Sin, Cos, Tan , ArcSin, ArcCos, ArcTan , Exp , Ln , Sqrt , Abs , Sign , Complex , Re , Im , I , Conjugate , Arg , ! , Bin , Sum , Average , Factorize , Min , Max , IsZero , IsRational , Numer , Denom , Commutator , Taylor , InverseTaylor , ReversePoly , BigOh , Newton , D , Curl , Diverge , Integrate , Simplify , RadSimp , Rationalize , Solve , SuchThat , Eliminate , PSolve , Pi , Random , VarList , Limit , TrigSimpCombine , LagrangeInterpolant , Fibonacci .

Calculus

In this chapter, some functions for doing calculus are described. These include functions implementing differentiation, integration, standard mathematical functions, and the solving of equations.

Sin, Cos, Tan Trigonometric functions
ArcSin, ArcCos, ArcTan Inverse trigonometric functions
Exp Exponential function
Ln Natural logarithm
Sqrt Square root
Abs Absolute value or modulus
Sign Sign of a number
Complex Construct a complex number
Re Real part of a complex number
Im Imaginary part of a complex number
I Imaginary unit
Conjugate Complex conjugate
Arg Argument of a complex number
! Factorial
Bin Binomial coefficient
Sum Sum of a list of values
Average Average of a list of values
Factorize Product of a list of values
Min Minimum of a number of values
Max Maximum of a number of values
IsZero Test whether argument is zero
IsRational Test whether argument is a rational
Numer Numerator of an expression
Denom Denominator of an expression
Commutator Commutator of two objects
Taylor Univariate Taylor series expansion
InverseTaylor Taylor expansion of inverse
ReversePoly Solve "h(f(x)) = g(x) + O(x^n)" for h
BigOh Drop all terms of a certain order in a polynomial
Newton Solve an equation numerically with Newton's method
D Differentiation
Curl Curl of a vector field
Diverge Divergence of a vector field
Integrate Integration
Simplify Try to simplify an expression
RadSimp Simplify expression with nested radicals
Rationalize Convert floating point numbers to fractions
Solve Solve one or more algebraic equations
SuchThat Find a value which makes some expression zero
Eliminate Substitute and simplify
PSolve Solve a polynomial equation
Pi Numerical approximation of pi
Random Random number between 0 and 1
VarList List of variables appearing in some expression
Limit Limit of an expression
TrigSimpCombine Combine products of trigonometric functions
LagrangeInterpolant Polynomial interpolation
Fibonacci Fibonacci sequence


Sin, Cos, Tan -- Trigonometric functions

Standard math library
Calling Sequence:
Sin(x)
Cos(x)
Tan(x)
Parameters:
x - argument to the function, in radians
Description:
These functions represent the trigonometric functions sine, cosine, and tangent respectively. Yacas leaves them alone even if x is a number, trying to keep the result as exact as possible. The floating point approximations of these functions can be forced by using the N function.

Yacas knows some trigonometric identities, so it can simplify to exact results even if N is not used. This is the case when the arguments are multiples of Pi/6 or Pi/4.

These functions are threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.
Examples:
In> Sin(1)
Out> Sin(1);
In> N(Sin(1),20)
Out> 0.84147098480789650665;
In> Sin(Pi/4)
Out> Sqrt(2)/2;
See Also:
ArcSin, ArcCos, ArcTan , N , Pi .


ArcSin, ArcCos, ArcTan -- Inverse trigonometric functions

Standard math library
Calling Sequence:
ArcSin(x)
ArcCos(x)
ArcTan(x)
Parameters:
x - argument to the function
Description:
These functions represent the inverse trigonometric functions. For instance, the value of "ArcSin(x)" is the number "y" such that "Sin(y)" equals "x".

Note that the number "y" is not unique. For instance, "Sin(0)" and "Sin(Pi)" both equal 0, so what should "ArcSin(0)" be? In Yacas, it is agreed that the value of "ArcSin(x)" should be in the interval [-Pi/2,Pi/2]. The same goes for "ArcTan(x)". However, "ArcCos(x)" is in the interval [0,Pi].

Usually, Yacas leaves these functions alone unless it is forced to do a numerical evaluation by the N function. If the argument is -1. 0, or 1 however, Yacas will simplify the expression. If the argument is complex, the expression will be rewritten as a Ln function.

These functions are threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.
Examples:
In> ArcSin(1)
Out> Pi/2;

In> ArcSin(1/3)
Out> ArcSin(1/3);
In> Sin(ArcSin(1/3))
Out> 1/3;

In> N(ArcSin(0.75))
Out> 0.848062;
In> N(Sin(%))
Out> 0.7499999477;
See Also:
Sin, Cos, Tan , N , Pi , Ln .


Exp -- Exponential function

Standard math library
Calling Sequence:
Exp(x)
Parameters:
x - argument to the function
Description:
This function calculates e raised to the power "x", where e is the mathematic constant 2.71828... One can use Exp(1) to represent e.

This function is threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.
Examples:
In> Exp(0)
Out> 1;
In> Exp(I*Pi)
Out> -1;
In> N(Exp(1))
Out> 2.7182818284;
See Also:
Ln , Sin, Cos, Tan , N .


Ln -- Natural logarithm

Standard math library
Calling Sequence:
Ln(x)
Parameters:
x - argument to the function
Description:
This function calculates the natural logarithm of "x". This is the inverse function of the exponential function, Exp, ie. "Ln(x) = y" implies that "Exp(y) = x". For complex arguments, the imaginary part of the logarithm is in the interval (-Pi,Pi]. This is compatible with the branch cut of Arg.

This function is threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.
Examples:
In> Ln(1)
Out> 0;
In> Ln(Exp(x))
Out> x;
In> D(x) Ln(x)
Out> 1/x;
See Also:
Exp , Arg .


Sqrt -- Square root

Standard math library
Calling Sequence:
Sqrt(x)
Parameters:
x - argument to the function
Description:
This function calculates the square root of "x". If the result is not rational, the call is returned unevaluated unless a numerical approximation is forced with the N function. This function can also handle negative and complex arguments.

This function is threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.
Examples:
In> Sqrt(16)
Out> 4;
In> Sqrt(15)
Out> Sqrt(15);
In> N(Sqrt(15))
Out> 3.8729833462;
In> Sqrt(4/9)
Out> 2/3;
In> Sqrt(-1)
Out> Complex(0,1);
Exp , ^ , N .


Abs -- Absolute value or modulus

Standard math library
Calling Sequence:
Abs(x)
Parameters:
x - argument to the function
Description:
This function returns the absolute value (also called the modulus) of "x". If "x" is positive, the absolute value is "x" itself; if "x" is negative, the absolute value is "-x". For complex "x", the modulus is the "r" in the polar decomposition "x = r * Exp(I*phi)".

This function is connected to the Sign function by the identity "Abs(x) * Sign(x) = x" for real "x".

This function is threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.
Examples:
In> Abs(2);
Out> 2;
In> Abs(-1/2);
Out> -1/2;
In> Abs(3+4*I);
Out> 5;
See Also:
Sign , Arg .


Sign -- Sign of a number

Standard math library
Calling Sequence:
Sign(x)
Parameters:
x - argument to the function
Description:
This function returns the sign of the real number "x". It is "1" for positive numbers and "-1" for negative numbers. Somewhat arbitrarily, Sign(0) is defined to be 1.

This function is connected to the Abs function by the identity "Abs(x) * Sign(x) = x" for real "x".

This function is threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.
Examples:
In> Sign(2)
Out> 1;
In> Sign(-3)
Out> -1;
In> Sign(0)
Out> 1;
In> Sign(-3) * Abs(-3)
Out> -3;
See Also:
Arg , Abs .


Complex -- Construct a complex number

Standard math library
Calling sequence:
Complex(r, c)
Parameters:
r - real part
c - imaginary part
Description:
This function represents the complex number "r + I*c", where "I" is the imaginary unit. It is the standard representation used in Yacas to represent complex numbers. Both "r" and "c" are supposed to be real.

Note that, at the moment, many functions in Yacas assume that all numbers are real unless it is obvious that it is a complex number. Hence Im(Sqrt(x)) evaluates to 0 which is only true for nonnegative "x".
Examples:
In> I
Out> Complex(0,1);
In> 3+4*I
Out> Complex(3,4);
In> Complex(-2,0)
Out> -2;
See Also:
Re , Im , I , Abs , Arg .


Re -- Real part of a complex number

Standard math library
Calling sequence:
Re(x)
Parameters:
x - argument to the function
Description:
This function returns the real part of the complex number "x".
Examples:
In> Re(5)
Out> 5;
In> Re(I)
Out> 0;
In> Re(Complex(3,4))
Out> 3;
See Also:
Complex , Im .


Im -- Imaginary part of a complex number

Standard math library
Calling sequence:
Im(x)
Parameters:
x - argument to the function
Description:
This function returns the imaginary part of the complex number "x".
Examples:
In> Im(5)
Out> 0;
In> Im(I)
Out> 1;
In> Im(Complex(3,4))
Out> 4;
See Also:
Complex , Re .


I -- Imaginary unit

Standard math library
Calling sequence:
I
Parameters:
none
Description:
This symbol represents the imaginary unit, which equals the square root of -1. It evaluates immediately to Complex(0,1).
Examples:
In> I
Out> Complex(0,1);
In> I = Sqrt(-1)
Out> True;
See Also:
Complex .


Conjugate -- Complex conjugate

Standard math library
Calling sequence:
Conjugate(x)
Parameters:
x - argument to the function
Description:
This function returns the complex conjugate of "x". The complex conjugate of "a + I*b" is "a - I*b". This function assumes that all unbound variables are real.
Examples:
In> Conjugate(2)
Out> 2;
In> Conjugate(Complex(a,b))
Out> Complex(a,-b);
See Also:
Complex , Re , Im .


Arg -- Argument of a complex number

Standard math library
Calling sequence:
Arg(x)
Parameters:
x - argument to the function
Description:
This function returns the argument of "x". The argument is the angle with the positive real axis in the Argand diagram, or the angle "phi" in the polar representation "r * Exp(I*phi)" of "x". The result lies between -Pi and Pi, excluding -Pi but including Pi. The argument of 0 is Undefined.
Examples:
In> Arg(2)
Out> 0;
In> Arg(-1)
Out> Pi;
In> Arg(1+I)
Out> Pi/4;
See Also:
Abs , Sign .


! -- Factorial

Standard math library
Calling sequence:
n!
Parameters:
n - argument to the function
Description:
This function calculate "n" factorial. For nonnegative integers, n! equals "n*(n-1)*(n-2)*...*1". The factorial of half-integers, defined via the gamma function, is also implemented.

This function is threaded, meaning that if the argument "x" is a list, the function is applied to all the entries in the list.
Examples:
In> 5!
Out> 120;
In> 1 * 2 * 3 * 4 * 5
Out> 120;
In> (1/2)!
Out> Sqrt(Pi)/2;
See Also:
Bin , Factorize .


Bin -- Binomial coefficient

Standard math library
Calling sequence:
Bin(n, m)
Parameters:
n, m - integers
Description:
This function calculates the binomial coefficient "n" above "m", which equals "n! / (n! * (n-m)!)". This is 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".
Examples:
In> Bin(10, 4)
Out> 210;
In> 10! / (4! * 6!)
Out> 210;
See Also:
! .


Sum -- Sum of a list of values

Standard math library
Calling sequence:
Sum(list)
Sum(var, from, to, body)
Parameters:
list - list of values to sum
var - variable to iterate over
from - integer value to iterate from
to - integer value to iterate upto
body - expression to evaluate for each iteration
Description:
The first form of the Sum command simply adds all the entries in "list" and returns their sum.

If the second calling sequence is used, the expression "body" is evaluated while the variable "var" ranges over all integers from "from" upto "to", and the sum of all the results is returned. Obviously, "to" should be greater than or equal to "from".
Examples:
In> Sum({1,4,9});
Out> 14;
In> Sum(i, 1, 3, i^2);
Out> 14;
See Also:
Average , Factorize , Apply .


Average -- Average of a list of values

Standard math library
Calling sequence:
Average(list)
Parameters:
list - list of values to average
Description:
This command calculates the (arithmetical) average of all the entries in "list", which is the sum of all entries divided by the number of entries.
Examples:
In> Average({1,2,3,4,5});
Out> 3;
In> Average({2,6,7});
Out> 5;
See Also:
Sum .


Factorize -- Product of a list of values

Standard math library
Calling sequence:
Factorize(list)
Factorize(var, from, to, body)
Parameters:
list - list of values to multiply
var - variable to iterate over
from - integer value to iterate from
to - integer value to iterate upto
body - expression to evaluate for each iteration
Description:
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" upto "to", and the product of all the results is returned. Obviously, "to" should be greater than or equal to "from".
Examples:
In> Factorize({1,2,3,4});
Out> 24;
In> Factorize(i, 1, 4, i);
Out> 24;
See Also:
Sum , Apply .


Min -- Minimum of a number of values

Standard math library
Calling sequence:
Min(x,y)
Min(list)
Parameters:
x, y - pair of values to determine the minimum of
list - list of values from which the minimum is sought
Description:
This function returns the minimum value of its argument(s). If the first calling sequence is used, the smaller of "x" and "y" is returned. If one uses the second form, the smallest of the entries in "list" is returned. In both cases, this function can only be used with numerical values and not with symbolic arguments.
Examples:
In> Min(2,3);
Out> 2;
In> Min({5,8,4});
Out> 4;
See Also:
Max , Sum , Average .


Max -- Maximum of a number of values

Standard math library
Calling sequence:
Max(x,y)
Max(list)
Parameters:
x, y - pair of values to determine the maximum of
list - list of values from which the maximum is sought
Description:
This function returns the maximum value of its argument(s). If the first calling sequence is used, the larger of "x" and "y" is returned. If one uses the second form, the largest of the entries in "list" is returned. In both cases, this function can only be used with numerical values and not with symbolic arguments.
Examples:
In> Max(2,3);
Out> 3;
In> Max({5,8,4});
Out> 8;
See Also:
Min , Sum , Average .


IsZero -- Test whether argument is zero

Standard math library
Calling Sequence:
IsZero(n)
Parameters:
n - number to test
Description:
IsZero(n) evaluates to True if "n" is zero. In case "n" is not a number, the function returns False.
Examples:
In> IsZero(3.25)
Out> False;
In> IsZero(0)
Out> True;
In> IsZero(x)
Out> False;
See Also:
IsNumber , IsNotZero .


IsRational -- Test whether argument is a rational

Standard math library
Calling sequence:
IsRational(expr)
Parameters:
expr - expression to test
Description:
This commands tests whether the expression "expr" is a rational number. This is the case if the top-level operator of "expr" is /.
Examples:
In> IsRational(5)
Out> False;
In> IsRational(2/7)
Out> True;
In> IsRational(a/b)
Out> True;
In> IsRational(x + 1/x)
Out> False;
See Also:
Numer , Denom .


Numer -- Numerator of an expression

Standard math library
Calling sequence:
Numer(expr)
Parameters:
expr - expression to determine numerator of
Description:
This function determines the numerator of the rational expression "expr" and returns it. As a special case, if its argument is numeric but not rational, it returns this number. If "expr" is neither rational nor numeric, the function returns unevaluated.
Examples:
In> Numer(2/7)
Out> 2;
In> Numer(a / x^2)
Out> a;
In> Numer(5)
Out> 5;
See Also:
Denom , IsRational , IsNumber .


Denom -- Denominator of an expression

Standard math library
Calling sequence:
Denom(expr)
Parameters:
expr - expression to determine denominator of
Description:
This function determines the denominator of the rational expression "expr" and returns it. As a special case, if its argument is numeric but not rational, it returns 1. If "expr" is neither rational nor numeric, the function returns unevaluated.
Examples:
In> Denom(2/7)
Out> 7;
In> Denom(a / x^2)
Out> x^2;
In> Denom(5)
Out> 1;
See Also:
Numer , IsRational , IsNumber .


Commutator -- Commutator of two objects

Standard math library
Calling sequence:
Commutator(a, b)
Parameters:
a, b - two objects whose commutator should be computed
Description:
This command computes the commutator of 'a" and "b", ie. the expression "a b - b a". For numbers and other objects for which multiplication is commutative, the commutator is zero. But this is not necessarily the case for matrices.
Examples:
In> Commutator(2,3)
Out> 0;
In> PrettyPrinter("PrettyForm");

True

Out> 
In> A := { {0,x}, {0,0} }

/              \
| ( 0 ) ( x )  |
|              |
| ( 0 ) ( 0 )  |
\              /

Out> 
In> B := { {y,0}, {0,z} }

/              \
| ( y ) ( 0 )  |
|              |
| ( 0 ) ( z )  |
\              /

Out> 
In> Commutator(A,B)

/                          \
| ( 0 ) ( x * z - y * x )  |
|                          |
| ( 0 ) ( 0 )              |
\                          /

Out> 


Taylor -- Univariate Taylor series expansion

Standard math library
Calling Sequence:
Taylor(var, at, order) expr
Parameters:
var - variable
at - point to get Taylor series around
order - order of approximation
expr - expression to get Taylor series for
Description:
This function returns the Taylor series expansion of the expression "expr" with respect to the variable "var" around "at" upto 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".
Examples:
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 -- Taylor expansion of inverse

Standard math library
Calling Sequence:
InverseTaylor(var, at, order) expr
Parameters:
var - variable
at - point to get inverse Taylor series around
order - order of approximation
expr - expression to get inverse Taylor series for
Description:
This function builds the Taylor series expansion of the inverse of the expression "expr" with respect to the variable "var" around "at" upto order "order". It uses the function ReversePoly to perform the task.
Examples:
In> PrettyPrinter("PrettyForm")

True

Out> 
In> exp1 := Taylor(x,0,7) Sin(x)

     3    5      7 
    x    x      x  
x - -- + --- - ----
    6    120   5040

Out> 
In> exp2 := InverseTaylor(x,0,7) ArcSin(x)

 5      7     3    
x      x     x     
--- - ---- - -- + x
120   5040   6     

Out> 
In> Simplify(exp1-exp2)

0

Out> 
See Also:
ReversePoly , Taylor , BigOh .


ReversePoly -- Solve "h(f(x)) = g(x) + O(x^n)" for h

Standard math library
Calling Sequence:
ReversePoly(f, g, var, newvar, degree)
Parameters:
f, g - expressions in "var"
var - a variable
newvar - a new variable to express the result in
degree - the degree of the required solution
Description:
This function returns a polynomial in "newvar", say "h(newvar)", with the property that "h(f(var))" equals "g(var)" upto order "degree". The degree of the result will be at most "degree-1". 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" (upto order "degree"), so "h" will be the inverse of "f".
Examples:
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*(x-1)^7)/131072+(791*(x-1)^6)/32768+(-119*(x-1)^5)/4096+(37*(x-1)^4)
/1024+(-3*(x-1)^3)/64+(x-1)^2/16;
See Also:
InverseTaylor , Taylor , BigOh .


BigOh -- Drop all terms of a certain order in a polynomial

Standard math library
Calling Sequence:
BigOh(poly, var, degree)
Parameters:
poly - a univariate polynomial
var - a free variable
degree - positive integer
Description:
This function drops all terms of order "degree" or higher in "poly", which is a polynomial in the variable "var".
Examples:
In> BigOh(1+x+x^2+x^3,x,2)
Out> x+1;
See Also:
Taylor , InverseTaylor .


Newton -- Solve an equation numerically with Newton's method

Standard math library
Calling Sequence:
Newton(expr, var, initial, accuracy)
Parameters:
expr - an expression to find a zero for
var - free variable to adjust to find a zero
initial - initial value for "var" to use in the search
accuracy - minimum required accuracy of the result
Description:
This function tries to numerically find a zero of the expression "expr", which should depend only on the variable "var". It uses the value "initial" as an initial guess.

The function will iterate using Newton's method until it estimates that it has come within a distance "accuracy" of the correct solution, and then it will return its best guess. In particular, it may loop forever if the algorithm does not converge.
Examples:
In> Newton(Sin(x),x,3,0.0001)
Out> 3.1415926535;
See Also:
Solve .


D -- Differentiation

Standard math library
Calling Sequence:
D(var) expr
D(list) expr
D(var,n) expr
Parameters:
var - variable
list - a list of variables
expr - expression to take derivative of
n - order of derivative
Description:
This function calculates the derivative of the expression "expr" with respect to the variable "var" and returns it. If the third calling sequence is used, the "n"-th derivative is determined. Yacas knows how the differentiate standard functions like 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.
Examples:
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 -- Curl of a vector field

Standard math library
Calling sequence:
Curl(vector, basis)
Parameters:
vector - vector field to take the curl of
basis - list of variables forming the basis
Description:
This function takes the curl of the vector field "vector" with respect to the variables "basis". The curl is defined as

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] }

Both "vector" and "basis" should be lists of length 3.
Examples:
In> Curl({x*y,x*y,x*y},{x,y,z})
Out> {x,-y,y-x};
See Also:
D , Diverge .


Diverge -- Divergence of a vector field

Standard math library
Calling sequence:
Diverge(vector, basis)
Parameters:
vector - vector field to calculate the divergence of
basis - list of variables forming the basis
Description:
This function calculates the divergence of the vector field "vector" with respect to the variables "basis". The divergence is defined as

Diverge(f,x) = D(x[1]) f[1] + ... + D(x[n]) f[n],

where n is the length of the lists "vector" and "basis". These lists should have equal length.
Examples:
In> Diverge({x*y,x*y,x*y},{x,y,z})
Out> y+x;
See Also:
D , Curl .


Integrate -- Integration

Standard math library
Calling Sequence:
Integrate(var, from, to) expr
Integrate(var) expr
Parameters:
var - variable to integrate over
from - begin of interval to integrate over
to - end of interval to integrate over
expression - expression to integrate
Description:
This function integrates the expression "expr" with respect to the variable "var". The first calling sequence is used to perform definite integration: the integration is carried out from "var=form" to "var=to". The second form signifies indefinite integration. In this case, the function UniqueConstant is called to get a variable of the form Cn (where "n" is an integer) which represent the integration constant.

Some simple integration rules have currently been implemented. Polynomials, quotients of polynomials, the transcendental functions Sin, Cos, Exp, and Ln, and products of these functions with polynomials can all be integrated.
Examples:
In> Integrate(x,a,b) Cos(x)
Out> Sin(b)-Sin(a);
In> Integrate(x) Cos(x)
Out> Sin(x)+C9;
See Also:
D , UniqueConstant .


Simplify -- Try to simplify an expression

Standard math library
Calling Sequence:
Simplify(expr)
Parameters:
expression - expression to simplify
Description:
This function tries to simplify the expression "expr" as much as possible. It does this by grouping powers within terms, and then grouping like terms.
Examples:
In> a*b*a^2/b-a^3
Out> (b*a^3)/b-a^3;
In> Simplify(a*b*a^2/b-a^3)
Out> 0;
See Also:
TrigSimpCombine , RadSimp .


RadSimp -- Simplify expression with nested radicals

Standard math library
Calling Sequence:
RadSimp(expr)
Parameters:
expr - an expression containing nested radicals
Description:
This function tries to write the expression "expr" as a sum of roots of integers: "Sqrt(e1) + Sqrt(e2) + ...", where "e1", "e2" and so on are natural numbers. The expression "expr" may not contain free variables.

It does this by trying all possible combinations for "e1", "e2", etcetera. Every possibility is numerically evaluated using N and compared with the numerical evaluation of "expr". If the approximations are equal (upto a certain margin), this possibility is returned. Otherwise, the expression is returned unevaluated.

Note that due to the use of numerical approximations, there is a small chance that the expression returned by RadSimp is close but not equal to "expr". The last example underneath illustrates this problem. Furthermore, if the numerical value of "expr" is large, the number of possibilities becomes exorbitantly big so the evaluation may take very long.
Examples:
In> RadSimp(Sqrt(9+4*Sqrt(2)))
Out> Sqrt(8)+1;
In> RadSimp(Sqrt(5+2*Sqrt(6))+Sqrt(5-2*Sqrt(6)))
Out> Sqrt(12);
In> RadSimp(Sqrt(14+3*Sqrt(3+2*Sqrt(5-12*Sqrt(3-2*Sqrt(2)))))) 
Out> Sqrt(2)+3;

But this command may yield incorrect results:
In> RadSimp(Sqrt(1+10^(-6)))
Out> 1;
See Also:
Simplify , N .


Rationalize -- Convert floating point numbers to fractions

Standard math library
Calling Sequence:
Rationalize(expr)
Parameters:
expr - an expression containing real numbers
Description:
This command converts every real number in the expression "expr" into a rational number. This is useful when a calculation needs to be done on floating point numbers and the algorithm is unstable. Converting the floating point numbers to rational numbers will force calculations to be done with infinite precision (by using rational numbers as representations).

It does this by finding the smallest integer n such that multiplying the number with 10^n is an integer. Then it divides by 10^n again, depending on the internal gcd calculation to reduce the resulting division of integers.
Examples:
In> {1.2,3.123,4.5}
Out> {1.2,3.123,4.5};
In> Rationalize(%)
Out> {6/5,3123/1000,9/2};
See Also:
IsRational .


Solve -- Solve one or more algebraic equations

Standard math library
Calling Sequence:
Solve(eq, var)
Solve(eqlist, varlist)
Parameters:
eq - single identity equation
var - single variable
eqlist - list of identity equations
varlist - list of variables
Description:
This command tries to solve one or more equations. Use the first form to solve a single equation and the second one for systems of equations.

The first calling sequence solves the equation "eq" for the variable "var". Use the == operator to form the equation. The value of "var" which satisfies the equation, is returned. Note that only one solution is found and returned.

To solve a system of equations, the second form should be used. It solves the system of equations contained in the list "eqlist" for the variables appearing in the list "varlist". A list of results is returned, and each result is a list containing the values of the variables in "varlist". Again, at most a single solution is returned.

The task of solving a single equation is simply delegated to SuchThat. Multiple equations are solved recursively: firstly, an equation is sought in which one of the variables occurs exactly once; then this equation is solved with SuchThat; and finally the solution is substituted in the other equations by Eliminate decreasing the number of equations by one. This suffices for all linear equations and a large group of simple nonlinear equations.
Examples:
In> Solve(a+x*y==z,x)
Out> (z-a)/y;
In> Solve({a*x+y==0,x+z==0},{x,y})
Out> {{-z,z*a}};
This means that "x = (z-a)/y" is a solution of the first equation and that "x = -z", "y = z*a" is a solution of the systems of equations in the second command.

An example which Solve cannot solve:
In> Solve({x^2-x == y^2-y, x^2-x == y^3+y}, {x,y});
Out> {};
See Also:
SuchThat , Eliminate , PSolve , == .


SuchThat -- Find a value which makes some expression zero

Standard math library
Calling Sequence:
SuchThat(expr, var)
Parameters:
expr - expression to make zero
var - variable (or subexpression) to solve for
Description:
This functions tries to find a value of the variable "var" which makes the expression "expr" zero. It is also possible to pass a subexpression as "var", in which case SuchThat will try to eliminate for that subexpression.

Basically, only expressions in which "var" occurs only once are handled; and in fact, SuchThat may even give wrong results if the variables occurs more than once. This is a consequence of the implementation, which repeatedly applies the inverse of the top function until the variable "var" is reached.
Examples:
In> SuchThat(a+b*x, x)
Out> (-a)/b;
In> SuchThat(Cos(a)+Cos(b)^2, Cos(b))
Out> Cos(a)^(1/2);
In> Expand(a*x+b*x+c, x)
Out> (a+b)*x+c;
In> SuchThat(%, x)
Out> (-c)/(a+b);
See Also:
Solve , Subst , Simplify .


Eliminate -- Substitute and simplify

Standard math library
Calling Sequence:
Eliminate(var, value, expr)
Parameters:
var - variable (or subexpression) to substitute
value - new value of "var"
expr - expression in which the substitution should take place
Description:
This function uses Subst to replace all instances of the variable (or subexpression) "var" in the expression "expr" with "value", calls Simplify to simplify the resulting expression, and returns the result.
Examples:
In> Subst(Cos(b), c) (Sin(a)+Cos(b)^2/c)
Out> Sin(a)+c^2/c;
In> Eliminate(Cos(b), c, Sin(a)+Cos(b)^2/c)
Out> Sin(a)+c;
See Also:
SuchThat , Subst , Simplify .


PSolve -- Solve a polynomial equation

Standard math library
Calling Sequence:
PSolve(poly, var)
Parameters:
poly - a polynomial in "var"
var - a variable
Description:
This commands returns a list containing the roots of "poly", considered as a polynomial in the variable "var". If there is only one root, it is not returned as a one-entry list but just by itself. A double root occurs twice in the result, and similarly for roots of higher multiplicity. All polynomials of degree upto 4 are handled.
Examples:
In> PSolve(b*x+a,x)
Out> -a/b;
In> PSolve(c*x^2+b*x+a,x)
Out> {(Sqrt(b^2-4*c*a)-b)/(2*c),(-(b+Sqrt(b^2-4*c*a)))/(2*c)};
See Also:
Solve , Factor .


Pi -- Numerical approximation of pi

Internal function
Calling Sequence:
Pi()
Parameters:
none
Description:
This commands returns the mathematical constant pi to the current precision, as set by Precision. Usually this function will not be called directly. The constant Pi can (and should) be used to represent pi, as it is recognized by the simplification rules. Then when the function N is invoked, Pi will be replaced with the value returned by Pi().
Examples:
In> Pi()
Out> 3.14159265358979323846;
In> Precision(40)
Out> True;
In> Pi()
Out> 3.1415926535897932384626433832795028841971;
See Also:
N , Pi , Precision .


Random -- Random number between 0 and 1

Standard math library
Calling Sequence:
Random()
Parameters:
none
Description:
This function returns a random number, uniformly distributed in the interval between 0 and 1. The same sequence of random numbers is generated in every Yacas session.
See Also:
RandomInteger , RandomPoly .


VarList -- List of variables appearing in some expression

Standard math library
Calling Sequence:
VarList(expr)
Parameters:
expr - an expression
Description:
This command returns a list of all the variables that appear in the expression "expr".
Examples:
In> VarList(Sin(x))
Out> {x};
In> VarList(x+a*y)
Out> {x,a,y};
See Also:
IsFreeOf , IsVariable .


Limit -- Limit of an expression

Standard math library
Calling Sequence:
Limit(var, val) expr
Limit(var, val, dir) expr
Parameters:
var - a variable
val - a number
dir - a direction (Left or Right)
expr - an expression
Description:
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.
Examples:
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;


TrigSimpCombine -- Combine products of trigonometric functions

Standard math library
Calling Sequence:
TrigSimpCombine(expr)
Parameters:
expr - expression to simplify
Description:
This function applies the product rules of trigonometry, like Cos(u)*Sin(v) = (Sin(v-u) + Sin(v+u)) / 2. As a result, all products of the trigonometric functions Cos and Sin disappear. The function also tries to simplify the resulting expression as much as possible by combining all like terms.

This function is used in for instance Integrate, to bring down the expression into a simpler form that hopefully can be integrated easily.
Examples:
In> PrettyPrinter("PrettyForm");

True

Out> 
In> TrigSimpCombine(Cos(a)^2+Sin(a)^2)

1

Out> 
In> TrigSimpCombine(Cos(a)^2-Sin(a)^2)

Cos( -2 * a )

Out> 
In> TrigSimpCombine(Cos(a)^2*Sin(b))

Sin( b )   Sin( -2 * a + b )   Sin( -2 * a - b )
-------- + ----------------- - -----------------
   2               4                   4        

Out> 
See Also:
Simplify , Integrate , Expand , Sin, Cos, Tan .


LagrangeInterpolant -- Polynomial interpolation

Standard math library
Calling Sequence:
LagrangeInterpolant(xlist, ylist, var)
Parameters:
xlist - list of argument values
ylist - list of function values
var - free variable for resulting polynomial
Description:
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.
Examples:
In> f := LagrangeInterpolant({0,1,2}, {0,1,1}, x);
Out> (x*(x-1))/2-x*(x-2);
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("PrettyForm");

True

Out> 
In> LagrangeInterpolant({x1,x2,x3}, {y1,y2,y3}, x)

y1 * ( x - x2 ) * ( x - x3 )   y2 * ( x - x1 ) * ( x - x3 ) 
---------------------------- + ---------------------------- 
 ( x1 - x2 ) * ( x1 - x3 )      ( x2 - x1 ) * ( x2 - x3 )   

  y3 * ( x - x1 ) * ( x - x2 )
+ ----------------------------
   ( x3 - x1 ) * ( x3 - x2 )  

Out> 
See Also:
Subst .


Fibonacci -- Fibonacci sequence

Standard math library
Calling Sequence:
Fibonacci(n)
Parameters:
n - an integer
Description:
This command calculates and returns the "n"-th Fibonacci number.

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, ..., where every number is the sum of the two preceding numbers. Formally, it is defined by F(1) = 1, F(2) = 1, and F(n+1) = F(n) + F(n-1), where F(n) denotes the n-th Fibonacci number.
Examples:
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};