Sympy¶

import sympy as smp

x = smp.symbols("x")
x = smp.symbols("x", real=True, positive=True)

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

f = smp.symbols("f", cls=smp.Function)

x = smp.Rational(5, 1)
frac = smp.Rational(1, 2)

y = smp.sin(x)
z = x**2 + y**2

z.factor()
z.expand()
z.simplify()

smp.solve(z, x) ## find value of x that makes z(x) = 0 
smp.solve(z, y)

expr = smp.sin(x) + smp.sin(y)
expr_f = smp.lambdify([x, y], expr)

expr = smp.sin(x) + smp.sin(y)
expr.subs([
    (x, 10)
])
expr.subs([
  (x, 10),
  (y, 5)
])
expr.subs([
  (x, 10),
  (y, smp.sin(x))
])

dfdx = smp.diff(f) ## f = function symbol, which is a function of  
dfdx_sub = dfdx.sub([
    (g, smp.sin(x))
])

dfdx_sub_value = dfdx_sub.doit()

## does not give +c
smp.integrate(
  expr,
  x
)

smp.integrate(
  expr,
  (x, 0, smp.log(4))
)

u1, u2, u3 = smp.symbols("u1 u2 u3")
u = smp.Matrix([u1, u2, u3])

v1, v2, v3 = smp.symbols("v1 v2 v3")
v = smp.Matrix([v1, v2, v3])

2*u + v
u.norm()

u.dot(v)
u.cross(v)

# symbols need to be defined with correct characteristics

t, f = smp.symbols("t, f", real=True)
k = smp.symbols("k", real=True, positive=True)
x = smp.exp(-k * t**2) * k * t
x

from sympy.integrals.transforms import fourier_transform as ft
x_FT = ft(x, t, f)

t = smp.symbols("t", real=True)
k, n, T = smp.symbols("k, n, T", real=True, positive=True)
fn = n/T
x = smp.exp(-k * t)

x_FT = smp.integrate(
  1/T * x * smp.exp(-2*smp.pi*smp.I*fn*t),
  (t, 0, T)
).simplify()

get_FT = smp.lambdify([k, T, n], x_FT)
ns = np.arange(0, 20, 1)
xFT = get_FT(k=1, T=4, n=ns)