1.5.3.2 List of functions

If derivative is not defined for some function, you can still use it in expressions and get symbolic derivative for this expression in the case when arguments/parameters for the function do not depend on variable.

Supported types

  • Real — real numbers
  • Complex — complex numbers
  • Matrix — vector / matrix
  • Physical — physical quantity with units of measurement
Якщо для функції не визначено похідну, ви можете використовувати її у виразах і отримати символьну похідну (у випадку, якщо аргументи/параметри функції не залежать від змінної).

Які типи даних підтримують функції

  • Real — дійсні числа
  • Complex — комплексні числа
  • Matrix — вектор / матриця
  • Physical — фізична величина з одиницями вимірювання

Algebraic functions

Function Name Type(s) Example(s) Derivative Note
Absolute value abs Real, Complex abs(x) sgn(x)
Signum sgn Real sgn(x) 2*delta(x)
Dirac delta function delta Real delta(x) not defined (under development)
Heaviside step function H Real H(x) delta(x) (under development)
If (conditional) function if Real, Complex, Physical if{x>0}(x x^2) if{x>0}(1 2*x)
Ceiling function ceil Real ceil(x) not defined
Floor function floor Real floor(x) not defined
Fractional part frac Real frac(x) not defined
Rounding roundto Real, Physical roundto{-3}(3.14159) not defined available in TE 3.7+
Rounding roundup Real, Physical roundup{-3}(3.14159) not defined available in TE 3.7+
Rounding rounddown Real, Physical rounddown{-3}(3.14159) not defined available in TE 3.7+
Minimum function Min Real, Complex, Matrix, Physical Min(M); Min(a b)
Maximum function Max Real, Complex, Matrix, Physical Max(M); Max(a b)

Trigonometric functions

Function Name Type(s) Example(s) Derivative Note
Sine sin Real, Complex sin(x) cos(x)
Cosine cos Real, Complex cos(x) -sin(x)
Tangent tan Real, Complex tan(x) 1/cos(x)^2
Cotangent cotan Real, Complex cotan(x) -1/sin(x)^2
Secant sec Real, Complex sec(x) sin(x)/cos(x)^2
Cosecant cosec Real, Complex cosec(x) -cos(x)/sin(x)^2
Inverse sine arcsin Real, Complex arcsin(x) 1/(1-x^2)^(1/2)
Inverse cosine arccos Real, Complex arccos(x) -1/(1-x^2)^(1/2)
Inverse tangent arctan Real, Complex arctan(x) 1/(1+x^2)
Inverse cotangent arccot Real, Complex arccot(x) -1/(1+x^2)
Inverse secant arcsec Real, Complex arcsec(x) 1/(x^2*(1-1/x^2)^(1/2))
Inverse cosecant arccsc Real, Complex arccsc(x) -1/(x^2*(1-1/x^2)^(1/2))
Hyperbolic sine sinh Real, Complex sinh(x) cosh(x)
Hyperbolic cosine cosh Real, Complex cosh(x) sinh(x)
Hyperbolic tangent tanh Real, Complex tanh(x) 1/cosh(x)^2
Hyperbolic cotangent coth Real, Complex coth(x) -1/sinh(x)^2
Hyperbolic secant sech Real, Complex sech(x) -tanh(x)*sech(x)
Hyperbolic cosecant cosech Real, Complex cosech(x) -coth(x)*cosech(x)
Inverse hyperbolic sine Real, Complex arsinh arsinh(x) 1/(x^2+1)^(1/2)
Inverse hyperbolic cosine Real, Complex arcosh arcosh(x) 1/(x^2-1)^(1/2)
Inverse hyperbolic tangent artanh Real, Complex artanh(x) 1/(1-x^2)
Inverse hyperbolic cotangent arcoth Real, Complex arcoth(x) 1/(1-x^2)
Inverse hyperbolic secant arsech Real, Complex arsech(x) -1/((x^2*(1/x-1)^(1/2))*(1/x+1)^(1/2))
Inverse hyperbolic cosecant arcsch Real, Complex arcsch(x) -1/(x^2*(1+1/x^2)^(1/2))

Logarithmic functions

Function Name Type(s) Example(s) Derivative Note
Logarithm to base log Real, Complex log{a}(x) 1/(ln(a)*x)
Natural logarithm ln Real, Complex ln(x) 1/x
Decimal logarithm lg Real, Complex lg(x) 1/(ln(10)*x)
Binary logarithm lb Real, Complex lb(x) 1/(ln(2)*x)
Exponent exp Real, Complex exp(x) exp(x)
Square root sqrt Real, Complex sqrt(x) 1/(2*x^(1/2))
Root (with index) root Real, Complex root{a}(x) 1/a*x^(1/a-1)
Power pow Real, Complex pow{a}(x) a*x^(a-1)

Special functions

NOTE: These functions is under development.

Function Name Type(s) Example(s) Derivative Note
Beta function Β Real, Complex Β(x y) Β(x y)*(ψ(x)-ψ(x+y)) Symbolic derivative only 1
Incomplete Beta Β Real, Complex Β{n m}(x) x^(n-1)*(1-x)^(m-1) Symbolic derivative only 1
Gamma function Γ Real Γ(x) Γ(x)*ψ(x)
Logarithm of Gamma Γlog Real Γlog(x) ψ(x)
Incomplete gamma Γ Real, Complex Γ{n}(x) -(x^(n-1)*e^-x) Symbolic derivative only 1
Digamma function ψ Real, Complex ψ(x) ψ{1}(x) Symbolic derivative only 1
Polygamma function ψ Real, Complex ψ{n}(x) ψ{n+1}(x) Symbolic derivative only 1
Error function erf Real erf(x) (2/π^(1/2))*e^-(x^2)
Complementary error erfc Real erfc(x) (-2/π^(1/2))*e^-(x^2)
Inversed error function erfi Real erfi(x) √π/2*e^(erfi(x)^2)
Bessel function of order 0 J₀ Real J₀(x) -J₁(x)
Bessel function of order 1 J₁ Real J₁(x) J₀(x)-J₁(x)/x
Bessel function of the second kind, order 0 Y₀ Real Y₀(x) -Y₁(x)
Bessel function of the second kind, order 1 Y₁ Real Y₁(x) Y₀(x)-Y₁(x)/x
Modified Bessel function of order 0 I₀ Real I₀(x) I₁(x)
Modified Bessel function of order 1 I₁ Real I₁(x) I₀(x)-I₁(x)/x
Modified Bessel function, second kind, order 0 K₀ Real K₀(x) -K₁(x)
Modified Bessel function, second kind, order 1 K₁ Real K₁(x) -K₀(x)-K₁(x)/x
Bessel function of order n J Real, Complex J{n}(x) -J{n+1}(x)+n*J{n}(x)/x Symbolic derivative only 1
Bessel function of the second kind, order n Y Real, Complex Y{n}(x) -Y{n+1}(x)+n*Y{n}(x)/x Symbolic derivative only 1
Modified Bessel function of order n I Real, Complex I{n}(x) I{n+1}(x)+n*I{n}(x)/x Symbolic derivative only 1
Modified Bessel function, second kind, order n K Real, Complex K{n}(x) -K{n+1}(x)+n*K{n}(x)/x Symbolic derivative only 1
Legendre polynomial P Real P{n}(x) (n+1)/(x^2-1)*(P{n+1}(x)-x*P{n}(x)) Symbolic derivative only 1
Legendre polynomial of the second kind Q Real Q{n}(x) (n+1)/(x^2-1)*(Q{n+1}(x)-x*Q{n}(x)) Symbolic derivative only 1
Associated Legendre polynomial P Real, Complex P{n m}(x) ((n+1-m)*P{n+1 m}(x)-(n+1)*x*P{n m}(x))/(x^2-1) Symbolic derivative only 1
Associated Legendre polynomial of the second kind Q Real, Complex Q{n m}(x) ((n+1-m)*Q{n+1 m}(x)-(n+1)*x*Q{n m}(x))/(x^2-1) Symbolic derivative only 1

Physical functions

Function Name Type(s) Example(s) Note
Changes the units of the argument without changing its value. convert Physical convert(a {units})
Returns the numeric value of a physical quantity without units. empiric Real empiric(a)

Matrix/Vector functions

Function Name Type(s) Example(s) Note
Minimal component of an array or a matrix Min Real Min([a b c])
Maximal component of an array or a matrix Min Real Min([a b c])
Range of an array components from “i” to “j” inclusively Range Matrix Range{i j}(A)
Submatrix of “M” containing “i1..i2” rows an “j1..j2” columns inclusively Range Matrix Range{i1 i2 j1 j2}(M)
Number of rows of a matrix RowCount Real RowCount(M)
Number of columns of a matrix ColumnCount Real ColumnCount(M)
Main diagonal of a matrix Diagonal Matrix Diagonal(M)
Main antidiagonal of a matrix Antidiagonal Matrix Antidiagonal(M)
“N”-th row (array) of a matrix Row Matrix Row{N}(M)
“N”-th column (array) of a matrix Column Matrix Column{N}(M)
Minor (matrix) of the [i,j]-th matrix’ element Minor Matrix Minor{i j}(M)
Cumulative sum of the array CumSum Real, Matrix CumSum(X) For matrix it is evaluated for each row.
Cumulative product of the array CumProduct Real, Matrix CumProduct(X) For matrix it is evaluated for each row.
Outer product of two arrays, the result is the matrix Outer Matrix Outer(X Y)
Determinant of a square matrix det Real det(M)
Trace of a square matrix tr Matrix tr(M)
Adjoint of a square matrix adj Matrix adj(M)
Condition number of a square matrix (using L2 norm) cond Matrix cond(M)
Pseudo-inverse of a rectangular matrix pinv Matrix pinv(M)
Cross linear interpolation LInterp Real LInterp(M AX AY X Y)

1 For this function only symbolic derivatives defined, it cannot be evaluated.