```*Console
cls: Clear console screen.
help: Show / hide help window.
editor: Show / hide editor window.
Shift+Enter: Input multi lines in the console.
Shift+Tab: Switch focus between console and editor.
F9 or Ctrl+Enter: Run code in the editor.

*Calculation
clear: Clears all variables and formulas, and go to initial state.
last: Get last output value.
```
Here is a list of the built-in functions.
abs  adj  arccos  arccosh  arcsin  arcsinh  arctan  arctanh  binomial  break  ceiling  check  condense  conj  contract  cos  cosh  d  det  dim  display  do  dot  draw  eigen  eval  exp  expcos  expsin  factor  factorial  filter  float  floor  for  gcd  hermite  hilbert  inner  integral  inv  isprime  laguerre  lcm  legendre  log  mod  outer  prime  print  product  prog  quote  rank  rationalize  return  roots  simplify  sin  sinh  sqrt  stop  subst  sum  tan  tanh  taylor  test  trace  transpose  unit  wedge  zero

# 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
```

Returns the adjunct of matrix m.
```Enter
A = ((a,b),(c,d))

Result
d     -b

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

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 expression 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)
```

# 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).

# 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

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
```

# 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(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
H(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
```

# 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
```

# 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))

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
```
In many cases the result of simplify will be a denormalized form. Denormalized forms can fail in tests for equality. The eval function returns normalized forms and can be used to undo the result of simplify.

# 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,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
```