The Eigenmath manual and additional support materials are available at eigenmath.github.io
Returns the absolute value or vector length of x.
X = (x,y,z) abs(X)
1/2
2 2 2
(x + y + z )
Returns the adjunct of matrix m. Adjunct is equal to determinant times inverse.
A = ((a,b),(c,d)) adj(A) == det(A) inv(A)
1
Returns 1 if all arguments are true (nonzero). Returns 0 otherwise.
and(1=1,2=2)
1
Returns the arc cosine of x.
arccos(1/2)
1 --- pi 3
Returns the arc hyperbolic cosine of x.
Returns the arc sine of x.
arcsin(1/2)
1 --- pi 6
Returns the arc hyperbolic sine of x.
Returns the arc tangent of y over x. If x is omitted then x = 1 is used.
arctan(1,0)
1 --- pi 2
Returns the arc hyperbolic tangent of x.
Returns the angle of complex z.
arg(2 - 3i)
arctan(-3,2)
Returns a solution to the Bessel differential equation.
besselj(x,1/2)
1/2 2 sin(x) ------------- 1/2 1/2 pi x
The result of evaluating a symbol can differ from the symbol's binding. For example, the result may be expanded. The binding function returns the actual binding of a symbol.
p = quote((x + 1)^2) p
2
p = x + 2 x + 1
binding(p)
2
(x + 1)
Returns the coefficient of xk yn-k in (x + y)n.
binomial(52,5)
2598960
Note: binomial and choose are the same function.
Returns the smallest integer greater than or equal to x.
ceiling(1/2)
1
If x is true (nonzero) then continue in a script, else stop. Expression x can include the relational operators =, ==, <, <=, >, >=. Use the not function to test for inequality.
A = 1 B = 1 check(A=B) -- script stops here if A not equal to B
Returns the number of combinations of n items taken k at a time. The following example computes the number of poker hands.
choose(52,5)
2598960
Returns expression x with circular and hyperbolic functions converted to exponentials.
circexp(cos(x) + i sin(x))
exp(i x)
Clears all symbol definitions.
Returns complex z in polar form with base of negative 1 instead of e.
clock(2 - 3i)
1/2 arctan(-3,2)/pi 13 (-1)
Returns the coefficient of xn in polynomial p.
p = x^3 + 6x^2 + 12x + 8 coeff(p,x,2)
6
Returns a cofactor of matrix m. The cofactor matrix is the transpose of the adjunct of m. This function returns the cofactor component at row i and column j.
A = ((a,b),(c,d)) cofactor(A,1,2) == transpose(adj(A))[1,2]
1
Returns the complex conjugate of z.
conj(2 - 3i)
2 + 3 i
Returns tensor a summed over indices i and j. If i and j are omitted then 1 and 2 are used. The expression contract(m) computes the trace of matrix m.
A = ((a,b),(c,d)) contract(A)
a + d
Returns the cosine of x.
cos(pi/4)
1 ------ 1/2 2
Returns the hyperbolic cosine of x.
circexp(cosh(x))
1 1 --- exp(-x) + --- exp(x) 2 2
Returns the cross product of vectors u and v. It is OK to redefine cross. This is the default definition.
cross(u,v) = (u[2] v[3] - u[3] v[2],
u[3] v[1] - u[1] v[3],
u[1] v[2] - u[2] v[1])
Returns the curl of vector u. It is OK to redefine curl. This is the default definition.
curl(u) = (d(u[3],y) - d(u[2],z),
d(u[1],z) - d(u[3],x),
d(u[2],x) - d(u[1],y))
Returns the partial derivative of f with respect to x.
d(x^2,x)
2 x
Argument f can be a tensor of any rank. Argument x can be a vector. When x is a vector the result is the gradient of f.
F = (f(),g(),h()) X = (x,y,z) d(F,X)
d(f(),x) d(f(),y) d(f(),z) d(g(),x) d(g(),y) d(g(),z) d(h(),x) d(h(),y) d(h(),z)
It is OK to use d as a variable name. It will not conflict with function d.
It is OK to redefine d as a different function. The function derivative, a synonym for d, can still be used to obtain a partial derivative.
Returns the definite integral of f with respect to x evaluated from a to b. The argument list can be extended for multiple integrals as shown in the following example.
f = (1 + cos(theta)^2) sin(theta) defint(f, theta, 0, pi, phi, 0, 2pi) -- integrate over theta then over phi
16 ---- pi 3
Returns the degree of polynomial p(x).
p = (2x + 1)^3 deg(p,x)
3
Returns the denominator of expression x.
denominator(a/b)
b
Returns the determinant of matrix m.
A = ((a,b),(c,d)) det(A)
a d - b c
Returns the dimension of the nth index of tensor a. Index numbering starts with 1.
A = ((1,2),(3,4),(5,6)) dim(A,1)
3
Returns the divergence of vector u. It is OK to redefine div. This is the default definition.
div(u) = d(u[1],x) + d(u[2],y) + d(u[3],z)
Evaluates each argument from left to right. Returns the result of the final argument.
do(A=1,B=2,A+B)
3
Returns the dot or matrix product of vectors, matrices, and tensors.
-- solve for X in AX=B A = ((1,2),(3,4)) B = (5,6) X = dot(inv(A),B) X
-4
X =
9
---
2
Draws a graph of f(x). Drawing ranges can be set with xrange and yrange.
xrange = (0,1) yrange = (0,1) draw(x^2,x)
Symbol e is initialized to the natural number e.
e^x
exp(x)
Note: It is OK to clear or redefine e and use the symbol for something else.
Computes eigenvalues and eigenvectors numerically. Matrix m is required to be both numerical and symmetric. Eigenvectors are returned in Q and eigenvalues are returned in D. Each row of Q is an eigenvector. Each diagonal element of D is an eigenvalue.
A = ((1,2),(2,1)) eigen(A) dot(transpose(Q),D,Q)
1.0 2.0 2.0 1.0
Error function of x.
Complementary error function of x.
Returns f evaluated at x equals a.
eval(x^2 + 3,x,0)
3
Returns the exponential of x.
exp(i pi)
-1
Returns the partial fraction expansion of the ratio of polynomials r in x.
p = (x + 1)^2 q = (x + 2)^2 expand(p/q,x)
2 1
- ------- + -------------- + 1
x + 2 2
x + 4 x + 4
Returns the cosine of z in exponential form.
expcos(z)
1 1 --- exp(i z) + --- exp(-i z) 2 2
Returns the hyperbolic cosine of z in exponential form.
expcosh(z)
1 1 --- exp(-z) + --- exp(z) 2 2
Returns the sine of z in exponential form.
expsin(z)
1 1 - --- i exp(i z) + --- i exp(-i z) 2 2
Returns the hyperbolic sine of z in exponential form.
expsinh(z)
1 1 - --- exp(-z) + --- exp(z) 2 2
Returns the tangent of z in exponential form.
exptan(z)
i i exp(2 i z)
---------------- - ----------------
exp(2 i z) + 1 exp(2 i z) + 1
Returns the hyperbolic tangent of z in exponential form.
exptanh(z)
1 exp(2 z)
- -------------- + --------------
exp(2 z) + 1 exp(2 z) + 1
Factors numerical value n and returns the result.
factor(12/35)
2 2 3 ------ 5 7
If n is a floating point value then a rational approximation of n is factored and returned.
factor(float(pi))
5 71 ------ 113
factor(p,x)
Factors polynomial p of x and returns the result. The argument list can be extended for multivariate polynomials.
p = 2x + x y + y + 2 factor(p,x,y)
(x + 1) (y + 2)
Note: factor returns an unexpanded expression. If the result is assigned to a symbol, evaluating the symbol will expand the result. Use binding to retrieve the unexpanded expression.
q = factor(p,x,y) binding(q)
(x + 1) (y + 2)
Returns the factorial of n. The expression n! can also be used.
20!
2432902008176640000
Returns f excluding any terms containing a, b, etc.
p = x^2 + 3x + 2 filter(p,x^2)
3 x + 2
Returns expression x with rational numbers and integers converted to floating point values. The symbol pi and the natural number are also converted.
float(212^17)
39
3.52947 10
Returns the largest integer less than or equal to x.
floor(1/2)
0
For i equals j through k evaluate a, b, etc.
for(k,1,3,A=k,print(A))
A = 1 A = 2 A = 3
Note: The original value of i is restored after for completes. If symbol i is used for index variable i then the imaginary unit is overridden in the scope of for.
Returns the greatest common divisor of expressions.
gcd(x,x y)
x
Returns the nth Hermite polynomial in x.
hermite(x,3)
3 8 x - 12 x
Returns an n by n Hilbert matrix.
hilbert(3)
1 1
1 --- ---
2 3
1 1 1
--- --- ---
2 3 4
1 1 1
--- --- ---
3 4 5
Symbol i is initialized to the imaginary unit (−1)1/2.
exp(i pi)
-1
Note: It is OK to clear or redefine i and use the symbol for something else.
Returns the imaginary part of complex z.
imag(2 - 3i)
-3
Returns the inner product of tensors.
A = ((a,b),(c,d)) B = (x,y) inner(A,B)
a x + b y c x + d y
Note: inner and dot are the same function.
Returns the integral of f with respect to x.
integral(x^2,x)
1 3 --- x 3
Returns the inverse of matrix m.
A = ((1,2),(3,4)) inv(A)
-2 1 3 1 --- - --- 2 2
Returns 1 if n is a prime number. Returns zero otherwise.
isprime(2^31 - 1)
1
Set j=sqrt(-1) to use j for the imaginary unit instead of i.
j = sqrt(-1) 1/sqrt(-1)
-j
Returns the nth associated Laguerre polynomial in x of order m. If m is omitted then zero is used.
laguerre(x,3)
1 3 3 2 - --- x + --- x - 3 x + 1 6 2
The result of the previous calculation is stored in last.
212^17
3529471145760275132301897342055866171392
last^(1/17)
212
Note: Symbol last is an implied argument when a function has no argument list.
212^17
3529471145760275132301897342055866171392
float
39
3.52947 10
Returns the least common multiple of expressions.
lcm(x,x y)
x y
Returns the leading coefficient of polynomial p(x).
leading(3x^2 + 1,x)
3
Returns the nth associated Legendre polynomial in x of order m. If m is omitted then zero is used.
legendre(x,3)
5 3 3 --- x - --- x 2 2
Evaluates expression x and returns the result as a string in prefix notation. Useful for debugging scripts.
lisp(x^2 + 1)
(+ (^ x 2) 1)
Returns the natural logarithm of x.
log(x^y)
y log(x)
Returns the magnitude of complex z. Mag treats undefined symbols as real while abs does not.
mag(x + i y)
1/2
2 2
(x + y )
Returns the remainder of a divided by b.
mod(10,7)
3
Returns 0 if x is true (nonzero). Returns 1 otherwise.
not(1=1)
0
Returns all roots, both real and complex, of polynomial p(x). The roots are computed numerically. The coefficients of p can be real or complex.
Returns 1 if x is a rational or floating point number. Returns 0 otherwise.
number(1/2)
1
Returns the numerator of expression x.
numerator(a/b)
a
Returns 1 if at least one argument is true (nonzero). Returns 0 otherwise.
or(1=1,2=2)
1
Returns the outer product of tensors. Also known as the tensor product.
A = (a,b,c) B = (x,y,z) outer(A,B)
a x a y a z b x b y b z c x c y c z
Symbol for π.
exp(i pi)
-1
Returns complex z in polar form.
polar(x - i y)
1/2
2 2
(x + y ) exp(i arctan(-y,x))
Use ^ to raise something to a power. Use parentheses for negative powers.
x^(-2)
1 ---- 2 x
Returns the nth prime number. The domain of n is 1 to 10000.
prime(100)
541
Evaluate expressions and print the results. Useful for printing from inside a for loop.
for(j,1,3,print(j))
j = 1 j = 2 j = 3
For i equals j through k evaluate f. Returns the product of all f.
product(j,1,3,x + j)
3 2 x + 6 x + 11 x + 6
Note: The original value of i is restored after product completes. If symbol i is used for index variable i then the imaginary unit is overridden in the scope of product.
Returns expression x without evaluating it first.
quote((x + 1)^2)
2
(x + 1)
Returns the quotient of polynomial p(x) over q(x).
p = x^2 + 1 q = x + 3 quotient(p,q,x)
x - 3
Returns the number of indices that tensor a has.
A = ((a,b),(c,d)) rank(A)
2
Returns expression x with everything over a common denominator.
rationalize(1/a + 1/b + 1/2)
2 a + a b + 2 b
-----------------
2 a b
Note: rationalize returns an unexpanded expression. If the result is assigned to a symbol, evaluating the symbol will expand the result. Use binding to retrieve the unexpanded expression.
f = rationalize(1/a + 1/b + 1/2) binding(f)
2 a + a b + 2 b
-----------------
2 a b
Returns the real part of complex z.
real(2 - 3i)
2
Returns complex z in rectangular form.
rect(exp(i x))
cos(x) + i sin(x)
Returns the values of x such that polynomial p(x) equals zero. The polynomial should be factorable over integers. Returns a vector for multiple roots.
roots(x^2 + 3x + 2,x)
-2 -1
Run script file. Useful for importing function libraries.
run("Downloads/EVA.txt")
Note: file must be in the Downloads folder due to security requirements for apps distributed on the Mac App Store.
Returns expression x in a simpler form.
simplify(sin(x)^2 + cos(x)^2)
1
Returns the sine of x.
sin(pi/4)
1 ------ 1/2 2
Returns the hyperbolic sine of x.
circexp(sinh(x))
1 1 - --- exp(-x) + --- exp(x) 2 2
Returns the square root of x.
sqrt(10!)
1/2
720 7
Prints memory statistics.
status
block_count 1 free_count 99258 gc_count 1 bignum_count 370 string_count 0 tensor_count 5
In a script, it does what it says.
Evaluates expression x and returns the result as a string. Useful for testing scripts.
string((x + 1)^2) == "x^2 + 2 x + 1"
1
Substitutes a for b in c and returns the result.
subst(x,y,y^2)
2 x
For i equals j through k evaluate f. Returns the sum of all f.
sum(j,1,5,x^j)
5 4 3 2 x + x + x + x + x
Note: The original value of i is restored after sum completes. If symbol i is used for index variable i then the imaginary unit is overridden in the scope of sum.
Returns the tangent of x.
simplify(tan(x) - sin(x)/cos(x))
0
Returns the hyperbolic tangent of x.
circexp(tanh(x))
1 exp(2 x)
- -------------- + --------------
exp(2 x) + 1 exp(2 x) + 1
Returns the Taylor expansion of f(x) near x equals a. If argument a is omitted then zero is used. Argument n is the degree of the expansion.
taylor(sin(x),x,5)
1 5 1 3 ----- x - --- x + x 120 6
If argument a is true (nonzero) then b is returned, else if c is true then d is returned, etc. If the number of arguments is odd then the final argument is returned if all else fails. Expressions can include the relational operators =, ==, <, <=, >, >=. Use the not function to test for inequality. (The equality operator == is available for contexts in which = is the assignment operator.)
A = 1 B = 1 test(A=B,"yes","no")
yes
Set trace=1 in a script to print the script as it is evaluated. Useful for debugging.
trace = 1
Note: The contract function is used to obtain the trace of a matrix.
Returns the transpose of tensor a with respect to indices i and j. If i and j are omitted then 1 and 2 are used. Hence a matrix can be transposed with a single argument.
A = ((a,b),(c,d)) transpose(A)
a c b d
Set tty=1 to print results in a flat format.
tty = 1 (x + 1/2)^2
x^2 + x + 1/4
Returns an n by n identity matrix.
unit(3)
1 0 0 0 1 0 0 0 1
Returns a null tensor with dimensions i, j, etc. Useful for creating a tensor and then setting the component values.
A = zero(3,3) for(k,1,3,A[k,k]=k) A
1 0 0
A = 0 2 0
0 0 3