def test_li():
z = Symbol("z")
zr = Symbol("z", extended_real=True)
zp = Symbol("z", positive=True)
zn = Symbol("z", negative=True)
assert li(0) == 0
assert li(1) == -oo
assert li(oo) == oo
assert isinstance(li(z), li)
assert diff(li(z), z) == 1/log(z)
assert conjugate(li(z)) == li(conjugate(z))
assert conjugate(li(-zr)) == li(-zr)
assert conjugate(li(-zp)) == conjugate(li(-zp))
assert conjugate(li(zn)) == conjugate(li(zn))
assert li(z).rewrite(Li) == Li(z) + li(2)
assert li(z).rewrite(Ei) == Ei(log(z))
assert li(z).rewrite(uppergamma) == (-log(1/log(z))/2 - log(-log(z)) +
log(log(z))/2 - expint(1, -log(z)))
assert li(z).rewrite(Si) == (-log(I*log(z)) - log(1/log(z))/2 +
log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)))
assert li(z).rewrite(Ci) == (-log(I*log(z)) - log(1/log(z))/2 +
log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)))
assert li(z).rewrite(Shi) == (-log(1/log(z))/2 + log(log(z))/2 +
Chi(log(z)) - Shi(log(z)))
assert li(z).rewrite(Chi) == (-log(1/log(z))/2 + log(log(z))/2 +
Chi(log(z)) - Shi(log(z)))
assert li(z).rewrite(hyper) == (log(z)*hyper((1, 1), (2, 2), log(z)) -
log(1/log(z))/2 + log(log(z))/2 + EulerGamma)
assert li(z).rewrite(meijerg) == (-log(1/log(z))/2 - log(-log(z)) + log(log(z))/2 -
meijerg(((), (1,)), ((0, 0), ()), -log(z)))
def test_expint():
""" Test various exponential integrals. """
from diofant import (expint, unpolarify, Symbol, Ci, Si, Shi, Chi, sin,
cos, sinh, cosh, Ei)
assert simplify(
unpolarify(
integrate(exp(-z * x) / x**y, (x, 1, oo),
meijerg=True,
conds='none').rewrite(expint).expand(
func=True))) == expint(y, z)
assert integrate(exp(-z*x)/x, (x, 1, oo), meijerg=True,
conds='none').rewrite(expint).expand() == \
expint(1, z)
assert integrate(exp(-z*x)/x**2, (x, 1, oo), meijerg=True,
conds='none').rewrite(expint).expand() == \
expint(2, z).rewrite(Ei).rewrite(expint)
assert integrate(exp(-z*x)/x**3, (x, 1, oo), meijerg=True,
conds='none').rewrite(expint).expand() == \
expint(3, z).rewrite(Ei).rewrite(expint).expand()
t = Symbol('t', positive=True)
assert integrate(-cos(x) / x, (x, t, oo), meijerg=True).expand() == Ci(t)
assert integrate(-sin(x)/x, (x, t, oo), meijerg=True).expand() == \
Si(t) - pi/2
assert integrate(sin(x) / x, (x, 0, z), meijerg=True) == Si(z)
assert integrate(sinh(x) / x, (x, 0, z), meijerg=True) == Shi(z)
assert integrate(exp(-x)/x, x, meijerg=True).expand().rewrite(expint) == \
I*pi - expint(1, x)
assert integrate(exp(-x)/x**2, x, meijerg=True).rewrite(expint).expand() \
== expint(1, x) - exp(-x)/x - I*pi
u = Symbol('u', polar=True)
assert integrate(cos(u)/u, u, meijerg=True).expand().as_independent(u)[1] \
== Ci(u)
assert integrate(cosh(u)/u, u, meijerg=True).expand().as_independent(u)[1] \
== Chi(u)
assert integrate(
expint(1, x), x,
meijerg=True).rewrite(expint).expand() == x * expint(1, x) - exp(-x)
assert integrate(expint(2, x), x, meijerg=True
).rewrite(expint).expand() == \
-x**2*expint(1, x)/2 + x*exp(-x)/2 - exp(-x)/2
assert simplify(unpolarify(integrate(expint(y, x), x,
meijerg=True).rewrite(expint).expand(func=True))) == \
-expint(y + 1, x)
assert integrate(Si(x), x, meijerg=True) == x * Si(x) + cos(x)
assert integrate(Ci(u), u, meijerg=True).expand() == u * Ci(u) - sin(u)
assert integrate(Shi(x), x, meijerg=True) == x * Shi(x) - cosh(x)
assert integrate(Chi(u), u, meijerg=True).expand() == u * Chi(u) - sinh(u)
assert integrate(Si(x) * exp(-x), (x, 0, oo), meijerg=True) == pi / 4
assert integrate(expint(1, x) * sin(x), (x, 0, oo),
meijerg=True) == log(2) / 2
def test_specfun():
for f in [besselj, bessely, besseli, besselk]:
assert octave_code(f(n, x)) == f.__name__ + '(n, x)'
assert octave_code(hankel1(n, x)) == 'besselh(n, 1, x)'
assert octave_code(hankel2(n, x)) == 'besselh(n, 2, x)'
assert octave_code(airyai(x)) == 'airy(0, x)'
assert octave_code(airyaiprime(x)) == 'airy(1, x)'
assert octave_code(airybi(x)) == 'airy(2, x)'
assert octave_code(airybiprime(x)) == 'airy(3, x)'
assert octave_code(uppergamma(n, x)) == "gammainc(x, n, 'upper')"
assert octave_code(lowergamma(n, x)) == "gammainc(x, n, 'lower')"
assert octave_code(jn(
n, x)) == 'sqrt(2)*sqrt(pi)*sqrt(1./x).*besselj(n + 1/2, x)/2'
assert octave_code(yn(
n, x)) == 'sqrt(2)*sqrt(pi)*sqrt(1./x).*bessely(n + 1/2, x)/2'
assert octave_code(Chi(x)) == 'coshint(x)'
assert octave_code(Ci(x)) == 'cosint(x)'
assert octave_code(laguerre(n, x)) == 'laguerreL(n, x)'
assert octave_code(li(x)) == 'logint(x)'
assert octave_code(loggamma(x)) == 'gammaln(x)'
assert octave_code(polygamma(n, x)) == 'psi(n, x)'
assert octave_code(Shi(x)) == 'sinhint(x)'
assert octave_code(Si(x)) == 'sinint(x)'
assert octave_code(LambertW(x)) == 'lambertw(x)'
assert octave_code(LambertW(x, n)) == 'lambertw(n, x)'
assert octave_code(zeta(x)) == 'zeta(x)'
assert octave_code(zeta(
x, y)) == '% Not supported in Octave:\n% zeta\nzeta(x, y)'
def test_ei():
pos = Symbol('p', positive=True)
neg = Symbol('n', negative=True)
assert Ei(0) == -oo
assert Ei(+oo) == oo
assert Ei(-oo) == 0
assert Ei(-pos) == Ei(polar_lift(-1) * pos) - I * pi
assert Ei(neg) == Ei(polar_lift(neg)) - I * pi
assert tn_branch(Ei)
assert mytd(Ei(x), exp(x) / x, x)
assert mytn(Ei(x),
Ei(x).rewrite(uppergamma),
-uppergamma(0, x * polar_lift(-1)) - I * pi, x)
assert mytn(Ei(x),
Ei(x).rewrite(expint), -expint(1, x * polar_lift(-1)) - I * pi,
x)
assert Ei(x).rewrite(expint).rewrite(Ei) == Ei(x)
assert Ei(x * exp_polar(2 * I * pi)) == Ei(x) + 2 * I * pi
assert Ei(x * exp_polar(-2 * I * pi)) == Ei(x) - 2 * I * pi
assert mytn(Ei(x), Ei(x).rewrite(Shi), Chi(x) + Shi(x), x)
assert mytn(Ei(x * polar_lift(I)),
Ei(x * polar_lift(I)).rewrite(Si),
Ci(x) + I * Si(x) + I * pi / 2, x)
assert Ei(log(x)).rewrite(li) == li(x)
assert Ei(2 * log(x)).rewrite(li) == li(x**2)
assert Ei(x).series(x) == (EulerGamma + log(x) + x + x**2 / 4 + x**3 / 18 +
x**4 / 96 + x**5 / 600 + O(x**6))
assert Ei(1 + x).series(x) == (Ei(1) + E * x + E * x**3 / 6 -
E * x**4 / 12 + 3 * E * x**5 / 40 + O(x**6))
pytest.raises(ArgumentIndexError, lambda: Ei(x).fdiff(2))
def test_messy():
assert laplace_transform(Si(x), x, s) == ((-atan(s) + pi / 2) / s, 0, True)
assert laplace_transform(Shi(x), x, s) == (acoth(s) / s, 1, True)
# where should the logs be simplified?
assert laplace_transform(Chi(x), x, s) == \
((log(s**(-2)) - log((s**2 - 1)/s**2))/(2*s), 1, True)
# TODO maybe simplify the inequalities?
assert laplace_transform(besselj(a, x), x, s)[1:] == \
(0, And(Integer(0) < re(a/2) + Rational(1, 2), Integer(0) < re(a/2) + 1))
# NOTE s < 0 can be done, but argument reduction is not good enough yet
assert fourier_transform(besselj(1, x)/x, x, s, noconds=False) == \
(Piecewise((0, 4*abs(pi**2*s**2) > 1),
(2*sqrt(-4*pi**2*s**2 + 1), True)), s > 0)
# TODO FT(besselj(0,x)) - conditions are messy (but for acceptable reasons)
# - folding could be better
assert integrate(E1(x)*besselj(0, x), (x, 0, oo), meijerg=True) == \
log(1 + sqrt(2))
assert integrate(E1(x)*besselj(1, x), (x, 0, oo), meijerg=True) == \
log(Rational(1, 2) + sqrt(2)/2)
assert integrate(1/x/sqrt(1 - x**2), x, meijerg=True) == \
Piecewise((-acosh(1/x), 1 < abs(x**(-2))), (I*asin(1/x), True))
def test_ei():
pos = Symbol('p', positive=True)
neg = Symbol('n', negative=True)
assert Ei(-pos) == Ei(polar_lift(-1) * pos) - I * pi
assert Ei(neg) == Ei(polar_lift(neg)) - I * pi
assert tn_branch(Ei)
assert mytd(Ei(x), exp(x) / x, x)
assert mytn(Ei(x),
Ei(x).rewrite(uppergamma),
-uppergamma(0, x * polar_lift(-1)) - I * pi, x)
assert mytn(Ei(x),
Ei(x).rewrite(expint), -expint(1, x * polar_lift(-1)) - I * pi,
x)
assert Ei(x).rewrite(expint).rewrite(Ei) == Ei(x)
assert Ei(x * exp_polar(2 * I * pi)) == Ei(x) + 2 * I * pi
assert Ei(x * exp_polar(-2 * I * pi)) == Ei(x) - 2 * I * pi
assert mytn(Ei(x), Ei(x).rewrite(Shi), Chi(x) + Shi(x), x)
assert mytn(Ei(x * polar_lift(I)),
Ei(x * polar_lift(I)).rewrite(Si),
Ci(x) + I * Si(x) + I * pi / 2, x)
assert Ei(log(x)).rewrite(li) == li(x)
assert Ei(2 * log(x)).rewrite(li) == li(x**2)
assert Ei(x).series(x) == EulerGamma + log(x) + x + x**2/4 + \
x**3/18 + x**4/96 + x**5/600 + O(x**6)
def test_expint():
assert mytn(expint(x, y),
expint(x, y).rewrite(uppergamma),
y**(x - 1) * uppergamma(1 - x, y), x)
assert mytd(expint(x, y),
-y**(x - 1) * meijerg([], [1, 1], [0, 0, 1 - x], [], y), x)
assert mytd(expint(x, y), -expint(x - 1, y), y)
assert mytn(expint(1, x),
expint(1, x).rewrite(Ei), -Ei(x * polar_lift(-1)) + I * pi, x)
assert expint(-4, x) == exp(-x)/x + 4*exp(-x)/x**2 + 12*exp(-x)/x**3 \
+ 24*exp(-x)/x**4 + 24*exp(-x)/x**5
assert expint(-Rational(3, 2), x) == \
exp(-x)/x + 3*exp(-x)/(2*x**2) - 3*sqrt(pi)*erf(sqrt(x))/(4*x**Rational(5, 2)) \
+ 3*sqrt(pi)/(4*x**Rational(5, 2))
assert tn_branch(expint, 1)
assert tn_branch(expint, 2)
assert tn_branch(expint, 3)
assert tn_branch(expint, 1.7)
assert tn_branch(expint, pi)
assert expint(y, x*exp_polar(2*I*pi)) == \
x**(y - 1)*(exp(2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x)
assert expint(y, x*exp_polar(-2*I*pi)) == \
x**(y - 1)*(exp(-2*I*pi*y) - 1)*gamma(-y + 1) + expint(y, x)
assert expint(2,
x * exp_polar(2 * I * pi)) == 2 * I * pi * x + expint(2, x)
assert expint(2, x *
exp_polar(-2 * I * pi)) == -2 * I * pi * x + expint(2, x)
assert expint(1, x).rewrite(Ei).rewrite(expint) == expint(1, x)
assert mytn(E1(x), E1(x).rewrite(Shi), Shi(x) - Chi(x), x)
assert mytn(E1(polar_lift(I) * x),
E1(polar_lift(I) * x).rewrite(Si),
-Ci(x) + I * Si(x) - I * pi / 2, x)
assert mytn(expint(2, x),
expint(2, x).rewrite(Ei).rewrite(expint), -x * E1(x) + exp(-x),
x)
assert mytn(expint(3, x),
expint(3, x).rewrite(Ei).rewrite(expint),
x**2 * E1(x) / 2 + (1 - x) * exp(-x) / 2, x)
assert expint(Rational(3, 2), z).nseries(z, n=10) == \
2 + 2*z - z**2/3 + z**3/15 - z**4/84 + z**5/540 - \
2*sqrt(pi)*sqrt(z) + O(z**6)
assert E1(z).series(z) == -EulerGamma - log(z) + z - \
z**2/4 + z**3/18 - z**4/96 + z**5/600 + O(z**6)
assert expint(4, z).series(z) == Rational(1, 3) - z/2 + z**2/2 + \
z**3*(log(z)/6 - Rational(11, 36) + EulerGamma/6) - z**4/24 + \
z**5/240 + O(z**6)
def test_si():
assert Si(I * x) == I * Shi(x)
assert Shi(I * x) == I * Si(x)
assert Si(-I * x) == -I * Shi(x)
assert Shi(-I * x) == -I * Si(x)
assert Si(-x) == -Si(x)
assert Shi(-x) == -Shi(x)
assert Si(exp_polar(2 * pi * I) * x) == Si(x)
assert Si(exp_polar(-2 * pi * I) * x) == Si(x)
assert Shi(exp_polar(2 * pi * I) * x) == Shi(x)
assert Shi(exp_polar(-2 * pi * I) * x) == Shi(x)
assert Si(oo) == pi / 2
assert Si(-oo) == -pi / 2
assert Shi(oo) == oo
assert Shi(-oo) == -oo
assert mytd(Si(x), sin(x) / x, x)
assert mytd(Shi(x), sinh(x) / x, x)
assert mytn(
Si(x),
Si(x).rewrite(Ei),
-I * (-Ei(x * exp_polar(-I * pi / 2)) / 2 +
Ei(x * exp_polar(I * pi / 2)) / 2 - I * pi) + pi / 2, x)
assert mytn(
Si(x),
Si(x).rewrite(expint),
-I * (-expint(1, x * exp_polar(-I * pi / 2)) / 2 +
expint(1, x * exp_polar(I * pi / 2)) / 2) + pi / 2, x)
assert mytn(Shi(x),
Shi(x).rewrite(Ei),
Ei(x) / 2 - Ei(x * exp_polar(I * pi)) / 2 + I * pi / 2, x)
assert mytn(
Shi(x),
Shi(x).rewrite(expint),
expint(1, x) / 2 - expint(1, x * exp_polar(I * pi)) / 2 - I * pi / 2,
x)
assert tn_arg(Si)
assert tn_arg(Shi)
assert Si(x).nseries(x, n=8) == \
x - x**3/18 + x**5/600 - x**7/35280 + O(x**9)
assert Shi(x).nseries(x, n=8) == \
x + x**3/18 + x**5/600 + x**7/35280 + O(x**9)
assert Si(sin(x)).nseries(
x, n=5) == x - 2 * x**3 / 9 + 17 * x**5 / 450 + O(x**7)
assert Si(x).series(x, 1, n=3) == \
Si(1) + (x - 1)*sin(1) + (x - 1)**2*(-sin(1)/2 + cos(1)/2) + O((x - 1)**3, (x, 1))
pytest.raises(ArgumentIndexError, lambda: Si(z).fdiff(2))