 7!!; Out> 105; In> 1/3 *** 10; Out> 17041024000/59049; In> Subfactorial(10) Out> 1334961;
See also
Bin()
,Factorize()
,Gamma()
,!()
,***()
,Subfactorial()

n
!!
¶ double factorial

x
***
y¶ whatever

Bin
(n, m)¶ binomial coefficients
 Param n}, {m
integers
This function calculates the binomial coefficient “n” above “m”, which equals \(n! / (m! * (nm)!)\) This is equal to the number of ways to choose “m” objects out of a total of “n” objects if order is not taken into account. The binomial coefficient is defined to be zero if “m” is negative or greater than “n”; {Bin(0,0)}=1.
 Example
In> Bin(10, 4) Out> 210; In> 10! / (4! * 6!) Out> 210;
See also
()
,Eulerian()

Eulerian
(n, m)¶ Eulerian numbers
The Eulerian numbers can be viewed as a generalization of the binomial coefficients, and are given explicitly by \(Sum(j,0,k+1,(1)^j*Bin(n+1,j)*(kj+1)^n)\).
 Example
In> Eulerian(6,2) Out> 302; In> Eulerian(10,9) Out> 1;
See also

KroneckerDelta
(i, j)¶ 
KroneckerDelta
({i, j, ...}) Kronecker delta
Calculates the Kronecker delta, which gives \(1\) if all arguments are equal and \(0\) otherwise.

LeviCivita
(list)¶ totally antisymmetric LeviCivita symbol
 Param list
a list of integers \(1,\ldots,n\) in some order
LeviCivita()
implements the LeviCivita symbol.list
should be a list of integers, and this function returns 1 if the integers are in successive order, eg.LeviCivita({1,2,3,...})
would return 1. Swapping two elements of this list would return 1. So,LeviCivita({2,1,3})
would evaluate to 1. Example
In> LeviCivita({1,2,3}) Out> 1; In> LeviCivita({2,1,3}) Out> 1; In> LeviCivita({2,2,3}) Out> 0;
See also

Permutations
(list)¶ get all permutations of a list
 Param list
a list of elements
Permutations returns a list with all the permutations of the original list.
 Example
In> Permutations({a,b,c}) Out> {{a,b,c},{a,c,b},{c,a,b},{b,a,c}, {b,c,a},{c,b,a}};
See also

Fibonacci
(n)¶ Fibonacci sequence
The function returns \(n\)th Fibonacci number
 Example
In> Fibonacci(4) Out> 3 In> Fibonacci(8) Out> 21 In> Table(Fibonacci(i), i, 1, 10, 1) Out> {1,1,2,3,5,8,13,21,34,55}