 1. abs		absolute value
 2. adj		adjunct of matrix
 3. arccos
 4. arccosh
 5. arcsin
 6. arcsinh
 7. arctan
 8. arctanh
 9. binomial
10. break	break out of a loop
11. ceiling
12. check	stop if not zero
13. condense	find subexpressions
14. conj	complex conjugate
15. contract	contract across tensor indices
16. cos
17. cosh
18. d		derivative and gradient
19. denominator
20. det		determinant of a matrix
21. dim
22. display	display an expression
23. do		evaluate multiple expressions
24. dot		inner product of tensors
25. draw	draw a graph
26. eigen
27. eval	normalize an expression
28. exp
29. expcos	exponential cosine
30. expsin	exponential sine
31. factor	factor a number or polynomial
32. factorial
33. filter
34. float
35. floor
36. for
37. gcd
38. hermite	emit a hermite polynomial
39. hilbert	emit a hilbert matrix
40. inner	inner product of tensors
41. integral
42. inv		invert a matrix
43. isprime
44. laguerre	emit a laguerre polynomial
45. lcm		least common multiple
46. legendre	emit a legendre polynomial
47. log		natural logarithm
48. mod
49. numerator
50. outer	outer product of tensors
51. prime
52. print
53. product	multiply over an index
54. prog	evaluate with scoped variables
55. quote
56. rank	rank of tensor
57. rationalize	combine fractions
58. return	early return from a function
59. roots	find roots of a polynomial
60. simfac	simplify factorials
61. simplify	simplify an expression
62. sin
63. sinh
64. sqrt	square root
65. stop	stop running a script
66. subst	substitute expressions
67. sum		sum over an index
68. tan
69. tanh
70. taylor	emit a taylor series
71. test	conditional evaluation
72. trace	trace of matrix
73. transpose	transpose tensor indices
74. unit	emit a unit matrix
75. wedge	wedge product of tensors
76. zero	emit a zero tensor

abs(x) Returns the absolute value (vector length, magnitude) of x. Enter abs(a - b) + abs(b - a) Result 2 abs(a - b) Enter abs(3 + 4 i) Result 5 Enter abs((a,b,c)) Result 1/2 2 2 2 (a + b + c ) Enter A = (1,i,-i) sqrt(dot(A,conj(A))) - abs(A) Result 0 Enter A = ((1,2),(3,4)) abs(A) Result stop: abs(tensor) with tensor rank > 1 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 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. 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 zero then continue, else stop. 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(x) Returns the complex conjugate of x. Enter conj(3+4i) 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. 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 Returns the gradient of f when x is a vector. Note that gradient raises the rank of f by 1. Enter u = x^2 + y^3 d(u,(x,y)) Result 2 x 2 3 y Functions with 0-arity are treated as dependent on all variables. 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) denominator(x) Returns the denominator of expression x. (Not in Mac version yet.) 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. Gamma Γ alpha α mu μ Delta Δ beta β nu ν Theta Θ gamma γ xi ξ Lambda Λ delta δ pi π Xi Ξ epsilon ε rho ρ Pi Π zeta ζ sigma σ Sigma Σ eta η tau τ Upsilon Υ theta θ upsilon υ Phi Φ iota ι phi φ Psi Ψ kappa κ chi χ Omega Ω lambda λ psi ψ omega ω 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). 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 = Q^TDQ. 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 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 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 Enter log(-10.0) Result 2.30259 + i π mod(a,b) Returns the remainder of a divided by b. 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 numerator(x) Returns the numerator of expression x. (Not in Mac version yet.) Enter numerator(a/b) Result a 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 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 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. (Not in Mac version yet.) 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 a is true then b is returned else if c is true then d is returned, etc. If none are true then the test function is returned unevaluated. An expression is true if it evaluates to a nonzero numerical value. If an expression evaluates to zero or a symbolic expression then it is not true. The following relational operators and their equivalent functions can be used: == testeq >= testge > testgt <= testle < testlt Each relational operator evaluates to 1 if the relation is true and 0 if the relation is false. If the relation cannot be determined then the associated relational function is returned unevaluated. Enter 2 < 3 Result 1 Enter 3 < 2 Result 0 Enter a < b Result testlt(a,b) The AND and OR of relations can be implemented using multiplication and addition. For example, the following function returns 1 when x is between a and b, otherwise 0 is returned. pulse(x) = test( (x >= a) * (x <= b), 1, (x < a) + (x > b), 0) # could use else clause "1, 0)" Relational operators have lower precedence than addition and multiplication so the relational expressions are parenthesized in this case. In this example the else clause is not used so that pulse returns unevaluated if the relation cannot be determined. 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 Result 0 0 0 0 Enter A[1,2] = a A Result 0 a 0 0

Gamma	Γ	alpha	α	mu	μ
Delta	Δ	beta	β	nu	ν
Theta	Θ	gamma	γ	xi	ξ
Lambda	Λ	delta	δ	pi	π
Xi	Ξ	epsilon	ε	rho	ρ
Pi	Π	zeta	ζ	sigma	σ
Sigma	Σ	eta	η	tau	τ
Upsilon	Υ	theta	θ	upsilon	υ
Phi	Φ	iota	ι	phi	φ
Psi	Ψ	kappa	κ	chi	χ
Omega	Ω	lambda	λ	psi	ψ
				omega	ω