Calculus¶

General Functions¶

Calculus functions.

class simula.calculus.functions.Function(var, expression=None)¶

Definition of a mathematical function.

EXAMPLES:

simula : x = var('x')
simula : f = function(x, x^2-2x+1); f
Function defined by x |--> x^2 - 2x + 1
simula : f(2x)
4x^2 - 4x + 1
simula : 3f
Function defined by x |--> 3x^2 - 6x + 3
simula : factor(f(x))
(x - 1)^2
simula : g = function("g"); g
X |--> g(X)
# We can simplify the definition of a function
simula : x, y = var('x, y')
simula : h(x, y) = x^2-y^2-x*y-2; h
Function defined by (x, y) |--> x^2 - x*y - y^2 - 2
simula : h(1,-2)
-3
simula : h(10, 0)
98
simula : diff(h(x, y), x)
2x - y

as_expr()¶: Returns the expression of self.

compose(g, *args)¶

Composition function of f and g i.e x –> f o g (x)

Parameters: g – Function

critical_points(field=RR)¶: Returns the critical points of self.

critical_points_ambigus(field=RR)¶: Returns the critical points ambigus of self.

gradient()¶: Returns the gradient function of self.

hessian()¶: Returns the hessian function of self.

hessian_matrix()¶: Returns the hessian matrix of self.

jacobian()¶: Returns the jacobian function of self.

jacobian_matrix()¶: Returns the jacobian matrix of self.

local_extrema(field=RR)¶: Returns the local extrema of self.

local_maxima(field=RR)¶: Returns the local maxima of self.

local_minima(field=RR)¶: Returns the local minima of self.

saddle_points(field=RR)¶: Returns the saddle points of self.

class simula.calculus.functions.FunctionPiecewise(var, *expr)¶

Representation of a piecewise function.

EXAMPLES:

simula : x = var("x")
simula : f = FunctionPiecewise(x, (2x-1, x<0), (3x, True)); f
x |--> 2x - 1 if x < 0  and  3x if True
simula : f(1)
3
simula : f(3)
27
simula : f(-2)
-5
simula : n = var('n', "NN")
simula : f(n)
3n
simula : y = var('y', "ZZ*-")
simula : f(y)
2y - 1

class simula.calculus.functions.O(expr, *args, **kwargs)¶

Big O notation.

EXAMPLES:

simula : O(x + x^2)
O(x)
simula : O(x^3-x^2)
O(x^2)
simula : O(5x^5)
O(x^5)
simula : O(x^5-x-4)
O(1)

simula.calculus.functions.canonical_form(expr, var=None)¶: Returns the canonical form of a second degree equation.

simula.calculus.functions.coefficients(expr, *args)¶: Returns the coefficients of a polynomial.

simula.calculus.functions.cyclotomic_polynomial(n, var=None)¶

Returns the cyclotomic polynomial of order n.

Parameters

n – an integer
var – (optional) a variable

simula.calculus.functions.degree(expr, *args)¶: Returns the degree of a polynomial.

simula.calculus.functions.derivative(f, var=None, *args)¶: Returns the derivative of f with respect to var.

simula.calculus.functions.derivative_number(func, var, point=None)¶: Returns the derivative number of func at point point.

simula.calculus.functions.diff(f, var, *args, **kwargs)¶: Returns the differentiation of f with respect to symbols.

simula.calculus.functions.discriminant(f, *args, **kwargs)¶: Returns the discriminant of``f``.

simula.calculus.functions.div(f, g, *args, **kwargs)¶: Returns the quotient and remainder of the division of f by g.

simula.calculus.functions.exp(x)¶: Returns the exponential of x i.e. \(e^x\).

simula.calculus.functions.expand(f, **kwargs)¶: Returns the expansion of f.

simula.calculus.functions.expand_trig(f, *args)¶: Returns the trigonometric expansion of f.

simula.calculus.functions.factor(expr, *args, **kwargs)¶: Factors an expression f.

simula.calculus.functions.function¶: alias of simula.calculus.functions.Function

simula.calculus.functions.function_composition(f, g, *args)¶

Composite function of f and g i.e x—-> f o g.

Parameters

f (simula.calculus.functions.Function) – Function
g ((<class 'simula.calculus.functions.Function'>, <class 'sympy.core.expr.Expr'>)) – Function

simula.calculus.functions.gcd(f, g, **kwargs)¶: Returns the gcd of f and g.

simula.calculus.functions.gcdex(f, g, **kwargs)¶: Extended Euclidean algorithm of f and g.

simula.calculus.functions.homogenize(f, var, *, symbols=None)¶: Returns the homogenization of f with respect to var.

simula.calculus.functions.inflection_point(func, var=None)¶: Returns Points d’inflexion de la fonction “func”

simula.calculus.functions.integrate(f, *symbols, **kwargs)¶: Integrates f with respect to symbols.

simula.calculus.functions.lcm(f, g, **kwargs)¶: Returns the lcm of f and g.

simula.calculus.functions.limit(f, x, x0, *, dir='+-')¶: Returns the limit of f when x --> x0.

simula.calculus.functions.limit_left(f, x, x0)¶: Returns the limit from the left of f when x --> x0.

simula.calculus.functions.limit_right(f, x, x0)¶: Returns the limit from the right of f when x --> x0.

simula.calculus.functions.ln(a, *args)¶: Returns the natural logarithm \(\ln(a)\) or \(\log(a)\).

simula.calculus.functions.log(a, *args)¶: Returns the natural logarithm \(\ln(a)\) or \(\log(a)\).

simula.calculus.functions.logb(a, b)¶: Returns the logarithm of a in base b.

simula.calculus.functions.partial(f, var, *args, **kwargs)¶: Returns the differentiation of f with respect to symbols.

simula.calculus.functions.poly(expr, *args, **kwargs)¶: Returns expr as a polynomial.

simula.calculus.functions.primitive(f, var=None)¶: Returns the primitive of f.

simula.calculus.functions.product(expr, *symbols, **kwargs)¶: Computes the product of expr with respect to symbols.

simula.calculus.functions.real_roots(func, var=None)¶: Returns the real roots of func.

simula.calculus.functions.roots(f, *args, **kwargs)¶: Returns the complex roots of f.

simula.calculus.functions.series(f, x=None, x0=0, n=6, *, dir='+')¶: Returns the series expansion of f of order n around point x = x0.

simula.calculus.functions.simplify(f, **kwargs)¶: Reduces f.

simula.calculus.functions.sqrt(expr)¶: Returns the square root of f.

simula.calculus.functions.summation(expr, *symbols, **kwargs)¶: Computes the summation of expr with respect to symbols.

simula.calculus.functions.taylor_polynomial(f, x=None, x0=0, n=6, *, dir='+')¶: Returns the taylor polynomial of f of order n around point x = x0.

simula.calculus.functions.trigsimp(f, **kwargs)¶: Reduces the trigonometric expression f.

simula.calculus.functions.trunc(f, n, *gens, **args)¶: Reduces f modulo a constant n.

Sequences¶

Representation of sequences :

Sequence
ArithmeticSequence
GeometricSequence
ArithmeticoGeometricSequence

class simula.calculus.sequense.ArithmeticGeometricSequence(a, b, *, ics=None)¶

Representation of a arithmetic-geometric sequence U(n+1) = aU(n) + b.

EXAMPLES:

simula :  n = var('n')
simula : U = ArithmeticGeometricSequence(1, 3, ics=(1, 5)); U
n |--> 3n + 2
simula : U(n+1) == U(n) + 3
True
simula : U(1)
5

class simula.calculus.sequense.ArithmeticSequence(terms=None, *, r=None)¶

Representation of an arithmetic sequence U(n+1) = U(n) + r.

EXAMPLES:

simula :  n = var('n')
simula : U = ArithmeticSequence((0, 1), r=5)
simula : U
n |--> 5n + 1
simula : U(n+1) - U(n)
5
simula : V = ArithmeticSequence(((0, 1), (2, 10))); V
n |--> 9n/2 + 1
simula : V(2)
10

class simula.calculus.sequense.GeometricSequence(terms=None, q=None)¶

Representation of a geometric sequence U(n+1) = qU(n).

EXAMPLES:

simula :  n = var('n')
simula : U = GeometricSequence((1, 3), q=5);
n |--> 3*5^n/5
simula : simplify(U(n+1) / U(n))
5
simula : V = GeometricSequence(((1, 3), (2, 10))); V
n |--> 10^(n - 1)*3^(2 - n)
simula : simplify(V(n+1) / V(n))
10/3

class simula.calculus.sequense.Sequence(var, expression=None)¶

Representation of a sequence.

EXAMPLES:

simula :  n = var('n')
simula : Un = sequence(n, n^3-3n-1)
simula : Un
n |--> n^3 - 3n - 1
simula : Un(1), Un(2)
(-3, 1)
simula : Un.is_convergente()
False
simula : Un.limit()
oo

is_convergente()¶

Tests if self is convergent.

EXAMPLES:

simula :  n = var('n')
simula : Vn = sequence(n, (n-1)/(2n^2-1))
simula : Vn
n |--> (n - 1)/(2n^2 - 1)
simula : Vn.is_convergente()
True

limit()¶

Returns the limit of self.

EXAMPLES:

simula :  n = var('n')
simula : Vn = sequence(n, (n-1)/(2n^2-1))
simula : Vn
n |--> (n - 1)/(2n^2 - 1)
simula : Un.limit()
oo

simula.calculus.sequense.limit_sequence(Un, *args)¶: Returns the limit of Un at infinity.

simula.calculus.sequense.sequence¶: alias of simula.calculus.sequense.Sequence