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Ask your administrator if you think this is wrong. ====== 1.5.3.2 List of functions ====== <tabs> * [[#tab-en|EN]] * [[#tab-ua|UA]] <pane id="tab-en"> 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 </pane> <pane id="tab-ua"> 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 </pane> </tabs> ==== 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 ==== <text type="warning">NOTE: These functions is under development.</text> ^ 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 <sup>1</sup> | | Incomplete Beta | Β | Real, Complex | Β{n m}(x) | %%x^(n-1)*(1-x)^(m-1)%% | Symbolic derivative only <sup>1</sup> | | 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 <sup>1</sup> | | Digamma function | ψ | Real, Complex | ψ(x) | %%ψ{1}(x)%% | Symbolic derivative only <sup>1</sup> | | Polygamma function | ψ | Real, Complex | ψ{n}(x) | %%ψ{n+1}(x)%% | Symbolic derivative only <sup>1</sup> | | 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 <sup>1</sup> | | 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 <sup>1</sup> | | Modified Bessel function of order n | I | Real, Complex | I{n}(x) | %%I{n+1}(x)+n*I{n}(x)/x%% | Symbolic derivative only <sup>1</sup> | | 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 <sup>1</sup> | | Legendre polynomial | P | Real | P{n}(x) | %%(n+1)/(x^2-1)*(P{n+1}(x)-x*P{n}(x))%% | Symbolic derivative only <sup>1</sup> | | 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 <sup>1</sup> | | 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 <sup>1</sup> | | 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 <sup>1</sup> | ==== 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)%% | | ---- <sup>1</sup> For this function only symbolic derivatives defined, it cannot be evaluated. Last modified: 20.08.2023 19:%iby dystlab-wiki-admin