Math plus ultra is a graphing programable calculator with a large set of functions that features:
- The manipulation of numeric and symbolic expressions.
- The development of customized functions.
- The creation of 2D and 3D graphs, including plots, contours and surfaces.
- Matrix and complex calculus.
- Integration and differentiation of numeric and symbolic functions.
Contents
Arithmetic
abs
abs(x)
Calculates the absolute value of a number, returns the magnitude of a complex number or returns the length of a vector.
ceiling
ceiling(x)
Returns the smallest integer value no less than x.
eval
eval(f,x,a)
Returns the function f evaluated at x=a.
exp
exp(x)
Calculates the exponential of x.
factor
factor(n)
Returns the factors of integer n.
float
float(x)
Takes rational numbers or integers and converts them to floating point values.
floor
floor(x)
Rounds a value to the nearest integer less or equal to it.
gcd
gcd(a,b,…)
Returns the greatest common divisor between two or more values.
isprime
isprime(n)
Determines if a value is a prime number. Returns 1 if n is a prime number or returns zero if it is not.
lcm
lcm(a,b,…)
Returns the least common multiple between two or more values.
log
log(x)
Calculates the natural logarithm of x.
mod
mod(a,b)
Returns the remainder after dividing a by b.
prime
prime(n)
Calculates the nth prime number. The domain of n is 1 to 10000.
product
product(f,i,j,k)
Evaluates f for i from j to k and returns the product of all f.
simplify
simplify(x)
Simplifies x.
sqrt
sqrt(x)
Calculates the square root of x
subst
subst(a,b,c)
Substitutes a for b in c and calculates the result.
sum
sum(f,i,j,k)
Evaluates f for i equals j through k and returns the sum of all f.
Calculus
circexp
circexp(x)
Converts circular and hyperbolic functions into exponential forms.
d
d(f,x)
Calculates the partial derivative of f with respect to x
defint
defint(f,x,a,b)
Calculates the definite integral of f with respect to x and evaluated from a to b. The arguments can be extended for multiple integrals or iterated integrals, for example, defint(f,x,a,b,y,c,d)
integral
integral(f,x)
Calculates the integral of f with respect to x.
Complex
arg
arg(z)
Calculates the argument of a complex number.
conj
conj(z)
Calculates the conjugate of a complex number.
imag
imag(z)
Returns the imaginary part of a complex number.
mag
mag(z)
Returns the magnitude of a complex number.
polar
polar(z)
Returns a complex number in polar form.
real
real(z)
Returns the real part of a complex number.
rect
rect(z)
Returns a complex number in rectangular form
Fractions
denominator
denominator(x)
Calculates the denominator of the expression x.
numerator
numerator(x)
Calculates the numerator of the expression x.
rationalize
rationalize(x)
Rationalize x.
Logical
and
and(a,b,…)
Logical operator AND.
check
check(x)
If x is true then continue, otherwise stop. Use A=B to verify if A equals B.
not
not(x)
Logical negation of x.
or
or(a,b,…)
Logical operator OR.
Matrix
adj
adj(m)
Calculates the adjunct of matrix m. The inverse of matrix m is equal to adj(m) divided by det(m).
cofactor
cofactor(m,i,j)
Returns the cofactor ij of matrix m. Let c be the cofactor matrix of m, then transpose(c )= adj(m).
contract
contract(a,i,j)
It is the equivalent to the trace of matrix m.
cross
cross(u,v)
Calculates the cross product between vectors u and v.
curl
curl(u)
Calculates the curl of vector u.
det
det(m)
Calculates the determinant of matrix m.
dot
dot(a,b,…)
Calculates the dot product of vectors.
eigen
eigen(m)
Compute eigenvalues and eigen vectors numerically. Argument m must be a numerical and symmetric matrix. This function stores the eigenvalue matrix in D and the eigenvector matrix in Q, so it does not return anything.
eigenval
eigenval(m)
Compute eigenvalues and eigen vectors numerically. Argument m must be a numerical and symmetric matrix. This function calculates a matrix with the eigenvalues along the diagonal.
eigenvec
eigenvec(m)
Compute eigenvalues and eigen vectors numerically. Argument m must be a numerical and symmetric matrix. This function calculates the eigenvectors and arranges them as row vectors.
hilbert
hilbert(n)
Calculates a n by n Hilbert matrix.
inv
inv(m)
Calculates the inverse of matrix m.
transpose
transpose(a,i,j)
Returns the transpose of matrix a with respect to indices i and j. When indices are omitted, 1 and 2 are assumed.
unit
unit(n)
Returns an identity matrix with n by n dimensions.
zero
zero(i,j,…)
Creates a matrix of all zeros with dimensions i,j, etc.
Polynomial
coeff
coeff(p,x,n)
Returns the nth coefficient of the polynomial p in the variable x. The argument x can be omitted for single variable polynomials in x.
deg
deg(p,x)
Calculates the degree of polynomial p(x).
expand
expand(r,x)
Expands the polynomial p with respect to x.
factor
factor(p,x)
Factors polynomial p with respect to x. If the polynomial is in x then x can be omitted. The polynomial must be factorable over integers. The argument list can be extended for multivariable polynomials, e.g., factor(p,x,y) so it factors p over x and then over y.
hermite
hermite(x,n)
Calculate the nth Hermite polynomial in x.
laguerre
laguerre(x,n,a)
Calculates the nth Laguerre polynomial in x, but if a is omitted, a=0 is assumed.
leading
leading(p,x)
Returns the leading coefficient of polynomial p in variable x.
legendre
legendre(x,n,m)
Calculates the nth Legendre polynomial in x. If m is omitted, m=0 is assumed.
nroots
nroots(p,x)
Calculates all the roots, real and complex of polynomial p in x. These roots are computed numerically. The coefficients of the polynomial can be real or complex.
quotient
quotient(p,q,x)
Calculates the quotient of two polynomials, p(x) over q(x). If the polynomials are in x, then the last argument can be omitted. The remainder can be calculated as p-q*quotient(p,q)
roots
roots(p,x)
Calculates the values of x such that polynomial p(x)=0. The polynomial must be factorable over integers. For multiple roots the result is a vector. Individual roots can be obtained by component rotation, e.g., r1=roots(p,x)[1].
Probability
choose
choose(n,k)
Returns the number of possible combinations of obtaining an unordered subset of k elements from a set of n objects.
factorial
factorial(x)
Returns the factorial of x. It can be entered as x!
Program
do
do(a,b,…)
Evaluates all the arguments from left to right and returns the result of the last argument.
for
for(do(a,b,…),i,j,k)
This function works as a loop to repeat a specified number of times according to a counter. The counter is i and repeats the cycle from the integer j through the integer k, evaluating arguments a, then b, etc.
stop
stop()
Stops the execution of a script.
test
test(a,b,c,d,…)
In this function if the argument ‘a’ is true then ‘b’ is returned else if argument ‘c’ is true then ‘d’ is returned and so on. But if the number of arguments is odd then the last argument will be returned after all else fails. Use A=B to check if A equals B.
Special
besselj
besselj(x,n)
Bessel function of the first kind.
bessely
bessely(x,n)
Bessel function of the second kind.
erf
erf(x)
Returns the error function of x.
erfc
erfc(x)
Returns the complementary error function of x
filter
filter(f,a,b,…)
Filters f by excluding any terms containing a,b,etc.
taylor
taylor(f,x,n,a)
Returns the Taylor expansion of f(x) at x=a. If a is omitted, then a=0 is assumed. The argument n is the degree of the Taylor expansion.
Tensor
dim
dim(a,n)
Calculates the cardinality of the nth index of tensor a.
dot
dot(a,b,…)
Returns the dot product of tensors.
inner
inner(a,b,…)
Returns the inner product of tensors, is the same as the dot product.
outer
outer(a,b,…)
Returns the tensor product or outer product of tensors.
rank
rank(a)
Returns the number of indices of tensor a.
shape
shape(x)
Returns a vector with the shape of the input tensor. If the tensor is a row vector it will return its length, but if the input is a matrix it will return a vector with its dimensions.
Trigonometry
arccos
arccos(x)
Returns the inverse cosine or arccosine of x.
arccosh
arccosh(x)
Returns the hyperbolic arccosine or inverse hyperbolic cosine of x.
arcsin
arcsin(x)
Returns the inverse sine or arcsine of x.
arcsinh
arcsinh(x)
Returns the hyperbolic arcsine or inverse hyperbolic sine of x.
arctan
arctan(x)
Returns the arctangent or inverse tangent of x.
arctanh
arctanh(x)
Returns the hyperbolic arctangent or inverse hyperbolic tangent of x.
cos
cos(x)
Calculates the cosine of x.
cosh
cosh(x)
Calculates the hyperbolic cosine of x.
expcos
expcos(x)
Calculates the exponential cosine of x.
expsin
expsin(x)
Calculates the exponential sine of x.
sin
sin(x)
Calculates the sine of x.
sinh
sinh(x)
Calculates the hyperbolic sine of x.
tan
tan(x)
Calculates the tangent of x.
tanh
tanh(x)
Calculates the hyperbolic tangent of x.
sec
sec(x)
Returns the secant of x, which is defined as sec(x) = 1/cos(x).
cot
cot(x)
Returns the cotangent of x. It is defined as cot(x)=1/tan(x).
csc
csc(x)
Returns the cosecant of x, which is defined as csc(x) = 1/sin(x).
sech
sech(x)
Returns the hyperbolic secant of x, which is defined as sech(x) = 1 / cosh(x).
coth
coth(x)
Returns the hyperbolic cotangent of x, which is defined as coth(x) = 1 / tanh(x).
csch
csch(x)
Returns the hyperbolic cosecant of x, which is defined as csch(x) = 1 / sinh(x).
acot
acot(x)
Returns the inverse cotangent or arctangent of x, which is defined as acot(x) = atan(1/x).
acoth
acoth(x)
Returns the hyperbolic arc cotangent of x, which is defined as acoth(x) = atanh(1/x).
asec
asec(x)
Returns the inverse secant or arc secant of x.
asech
asech(x)
Returns the hyperbolic arc secant of x, which is defined as asech(x) = acosh(1/x).
acsc
acsc(x)
Returns the inverse cosecant or arc cosecant of x, which is defined as acsc(x) = asin(1/x).
acsch
acsch(x)
Returns the hyperbolic arc cosecant of x, which is defined as acsch(x) = asinh(1/x).
Variable
quote
quote(x)
Returns the expression without evaluating it first.