from __future__ import division, print_function   # Python 3
from sympy import init_printing
init_printing(use_latex='mathjax',use_unicode=False)  # Affichage des résultats

from sympy import limit, sin, S
from sympy.abc import x

limit((sin(x)-x)/x**3, x, 0)

limit(2*x+1, x, S(5)/2)   # la fonction S permet de créer un nombre rationel

limit(1/x, x, 0, dir="-")

limit(1/x, x, 0, dir="+")

limit(1/x, x, 0)

from sympy import oo
oo

5 - oo

oo - oo           # nan signifie "Not A Number"

limit(1/x, x, oo)

limit(4+x*exp(x), x, -oo)

limit((1+1/x)**x, x, oo)

sum([1,2,3,4,5])

from sympy import tan
from sympy.abc import x,z
sum([1,2,3,4,5,x,tan(z)])

from sympy import summation

from sympy.abc import n
summation(n, (n,1,5))

from sympy.abc import a,b
summation(n, (n,1,b))

summation(n, (n,a,b))

from sympy import oo
summation(1/n**2, (n, 1, oo))

summation(n, (n,1,oo))

summation((-1)**n, (n,1,oo))

from sympy.abc import m,n
summation(n*m, (n,1,m), (m,1,10))

summation(1/(n*m)**2, (n,1,oo), (m,1,oo))

from sympy import product
from sympy.abc import n,b
product(n, (n,1,5))

product(n, (n,1,b))

product(n*(n+1), (n, 1, b))

from sympy import diff

from sympy import sin,cos,tan,atan,pi
from sympy.abc import x,y

diff(sin(x), x)

diff(cos(x**3), x)

diff(atan(2*x), x)

diff(1/tan(x), x)

diff(sin(x), x, x, x)

diff(sin(x), x, 3)

diff(x**2*y**3, x, y, y)

from sympy import integrate

integrate(1/x, x)

integrate(1/x, (x, 1, 57))

from sympy import exp
integrate(cos(x)*exp(x), x)

integrate(x**2, (x,0,1))

integrate((x+1)/(x**2+4*x+4), x)

integrate(5*x**2 * exp(x) * sin(x), x)

from sympy import erf
integrate(exp(-x**2)*erf(x), x)

integrate(x**2 * cos(x), x)

integrate(x**2 * cos(x), (x, 0, pi/2))

from sympy import Sum, Product, Derivative, Integral, sin, oo
from sympy.abc import n, x
Sum(1/n**2, (n, 1, oo))

Product(n, (n,1,10))

Derivative(sin(x**2), x)

Integral(1/x**2, (x,1,oo))

Sum(1/n**2, (n, 1, oo)).doit()

Product(n, (n,1,10)).doit()

Derivative(sin(x**2), x).doit()

Integral(1/x**2, (x,1,oo)).doit()

A = Sum(1/n**2, (n, 1, oo))
B = Product(n, (n,1,10))
C = Derivative(sin(x**2), x)
D = Integral(1/x**2, (x,1,oo))
from sympy import Eq
Eq(A, A.doit())

Eq(B, B.doit())

Eq(C, C.doit())

Eq(D, D.doit())

from sympy.abc import x,y
integrate(integrate(x**2+y**2, x), y)

integrate(x**2+y**2, x, y)

integrate(x**2+y**2, (x,0,y), (y,0,10))

from sympy import series, cos
from sympy.abc import x
series(cos(x), x, 0, 14)

series(cos(x), x)

(1/cos(x**2)).series(x, 0, 14)

from sympy import log
series(log(x), x, 0)

series(log(x), x, 1)

from sympy import Function
f = Function("f")

f

from sympy.abc import x
f(x)

from sympy import Derivative
Derivative(f(x), x)             # ordre 1

Derivative(f(x), x, x)          # ordre 2

Eq(f(x), Derivative(f(x),x))

from sympy import dsolve
dsolve(Eq(f(x), Derivative(f(x),x)), f(x))

Eq(f(x), -Derivative(f(x),x,x))

dsolve(Eq(f(x), -Derivative(f(x),x,x)), f(x))

dsolve(Eq(Derivative(f(x),x,x) + 9*f(x), 1), f(x))

dsolve(Eq(f(x).diff(x, x) + 9*f(x), 1), f(x))

from sympy.abc import x,y,t
eq1 = Eq(Derivative(x(t),t), x(t)*y(t)*sin(t))
eq2 = Eq(Derivative(y(t),t), y(t)**2*sin(t))
systeme = [eq1, eq2]
systeme

dsolve(systeme)