%pylab inline

from sympy import *

init_printing()

# or with older versions of sympy/ipython, load the IPython extension
#%load_ext sympy.interactive.ipythonprinting
# or
#%load_ext sympyprinting

x = Symbol('x')

(pi + x)**2

# alternative way of defining symbols
a, b, c = symbols("a, b, c")

type(a)

x = Symbol('x', real=True)

x.is_imaginary

x = Symbol('x', positive=True)

x > 0

1+1*I

I**2

(x * I + 1)**2

r1 = Rational(4,5)
r2 = Rational(5,4)

r1

r1+r2

r1/r2

pi.evalf(n=50)

y = (x + pi)**2

N(y, 5) # same as evalf

y.subs(x, 1.5)

N(y.subs(x, 1.5))

y.subs(x, a+pi)

import numpy

x_vec = numpy.arange(0, 10, 0.1)

y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])

fig, ax = subplots()
ax.plot(x_vec, y_vec);

f = lambdify([x], (x + pi)**2, 'numpy')  # the first argument is a list of variables that
                                         # f will be a function of: in this case only x -> f(x)

y_vec = f(x_vec)  # now we can directly pass a numpy array and f(x) is efficiently evaluated

%%timeit

y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])

%%timeit

y_vec = f(x_vec)

(x+1)*(x+2)*(x+3)

expand((x+1)*(x+2)*(x+3))

sin(a+b)

expand(sin(a+b), trig=True)

factor(x**3 + 6 * x**2 + 11*x + 6)

# simplify expands a product
simplify((x+1)*(x+2)*(x+3))

# simplify uses trigonometric identities
simplify(sin(a)**2 + cos(a)**2)

simplify(cos(x)/sin(x))

f1 = 1/((a+1)*(a+2))

f1

apart(f1)

f2 = 1/(a+2) + 1/(a+3)

f2

together(f2)

simplify(f2)

y

diff(y**2, x)

diff(y**2, x, x)

diff(y**2, x, 2) # same as above

x, y, z = symbols("x,y,z")

f = sin(x*y) + cos(y*z)

diff(f, x, 1, y, 2)

f

integrate(f, x)

integrate(f, (x, -1, 1))

integrate(exp(-x**2), (x, -oo, oo))

n = Symbol("n")

Sum(1/n**2, (n, 1, 10))

Sum(1/n**2, (n,1, 10)).evalf()

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

Product(n, (n, 1, 10)) # 10!

limit(sin(x)/x, x, 0)

f

diff(f, x)

h = Symbol("h")

limit((f.subs(x, x+h) - f)/h, h, 0)

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

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

series(exp(x), x)

series(exp(x), x, 1)

series(exp(x), x, 1, 10)

s1 = cos(x).series(x, 0, 5)
s1

s2 = sin(x).series(x, 0, 2)
s2

expand(s1 * s2)

expand(s1.removeO() * s2.removeO())

(cos(x)*sin(x)).series(x, 0, 6)

m11, m12, m21, m22 = symbols("m11, m12, m21, m22")
b1, b2 = symbols("b1, b2")

A = Matrix([[m11, m12],[m21, m22]])
A

b = Matrix([[b1], [b2]])
b

A**2

A * b

A.det()

A.inv()

solve(x**2 - 1, x)

solve(x**4 - x**2 - 1, x)

solve([x + y - 1, x - y - 1], [x,y])

solve([x + y - a, x - y - c], [x,y])