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The following examples can be pasted into the Edit Script window.

Index

abs(x)

Returns the absolute value (vector length, magnitude) of x. 
Enter

     abs((a,b,c))

Result

                   1/2
       2    2    2
     (a  + b  + c )

adj(m)

Returns the adjunct of matrix m. 
Enter

     A = ((a,b),(c,d))

     adj(A)

Result

     d     -b
 
     -c    a
The inverse of a matrix is equal to the adjunct divided by the determinant. 
Enter

     inv(A) - adj(A) / det(A)

Result

     0

and(a,b,...)

Logical-and of predicate expressions. 
Example

     and(A = B, B = C)

arccos(x)

Returns the inverse cosine of x.

arccosh(x)

Returns the inverse hyperbolic cosine of x.

arcsin(x)

Returns the inverse sine of x.

arcsinh(x)

Returns the inverse hyperbolic sine of x.

arctan(x)

Returns the inverse tangent of x.

arctanh(x)

Returns the inverse hyperbolic tangent of x.

arg(z)

Returns the argument (angle) of complex z. Symbols in expression z are presumed to be positive and real. 
Enter
	arg(1 + exp(i pi/3))

Result
	1/6 pi

binomial(n,k)

Returns the binomial coefficient. 
Enter

     binomial(10,5)

Result

     252

break(x)

Causes an immediate return from a for function. Expression x is evaluated and returned as the for function value. A break with no argument returns the symbol nil. A break can be evaluated at any function level. For example, for() can evaluate f() which evaluates g() which evaluates break().

ceiling(x)

Returns the smallest integer not less than x.

check(x)

If x is true then continue, else stop. 
Example

    check(A = B)

coeff(p,x,n)

Returns the coefficient of x^n in polynomial p. The x argument can be omitted for polynomials in x.

cofactor(m,i,j)

Returns the cofactor of m for row i and column j.

condense(x)

Attempts to simplify x by factoring common terms. 
Enter

     2 a (x + 1)

Result

     2 a + 2 a x

Enter

     condense(last)

Result

     2 a (x + 1)

conj(z)

Returns the complex conjugate of z. 
Enter

     conj(3 + 4 i)

Result

     3 - 4 i

contract(a,i,j)

Returns the contraction of tensor a across indices i and j. If i and j are omitted then indices 1 and 2 are used. The following example shows how contract adds diagonal elements. 
Enter

     A = ((a,b),(c,d))

     contract(A,1,2)

Result

     a + d

cos(x)

Returns the cosine of x.

cosh(x)

Returns the hyperbolic cosine of x.

d(f,x)

Returns the partial derivative of f with respect to x. The second argument can be omitted in which case the computer will guess which symbol to use. Returns the gradient of f when x is a vector. Note that gradient raises the rank of f by 1. 
Enter
	d(x^2,x)

Result
	2 x
For tensor f the derivative of each element is computed. 
Enter

     d((x,x^2),x)

Result

      1
 
     2 x
Functions with no arguments are treated as dependent on any variable. 
Enter

     d(f(),(x,y))

Result

     d(f(),x)
 
     d(f(),y)
Since partial derivatives commute, multi-derivatives are ordered to produce a canonical form. 
Enter

     d(d(f(),y),x)

Result

     d(d(f(),x),y)
The following table shows the various forms that can be used to compute multiderivatives. 
	Form		Equivalent form
	d(f,2)		d(d(f,x),x)
	d(f,x,2)	d(d(f,x),x)
	d(f,x,x)	d(d(f,x),x)
	d(f,x,2,y,2)	d(d(d(d(f,x),x),y),y)
	d(f,x,x,y,y)	d(d(d(d(f,x),x),y),y)

deg(p,x)

Returns the degree of polynomial p in x. The x argument can be omitted. If omitted, the computer will guess x from the contents of p.

denominator(x)

Returns the denominator of expression x. 
Enter

     denominator(a/b)

Result

	b

det(m)

Returns the determinant of matrix m. 
Enter

     A = ((a,b),(c,d))

     det(A)

Result

     a d - b c

dim(a,i)

Returns the dimension of the ith index of tensor a. If i is omitted then the dimension of the first index is returned. 
Enter

     A = (1,2,3,4)

     dim(A)

Result

     4

Enter

     A = ((1,2,3),(4,5,6))

     dim(A,1)

Result

     2

Enter

     dim(A,2)

Result

     3

display(x)

Evaluates expression x and displays the result using Times and Symbol fonts. User symbols are scanned for the keywords shown below. Each keyword is replaced with its associated Greek letter glyph. Multiglyph symbols are displayed using subscripts. This function can be selected as the default display mode by setting tty = 0.

GammaΓ alphaα muμ
DeltaΔ betaβ nuν
ThetaΘ gammaγ xiξ
LambdaΛ deltaδ piπ
XiΞ epsilonε rhoρ
PiΠ zetaζ sigmaσ
SigmaΣ etaη tauτ
UpsilonΥ thetaθ upsilonυ
PhiΦ iotaι phiφ
PsiΨ kappaκ chiχ
OmegaΩ lambdaλ psiψ
omegaω

do(a,b,...)

Evaluates each argument from left to right. Returns the value of the last argument.

dot(a,b,...)

Returns the dot product of tensors (aka inner product). 
Enter

     A = (A1,A2,A3)

     B = (B1,B2,B3)

     dot(A,B)

Result

     A1 B1 + A2 B2 + A3 B3
The dot product is equivalent to an outer product followed by a contraction across the inner indices. 
Enter

     A = hilbert(10)

     dot(A,A) - contract(outer(A,A),2,3)

Result

     0

draw(f,x)

Draws a graph of f. The second argument can be omitted in which case the computer will guess what variable to use. Parametric drawing occurs when f returns a vector. Ranges are set with xrange and yrange. The defaults are xrange = (-10,10) and yrange = (-10,10). The parametric variable range is set with trange. The default is trange = (-pi,pi).

Example 1. 

Enter
	draw(5(cos(t),sin(t)))

Example 2. 

Enter
	draw(5(cos(3t),sin(5t)))

eigen(m)

eigenval(m)

eigenvec(m)

These functions compute eigenvalues and eigenvectors numerically. Matrix m must be both numerical and symmetric. The eigenval function returns a matrix with the eigenvalues along the diagonal. The eigenvec function returns a matrix with the eigenvectors arranged as row vectors. The eigen function does not return anything but stores the eigenvalue matrix in D and the eigenvector matrix in Q.

Example 1. Check the relation AX = lambda X where lambda is an eigenvalue and X is the associated eigenvector. 

Enter

     A = hilbert(3)

     eigen(A)

     lambda = D[1,1]

     X = Q[1]

     dot(A,X) - lambda X

Result

     -1.16435e-14
 
     -6.46705e-15
 
     -4.55191e-15

Example 2: Check the relation A = QTDQ. 

Enter

	A - dot(transpose(Q),D,Q)

Result
 
	6.27365e-12    -1.58236e-11   1.81902e-11
 
	-1.58236e-11   -1.95365e-11   2.56514e-12
 
	1.81902e-11    2.56514e-12    1.32627e-11

erf(x)

Error function of x. 
Enter
	draw(erf(x))

erfc(x)

Complementary error function of x. 
Enter
	draw(erfc(x))

eval(x)

Returns the evaluation of expression x. 
Enter

     A = quote(sin(pi/6))

     A

Result

          1
     sin(--- pi)
          6

Enter

     eval(A)

Result

      1
     ---
      2

exp(x)

Returns the exponential of x. The expression exp(1) should be used to represent the natural number e. 
Enter

     exp(1.0)

Result

     2.71828

Enter

     exp(a) exp(b)

Result

     exp(a + b)

expcos(x)

Returns the exponential cosine of x. 
Enter

     expcos(x)

Result

      1               1
     --- exp(-i x) + --- exp(i x)
      2               2

expsin(x)

Returns the exponential sine of x. 
Enter

     expsin(x)

Result

      1                 1
     --- i exp(-i x) - --- i exp(i x)
      2                 2

factor(n)

factor(p,x)

The first form returns the prime factors for integer n. 
Enter

     factor(12345)

Result

     3 5 823
The second form factors polynomial p in x. The argument x can be omitted in which case the computer will guess which symbol to use. 
Enter

     factor(x^3 + x^2 + x + 1)

Result

                   2
     (1 + x) (1 + x )

factorial(x)

Returns the factorial of x. The syntax x! can also be used. 
Enter

     100!

Result

     93326215443944152681699238856266700490715968264381621468592963895217599993229915
     608941463976156518286253697920827223758251185210916864000000000000000000000000

filter(f,a,b,...)

Returns f with terms containing a (or b or...) removed. Useful for implementing a "poor man's" Fourier transform. 
Enter

     Y = A exp(-i k x) + B exp(i k x)

     filter(Y exp(-i k x), x)

Result

     B

float(x)

Converts rational numbers and integers in x to floating point values. The symbol pi is also converted. 
Enter

     float(100!)

Result

     9.33262e+157

floor(x)

Returns the largest integer not greater than x.

for(a,i,j,b)

For a equals i through j evaluate b. Normally for() returns the symbol nil. A break() function can be used to return a different value. The variable a has local scope within the for function. The variable a remains unmodified after for returns. The expressions i and j must evaluate to integers. Usually b is a do() function. 
Enter

     for(k,1,4,print(1/k,tab(10),1/k^2))

Result
 
     1         1
 
      1         1
     ---       ---
      2         4
 
      1         1
     ---       ---
      3         9
 
      1         1
     ---       ----
      4         16

gcd(a,b,...)

Returns the greatest common divisor.

hermite(x,n)

Returns the nth Hermite polynomial in x. 
Enter

     hermite(x,3)

Result

                3
     -12 x + 8 x

hilbert(n)

Returns a Hilbert matrix of order n. 
Enter

     hilbert(3)

Result

             1      1
      1     ---    ---
             2      3
 
      1      1      1
     ---    ---    ---
      2      3      4
 
      1      1      1
     ---    ---    ---
      3      4      5

imag(z)

Returns the imaginary part of complex z. Symbols in expression z are presumed to be positive and real. 
Enter
	imag(1 + exp(i pi/3))

Result
	1/2 3^(1/2)

inner(a,b,...)

Returns the inner product of tensors. This is the same function as the dot product. 
Enter

     A = (A1,A2,A3)

     B = (B1,B2,B3)

     inner(A,B)

Result 

     A1 B1 + A2 B2 + A3 B3

integral(f,x)

Returns the integral of f with respect to x. The second argument can be omitted in which case the computer will guess which symbol to use. 
Enter

     integral(log(x),x)

Result

     -x + x log(x)

inv(m)

Returns the inverse of matrix m. 
Enter

     A = ((a,b),(c,d))

     inv(A)

Result

           d                 b
      -----------     - -----------
       a d - b c         a d - b c
 
            c               a
     - -----------     -----------
        a d - b c       a d - b c

isprime(n)

Returns 1 if integer n is a prime number. Returns 0 if n is not a prime number. 
Enter

     isprime(9007199254740991)

Result

     0

Enter

     isprime(2^53 - 111)

Result

     1

laguerre(x,n,a)

Returns the nth Laguerre polynomial in x. If the argument a is omitted or a equals zero then the function returns the non-associated Laguerre polynomial. 
Enter

     laguerre(x,2)

Result

                1   2
     1 - 2 x + --- x
                2

Enter

     laguerre(x,2,a)

Result

          3                   1   2    1   2
     1 + --- a - 2 x - a x + --- a  + --- x
          2                   2        2

lcm(a,b,...)

Returns the least common multiple. 
Enter

     lcm(2,3,4,x)

Result

     12 x

legendre(x,n,m)

Returns the nth Legendre polynomial in x. 
Enter

     legendre(x,2)

Result

        1     3   2
     - --- + --- x
        2     2

Enter

     legendre(x,2,0)

Result

        1     3   2
     - --- + --- x
        2     2

Enter

     legendre(x,2,1)

Result

                  1/2
                2
     -3 x (1 - x )

log(x)

Returns the natural logarithm of x. 
Enter

     log(-10.0)

Result

     2.30259 + i π

mag(z)

Returns the magnitude of complex z. Symbols in expression z are presumed to be positive and real. 
Enter
	mag(1 + exp(i pi/3))

Result
	3^(1/2)

not(x)

Negates a predicate expression. 
Example

     not(A = B)

mod(a,b)

Returns the remainder of a divided by b.

numerator(x)

Returns the numerator of expression x. 
Enter

     numerator(a/b)

Result

	a

or(a,b,...)

Logical-or of predicate expressions. 
Example

     or(A = B, A = C)

outer(a,b,...)

Returns the outer product of tensors (aka tensor product). 
Enter

     A = (A1,A2,A3)

     B = (B1,B2,B3)

     outer(A,B)

Result

     A1 B1    A1 B2    A1 B3

     A2 B1    A2 B2    A2 B3

     A3 B1    A3 B2    A3 B3

polar(z)

Converts complex z to polar form. 
Enter

     polar(1 + exp(i pi/3))

Result

         1/6  1/2
     (-1)    3

prime(n)

Returns the nth prime number, n ≤ 10000. 
Enter

     prime(10000)

Result

     104729

print(a,b,...)

Print args in tty mode.

product(i,j,k,y)

For i equals j through k evaluate y. Returns the product of all y. 
Enter

     product(k,1,3,1/(1-(1/prime(k)^s)))

Result

                     1
     ----------------------------------
            1          1          1
      (1 - ----) (1 - ----) (1 - ----)
             s          s          s
            2          3          5

prog(a,b,...,f)

The prog function evaluates f. The result of the evaluation of f is returned as the prog value. The variables a,b,... have local scope within prog. The variables a,b,... remain unmodified after prog returns. Usually f is a do function.

quote(x)

Returns expression x without evaluating symbols or functions. Can be used to clear symbolic values. 
Enter

     n = 3

     n

Result

     3

Enter

     n = quote(n)

     n

Result

     n

quotient(p,q,x)

Returns the quotient of polynomial p(x) over q(x). The last argument can be omitted when the polynomials are in x. The remainder can be calculated as p - q * quotient(p,q).

rank(a)

Returns the rank (number of indices) of tensor a. 
Enter

     U = (u1,u2,u3,u4)

     rank(U)

Result

     1

rationalize(x)

Puts terms in x over a common denominator. 
Enter

     rationalize(1/x + 1/y)

Result

      x + y
     -------
       x y
Rationalize can often simplify expressions. 
Enter

     A = ((a,b),(c,d))

     B = inv(A)

     dot(A,B)

Result

         a d           b c
     ----------- - -----------                0
      a d - b c     a d - b c
 
                                      a d           b c
                 0                ----------- - -----------
                                   a d - b c     a d - b c

Enter

     rationalize(last)

Result

     1    0
 
     0    1

real(z)

Returns the real part of complex z. Symbols in expression z are presumed to be positive and real. 
Enter
	real(1 + exp(i pi/3))

Result
	3/2

rect(z)

Returns complex z in rectangular form. Symbols in expression z are presumed to be positive and real. 
Enter
	rect(1 + exp(i pi/3))

Result
	3/2 + 1/2 i 3^(1/2)

return(x)

Evaluation of return causes an immediate exit from prog. Expression x is evaluated and returned as the prog value. A return with no argument returns the symbol nil. A return can be evaluated at any function level. For example, prog() can evaluate f() which evaluates g() which evaluates return().

roots(p,x)

Finds the values of x for which the polynomial p(x) = 0. Returns a vector for multiple roots. 
Enter

     (x + 1)(x + 2)

Result

      2
     x  + 3 x + 2

Enter

     roots

Result

     -2

     -1

simplify(x)

Evaluates expression x and then attempts to simplify the result. 
Enter

     (A-B)/(B-A)

Result

        A          B
     -------- - --------
      -A + B     -A + B
 
Enter

     simplify(last)

Result

     -1

Enter

     A = ((A11,A12),(A21,A22))

     det(A) inv(A) - adj(A)

Result

     ((-A22 + A11 A22^2 / (A11 A22 - A12 A21) - A12 A21 A22 / (A11 A22 - A12 A21),
     A12 - A11 A12 A22 / (A11 A22 - A12 A21) + A12^2 A21 / (A11 A22 - A12 A21)),
     (A21 - A11 A21 A22 / (A11 A22 - A12 A21) + A12 A21^2 / (A11 A22 - A12 A21),
     -A11 - A11 A12 A21 / (A11 A22 - A12 A21) + A11^2 A22 / (A11 A22 - A12 A21)))

Enter

     simplify(last)

Result

     0
The simplify function can return factored (unexpanded) expressions. Factored expressions can fail in tests for equality. The eval function can be used to expand factored expressions.

sin(x)

Returns the sine of x.

sinh(x)

Returns the hyperbolic sine of x.

sqrt(x)

Returns the square root of x.

stop()

When evaluated in a script, the stop function exits run mode and returns to interactive mode.

subst(a,b,c)

Substitutes a for b in c and returns the result. Note that this operation can return a denormalized expression. Use eval to normalize the result of subst. 
Enter

     f = x^2

     subst(sqrt(x),x,f)

Result

           2
       1/2
     (x   )

Enter

     eval(last)

Result

     x

sum(a,i,j,b)

For a equals i through j evaluate b. Returns the sum of all b. The variable a has local scope within the sum function, a remains unchanged after the sum function returns. The expressions i and j should evaluate to integers. 
Enter

     sum(k,1,3,1/k^s)

Result

          1      1
     1 + ---- + ----
           s      s
          2      3

tan(x)

Returns the tangent of x.

tanh(x)

Returns the hyperbolic tangent of x.

taylor(f,x,n,a)

Returns the Taylor expansion of f at a. If a is omitted then zero is used. The argument x is the free variable in f and n is the power of the expansion. 
Enter

     taylor(1/cos(x),x,6)

Result

          1   2    5    4    61    6
     1 + --- x  + ---- x  + ----- x
          2        24        720

test(a,b,c,d,...)

If argument a is true then b is returned else if c is true then d is returned, etc. If the number of arguments is odd then the last argument is the default case. 
Example

     f(x) = test(x < 0, g(x), h(x))

trace(m)

Returns the trace of matrix m. 
Enter

     A = ((a,b),(c,d))

     trace(A)

Result

     a + d
Note that trace is equivalent to contract. 
Enter

     trace(A) - contract(A,1,2)

Result

     0

transpose(a,i,j)

Returns the transpose of tensor a across indices i and j. If i and j are omitted then indices 1 and 2 are used.

unit(n)

Returns a unit matrix with dimension n. 
Enter

     unit(4)

Result

     1    0    0    0

     0    1    0    0

     0    0    1    0

     0    0    0    1

zero(i,j,...)

Returns a zero tensor with dimensions i,j,... The zero function is useful for creating a tensor and then setting the component values. 
Enter

     A = zero(2,2)

     A[1,2] = a

     A

Result

     0    a

     0    0