%pylab inline
import seaborn
style.use('seaborn-poster')
import sympy
sympy.init_printing()

1 + 1 / (1 + 1 / (1 + 1 / ( 1 + 1/2)))

1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / ( 1 + 1/2))))))))

sympy.GoldenRatio

sympy.GoldenRatio.evalf(100)

str(sympy.Rational(1, 97).evalf(1000))

phi = [sympy.Integer(1)]
for i in range(30):
    phi.append(1 + 1 / phi[-1])
phi, phi[-1].evalf(20)

1 + (1 / sympy.GoldenRatio).evalf(20)

(1/(sympy.GoldenRatio - 1)).evalf(20)

eps = sympy.Symbol(r"\varepsilon")
m2 = sympy.Matrix([[1, 0], [eps, 1]])
m2

m1 = sympy.Matrix([[1, -eps], [0, 1]])
m1

m3 = m2 * m1
m3

[m.det() for m in [m1, m2, m3]]

m3.LUdecomposition()

m3.eigenvects(simplify=True)

m3.inv()

m3.inv().LUdecomposition()

P, D = m3.diagonalize()
P, D, P.inv()

sympy.simplify(P.inv())

print(sympy.latex(m3.inv().LUdecomposition()))

m3.replace(eps, .01).eigenvects()

m3.subs(eps, 2).eigenvects()

m3.subs(eps, 2.01).eigenvects()

m4 = (m3*m3).expand()
m4

m5 = (m4 * m4).expand()
m5

m5.subs(eps, .01)

m6 = (m5 * m5).expand()
m6

m6.subs(eps, .01)

c = m6.dot(sympy.Matrix([[1, 0]]))[0]
c

sympy.plot(c, m6.dot(sympy.Matrix([[1, 0]]))[1], (eps, -1.9, 1.9));

v = sympy.Matrix([[1, 0]])
circle = [v]
mf = m3.subs(eps, 2**-5)
for i in range(256):
    v = mf.dot(v)
    circle.append(v)
    
plot(circle)
circle[:5]

plot([x for x, y in circle], [y for x, y in circle], 'o', markersize=1)
None

sympy.Integral(sympy.sqrt(1/eps), eps)

import sympy.printing.dot
import os
os.popen('dot -Tx11', 'w').write(sympy.printing.dot.dotprint(m6[0]))

x, y = sympy.symbols('x y')
x, y

integ = sympy.Integral(x**y * sympy.exp(-x), (x, 0, sympy.oo))
integ

integ.doit()

print(sympy.latex(integ.doit()))

print(sympy.pretty(integ.doit()))

sympy.srepr(integ.doit())

ido = integ.doit()
ido.func, ido.args, ido.args[0].func

from sympy.functions.elementary.piecewise import Piecewise, ExprCondPair

box = Piecewise(ExprCondPair(sympy.Integer(0), sympy.StrictLessThan(x, 0)),
                ExprCondPair(sympy.Integer(1), sympy.StrictLessThan(x, 1)),
                ExprCondPair(sympy.Integer(0), sympy.S.true))
box

print(sympy.latex(box))

[box.subs(x, i) for i in [-1, -0.5, 0, 0.5, 1, 1.5]]

sympy.plot(x**2, (x, -1, 2))

sympy.plot(box, (x, -1, 2))

sympy.var('ohshit')

ohshit

sympy.symbols('x0 y37 foo1')

x0

[sympy.Eq(f(x), f(x).rewrite(sympy.exp)) for f in [sympy.sin, sympy.cos, sympy.tan]]

sympy.cos(x).rewrite(sympy.exp).factor()

((x+y)**14).expand()

sympy.factorint(3801372, visual=True)

import sympy.utilities.autowrap

sympy.utilities.autowrap??