abs		absolute value
adj		adjunct of matrix
and             logical and
arccos
arccosh
arcsin
arcsinh
arctan
arctanh
arg		argument of complex
binomial
break		break out of a loop
ceiling		ceiling function
check           stop if false
coeff           coefficient of polynomial
cofactor        cofactor of matrix
condense	find subexpressions
conj		complex conjugate
contract	contract across tensor indices
cos
cosh
d		derivative and gradient
deg             degree of polynomial
denominator	denominator of expression
det		determinant of a matrix
dim		dimension of tensor index
display		display an expression
do		evaluate multiple expressions
dot		inner product of tensors
draw		draw a graph
eigen
erf		error function
erfc		complementary error function
eval		normalize an expression
exp		exponential function
expcos		exponential cosine
expsin		exponential sine
factor		factor a number or polynomial
factorial	factorial function
filter		remove subexpressions
float		convert to floating point
floor		floor function
for		for loop
gcd		greatest common denominator
hermite		hermite polynomial
hilbert		hilbert matrix
imag		imaginary part of complex
inner		inner product of tensors
integral
inv		invert a matrix
isprime
laguerre	laguerre polynomial
lcm		least common multiple
legendre	legendre polynomial
log		natural logarithm
mag		magnitude of complex
mod		modulo function
not             logical negation
numerator	numerator of expression
or              logical or
outer		outer product of tensors
polar           convert to polar form
prime
product		multiply over an index
prog		evaluate with scoped variables
quote		do not evaluate
quotient        divide polynomials
rank		rank of tensor
rationalize	combine fractions
real		real part of complex
rect		complex to rectangular
return		early return from a function
roots		find roots of a polynomial
simfac		simplify factorials
simplify	simplify an expression
sin
sinh
sqrt		square root
stop		stop running a script
subst		substitute expressions
sum		sum over an index
tan
tanh
taylor		taylor series
test		conditional evaluation
trace		trace of matrix
transpose	transpose tensor indices
unit		unit matrix
wedge		wedge product of tensors
zero		new zero tensor
Trivia
  • Press the up and down arrow keys to choose a previously entered command.
  • To use the symbol e for the natural number, first define it as e=exp(1).
  • Most functions can be used without arguments. The previous result is used as a default argument.
  • The simplify function converts tensor zero to scalar zero.
  • A=B evaluates B and binds the result to the symbol A. On the other hand, the function definition A(X)=B does not evaluate B until A itself is evaluated.
  • Enter bake=0 to get results in raw format. Enter bake=1/2 for half-baked code :)

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 statement in sequence. Returns the value of the last statement.

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

     factorial(100)

Result

     93326215443944152681699238856266700490715968264381621468592963895217599993229915
     608941463976156518286253697920827223758251185210916864000000000000000000000000

Enter

     factorial(100) - 100!

Result

     0

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 of a and b.

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 of a and b. The least common multiple is the smallest value or expression divisible by both a and b. 
Enter

     lcm(4,6)

Result

     12

Enter

     lcm(4 x, 6 x y)

Result

     12 x y

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.

prime(n)

Returns the nth prime number. The value of n must be greater than zero and less than 10,001. 
Enter

     prime(1)

Result

     2

Enter

     prime(10000)

Result

     104729

print(a,b,...)

The print function evaluates and prints each expression. The printing is done in tty mode. The symbol nil is returned as the function value. Spaces and other text can be printed by using quoted strings for print arguments. This function can be selected as the default printing mode by setting the symbol tty = 1.

product(a,i,j,b)

For a equals i through j evaluate b. Returns the product of all b. The variable a has local scope within the product function, a remains unchanged after the product function returns. The expressions i and j should evaluate to integers. 
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) equals zero. Graphically, the roots of p are the x coordinates where the curve p(x) crosses the horizontal line. The argument p may be expressed as an equality. The argument x can be omitted if it is literally x, y, z, t or r. If there is more than one root then a vector containing the roots is returned. If p cannot be factored then roots returns the symbol nil. Note: The symbol nil is not normally printed so no result is visible when p cannot be factored. 
Enter

     (x - 1/2) (x - 1/3) (x + 1/4) / x^3

Result

            1         1        7
     1 + ------- - ------- - ------
              3         2     12 x
          24 x      24 x

Enter

     roots(last,x)

Result

        1
     - ---
        4

       1
      ---
       3

       1
      ---
       2

Enter

     roots(a x = b)

Result

      b
     ---
      a

Enter

     roots(a x^2 + b x + c)

Result

                             1/2
                           2
         b      (-4 a c + b )
     - ----- - ------------------
        2 a           2 a

                             1/2
                           2
         b      (-4 a c + b )
     - ----- + ------------------
        2 a           2 a

simfac(x)

Simplifies factorials in expression x according to the following rules. 
	This			is converted to this

	n! / n			(n - 1)!

	n (n - 1)! 		n!

	(n + 1)! / n!		n + 1

	(n + 2)! / n!		(n + 1) (n + 2)
Example 1. 
Enter

     F(n,k) = k binomial(n,k)

     (F(n,k) + F(n,k-1)) / F(n+1,k)

Result

            k! n!             n! (1 - k + n)!              k! n!
     -------------------- + -------------------- - ----------------------
      (-1 + k)! (1 + n)!     (1 + n)! (-k + n)!     k (-1 + k)! (1 + n)!

Enter

     simfac

Result

        1       1 - k + n       1
     ------- + ----------- - -------
      1 + n       1 + n       1 + n

Enter

     simplify

Result

        n
     -------
      1 + n
Example 2. It may be necessary to rationalize or condense an expression first. 
Enter

     (n + 1) / (n + 1)!

Result

         n            1
     ---------- + ----------
      (1 + n)!     (1 + n)!

Enter

     simfac

Result

         n            1
     ---------- + ----------
      (1 + n)!     (1 + n)!

Enter

     rationalize

Result

       1 + n
     ----------
      (1 + n)!

Enter

     simfac

Result

      1
     ----
      n!

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

wedge(u,v)

wedge(u,v,w)

Returns the wedge product of tensors. 
Enter

     u = (u1,u2,u3,u4)

     v = (v1,v2,v3,v4)

     wedge(u,v)

Result

           0           u1 v2 - u2 v1     u1 v3 - u3 v1     u1 v4 - u4 v1

     -u1 v2 + u2 v1          0           u2 v3 - u3 v2     u2 v4 - u4 v2

     -u1 v3 + u3 v1    -u2 v3 + u3 v2          0           u3 v4 - u4 v3

     -u1 v4 + u4 v1    -u2 v4 + u4 v2    -u3 v4 + u4 v3          0

Enter

     wedge(u,v) + wedge(v,u)

Result

     0

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

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