p = x**2 + 4*x - 6
glg = Eq(p, 0)
solveset(glg)

p = x**3 - x**2 + 5
glg = Eq(p, 0)
lsg = solveset(glg)
lsg

w = list(lsg)[1]
w

w.simplify()

for w in lsg:
    display(w.n())
    print(w.is_real)

p2 = x**4 - x**3 - 5
glg2 = Eq(p2, 0)
lsg2 = solveset(glg2)
w = list(lsg2)[0]
w

w.radsimp()

for w in lsg2:
    display(w.n())
    print(w.is_real)

p3 = x**5 - x**4 - 5
glg3 = Eq(p3, 0)
lsg3 = solveset(glg3)
lsg3

summe = 0
for w in lsg3:
    display(w.n())
    print(w.is_real)
    summe += w
summe

summe.radsimp()

summe.n()

minimal_polynomial(summe)  

x = S('x')
ugl = x**2 < 5
ugl

solve(ugl)

solveset(ugl)

m1 = solveset(ugl, domain=Reals)
m1

print(m1)

m2 = solveset(exp(x) >= E**(-1), domain=Reals)
m2

print(m2)

m1 & m2

print(m1 & m2)

Interval(1, oo) == Interval.Ropen(1, oo)

Interval.open(1, oo) == Interval.Lopen(1, oo)

solveset(exp(x) > 0, domain=Reals)

solveset(exp(x) == 0)

print(_)  #  symbolische leere Menge, nicht zu verwechslen mit set()

solveset(3*log(x) < x, domain=Reals)

plot(3*log(x), x);

glg = Eq(3*log(x), x)
lsg = solve(glg)
lsg

print(lsg[0])

w = S('w')
glgW = Eq(x, w*exp(w))
glgW

lsg = solve(glgW, w, dict=True)
lsg

glgW.subs(lsg[0]).simplify()

glg = Eq(exp(x), 5*x**2)
lsg = solve(glg)
lsg

plot(exp(x), 5*x**2, (x, -1, 1));

a = S('a')
f = exp(-a*x**2)
I1 = Integral(f, (x, -oo, oo))
I1

I1.doit()

a = Symbol('a', positive=True)
I2 = I1.subs(S('a'), a)
I2

I2.doit()

z = exp(I*pi/2)
z

arg(z)

arg(-1), arg(1), arg(-1-I/100)

phi = Symbol('phi', real=True)
plot(arg(exp(I*phi)));

f = sin(x)**2
F = Integral(f, x).doit()
F

plot(f, F);

f = 1/(2-cos(x))
I2 = Integral(f, x)
I2

F = I2.doit()
F

print(F)

floor(3.5)

floor(-3.5)

plot(f, F);

g = atan(sqrt(3)*tan(x/2))
g

plot(g);

F.diff(x)

F.diff(x).simplify()

def null(x):
    return 0

F2 = F.replace(floor, null)
F2.diff().simplify()

a = S('a')
f = 1 / (a - sin(2*pi*x))
I1 = Integral(f, (x, 0, 1))
I1

I1.doit()

plot(f.subs(a, 2), 0);