def test_binomial(): n = Symbol('n', integer=True) k = Symbol('k', integer=True) u = Symbol('v', negative=True) v = Symbol('m', positive=True) assert binomial(0, 0) == 1 assert binomial(1, 1) == 1 assert binomial(10, 10) == 1 assert binomial(1, 2) == 0 assert binomial(1, -1) == 0 assert binomial(-1, 1) == -1 assert binomial(-10, 1) == -10 assert binomial(-10, 7) == -11440 assert binomial(n, -1) == 0 assert binomial(n, 0) == 1 assert expand_func(binomial(n, 1)) == n assert expand_func(binomial(n, 2)) == n*(n - 1)/2 assert expand_func(binomial(n, n - 2)) == n*(n - 1)/2 assert expand_func(binomial(n, n - 1)) == n assert binomial(n, 3).func == binomial assert binomial(n, 3).expand(func=True) == n**3/6 - n**2/2 + n/3 assert expand_func(binomial(n, 3)) == n*(n - 2)*(n - 1)/6 assert binomial(n, n) == 1 assert binomial(n, n + 1) == 0 assert binomial(n, u) == 0 assert binomial(n, v).func == binomial assert binomial(n, k).func == binomial assert binomial(n, n + v) == 0
def test_gamma(): assert gamma(nan) == nan assert gamma(oo) == oo assert gamma(-100) == zoo assert gamma(0) == zoo assert gamma(1) == 1 assert gamma(2) == 1 assert gamma(3) == 2 assert gamma(102) == factorial(101) assert gamma(Rational(1, 2)) == sqrt(pi) assert gamma(Rational(3, 2)) == Rational(1, 2) * sqrt(pi) assert gamma(Rational(5, 2)) == Rational(3, 4) * sqrt(pi) assert gamma(Rational(7, 2)) == Rational(15, 8) * sqrt(pi) assert gamma(Rational(-1, 2)) == -2 * sqrt(pi) assert gamma(Rational(-3, 2)) == Rational(4, 3) * sqrt(pi) assert gamma(Rational(-5, 2)) == -Rational(8, 15) * sqrt(pi) assert gamma(Rational(-15, 2)) == Rational(256, 2027025) * sqrt(pi) assert gamma(x).diff(x) == gamma(x) * polygamma(0, x) assert gamma(x - 1).expand(func=True) == gamma(x) / (x - 1) assert gamma(x + 2).expand(func=True, mul=False) == x * (x + 1) * gamma(x) assert expand_func(gamma(x + Rational(3, 2))) == (x + Rational(1, 2)) * gamma(x + Rational(1, 2)) assert expand_func(gamma(x - Rational(1, 2))) == gamma(Rational(1, 2) + x) / (x - Rational(1, 2))
def test_gamma(): assert gamma(nan) == nan assert gamma(oo) == oo assert gamma(-100) == zoo assert gamma(0) == zoo assert gamma(1) == 1 assert gamma(2) == 1 assert gamma(3) == 2 assert gamma(102) == factorial(101) assert gamma(Rational(1, 2)) == sqrt(pi) assert gamma(Rational(3, 2)) == Rational(1, 2)*sqrt(pi) assert gamma(Rational(5, 2)) == Rational(3, 4)*sqrt(pi) assert gamma(Rational(7, 2)) == Rational(15, 8)*sqrt(pi) assert gamma(Rational(-1, 2)) == -2*sqrt(pi) assert gamma(Rational(-3, 2)) == Rational(4, 3)*sqrt(pi) assert gamma(Rational(-5, 2)) == -Rational(8, 15)*sqrt(pi) assert gamma(Rational(-15, 2)) == Rational(256, 2027025)*sqrt(pi) assert gamma(Rational( -11, 8)).expand(func=True) == Rational(64, 33)*gamma(Rational(5, 8)) assert gamma(Rational( -10, 3)).expand(func=True) == Rational(81, 280)*gamma(Rational(2, 3)) assert gamma(Rational( 14, 3)).expand(func=True) == Rational(880, 81)*gamma(Rational(2, 3)) assert gamma(Rational( 17, 7)).expand(func=True) == Rational(30, 49)*gamma(Rational(3, 7)) assert gamma(Rational( 19, 8)).expand(func=True) == Rational(33, 64)*gamma(Rational(3, 8)) assert gamma(x).diff(x) == gamma(x)*polygamma(0, x) assert gamma(x - 1).expand(func=True) == gamma(x)/(x - 1) assert gamma(x + 2).expand(func=True, mul=False) == x*(x + 1)*gamma(x) assert conjugate(gamma(x)) == gamma(conjugate(x)) assert expand_func(gamma(x + Rational(3, 2))) == \ (x + Rational(1, 2))*gamma(x + Rational(1, 2)) assert expand_func(gamma(x - Rational(1, 2))) == \ gamma(Rational(1, 2) + x)/(x - Rational(1, 2)) # Test a bug: assert expand_func(gamma(x + Rational(3, 4))) == gamma(x + Rational(3, 4)) assert gamma(3*exp_polar(I*pi)/4).is_nonnegative is False assert gamma(3*exp_polar(I*pi)/4).is_nonpositive is True # Issue 8526 k = Symbol('k', integer=True, nonnegative=True) assert isinstance(gamma(k), gamma) assert gamma(-k) == zoo
def test_beta(): x, y = Symbol('x'), Symbol('y') assert isinstance(beta(x, y), beta) assert expand_func(beta(x, y)) == gamma(x)*gamma(y)/gamma(x + y) assert expand_func(beta(x, y) - beta(y, x)) == 0 # Symmetric assert expand_func(beta(x, y)) == expand_func(beta(x, y + 1) + beta(x + 1, y)).simplify() assert diff(beta(x, y), x) == beta(x, y)*(digamma(x) - digamma(x + y)) assert diff(beta(x, y), y) == beta(x, y)*(digamma(y) - digamma(x + y))
def test_harmonic_rewrite_polygamma(): n = Symbol("n") m = Symbol("m") assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma assert harmonic(n).rewrite(trigamma) == polygamma(0, n + 1) + EulerGamma assert harmonic(n).rewrite(polygamma) == polygamma(0, n + 1) + EulerGamma assert harmonic(n,3).rewrite(polygamma) == polygamma(2, n + 1)/2 - polygamma(2, 1)/2 assert harmonic(n,m).rewrite(polygamma) == (-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1) assert expand_func(harmonic(n+4)) == harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1) assert expand_func(harmonic(n-4)) == harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n assert harmonic(n, m).rewrite("tractable") == harmonic(n, m).rewrite(polygamma)
def test_issue1893(): from sympy import simplify, expand_func, polygamma, gamma a = Symbol("a", positive=True) assert simplify(expand_func(integrate(exp(-x) * log(x) * x ** a, (x, 0, oo)))) == (a * polygamma(0, a) + 1) * gamma( a )
def test_erf2(): assert erf2(0, 0) == S.Zero assert erf2(x, x) == S.Zero assert erf2(nan, 0) == nan assert erf2(-oo, y) == erf(y) + 1 assert erf2( oo, y) == erf(y) - 1 assert erf2( x, oo) == 1 - erf(x) assert erf2( x,-oo) == -1 - erf(x) assert erf2(x, erf2inv(x, y)) == y assert erf2(-x, -y) == -erf2(x,y) assert erf2(-x, y) == erf(y) + erf(x) assert erf2( x, -y) == -erf(y) - erf(x) assert erf2(x, y).rewrite('fresnels') == erf(y).rewrite(fresnels)-erf(x).rewrite(fresnels) assert erf2(x, y).rewrite('fresnelc') == erf(y).rewrite(fresnelc)-erf(x).rewrite(fresnelc) assert erf2(x, y).rewrite('hyper') == erf(y).rewrite(hyper)-erf(x).rewrite(hyper) assert erf2(x, y).rewrite('meijerg') == erf(y).rewrite(meijerg)-erf(x).rewrite(meijerg) assert erf2(x, y).rewrite('uppergamma') == erf(y).rewrite(uppergamma) - erf(x).rewrite(uppergamma) assert erf2(x, y).rewrite('expint') == erf(y).rewrite(expint)-erf(x).rewrite(expint) assert erf2(I, 0).is_real is False assert erf2(0, 0).is_real is True assert expand_func(erf(x) + erf2(x, y)) == erf(y) assert conjugate(erf2(x, y)) == erf2(conjugate(x), conjugate(y)) assert erf2(x, y).rewrite('erf') == erf(y) - erf(x) assert erf2(x, y).rewrite('erfc') == erfc(x) - erfc(y) assert erf2(x, y).rewrite('erfi') == I*(erfi(I*x) - erfi(I*y)) raises(ArgumentIndexError, lambda: erfi(x).fdiff(3))
def test_airyai(): z = Symbol('z', real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airyai(z), airyai) assert airyai(0) == 3**(S(1)/3)/(3*gamma(S(2)/3)) assert airyai(oo) == 0 assert airyai(-oo) == 0 assert diff(airyai(z), z) == airyaiprime(z) assert series(airyai(z), z, 0, 3) == ( 3**(S(5)/6)*gamma(S(1)/3)/(6*pi) - 3**(S(1)/6)*z*gamma(S(2)/3)/(2*pi) + O(z**3)) assert airyai(z).rewrite(hyper) == ( -3**(S(2)/3)*z*hyper((), (S(4)/3,), z**S(3)/9)/(3*gamma(S(1)/3)) + 3**(S(1)/3)*hyper((), (S(2)/3,), z**S(3)/9)/(3*gamma(S(2)/3))) assert isinstance(airyai(z).rewrite(besselj), airyai) assert airyai(t).rewrite(besselj) == ( sqrt(-t)*(besselj(-S(1)/3, 2*(-t)**(S(3)/2)/3) + besselj(S(1)/3, 2*(-t)**(S(3)/2)/3))/3) assert airyai(z).rewrite(besseli) == ( -z*besseli(S(1)/3, 2*z**(S(3)/2)/3)/(3*(z**(S(3)/2))**(S(1)/3)) + (z**(S(3)/2))**(S(1)/3)*besseli(-S(1)/3, 2*z**(S(3)/2)/3)/3) assert airyai(p).rewrite(besseli) == ( sqrt(p)*(besseli(-S(1)/3, 2*p**(S(3)/2)/3) - besseli(S(1)/3, 2*p**(S(3)/2)/3))/3) assert expand_func(airyai(2*(3*z**5)**(S(1)/3))) == ( -sqrt(3)*(-1 + (z**5)**(S(1)/3)/z**(S(5)/3))*airybi(2*3**(S(1)/3)*z**(S(5)/3))/6 + (1 + (z**5)**(S(1)/3)/z**(S(5)/3))*airyai(2*3**(S(1)/3)*z**(S(5)/3))/2)
def test_airybiprime(): z = Symbol('z', real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airybiprime(z), airybiprime) assert airybiprime(0) == 3**(S(1)/6)/gamma(S(1)/3) assert airybiprime(oo) == oo assert airybiprime(-oo) == 0 assert diff(airybiprime(z), z) == z*airybi(z) assert series(airybiprime(z), z, 0, 3) == ( 3**(S(1)/6)/gamma(S(1)/3) + 3**(S(5)/6)*z**2/(6*gamma(S(2)/3)) + O(z**3)) assert airybiprime(z).rewrite(hyper) == ( 3**(S(5)/6)*z**2*hyper((), (S(5)/3,), z**S(3)/9)/(6*gamma(S(2)/3)) + 3**(S(1)/6)*hyper((), (S(1)/3,), z**S(3)/9)/gamma(S(1)/3)) assert isinstance(airybiprime(z).rewrite(besselj), airybiprime) assert airyai(t).rewrite(besselj) == ( sqrt(-t)*(besselj(-S(1)/3, 2*(-t)**(S(3)/2)/3) + besselj(S(1)/3, 2*(-t)**(S(3)/2)/3))/3) assert airybiprime(z).rewrite(besseli) == ( sqrt(3)*(z**2*besseli(S(2)/3, 2*z**(S(3)/2)/3)/(z**(S(3)/2))**(S(2)/3) + (z**(S(3)/2))**(S(2)/3)*besseli(-S(2)/3, 2*z**(S(3)/2)/3))/3) assert airybiprime(p).rewrite(besseli) == ( sqrt(3)*p*(besseli(-S(2)/3, 2*p**(S(3)/2)/3) + besseli(S(2)/3, 2*p**(S(3)/2)/3))/3) assert expand_func(airybiprime(2*(3*z**5)**(S(1)/3))) == ( sqrt(3)*(z**(S(5)/3)/(z**5)**(S(1)/3) - 1)*airyaiprime(2*3**(S(1)/3)*z**(S(5)/3))/2 + (z**(S(5)/3)/(z**5)**(S(1)/3) + 1)*airybiprime(2*3**(S(1)/3)*z**(S(5)/3))/2)
def test_weibull_numeric(): # Test for integers and rationals a = 1 bvals = [S.Half, 1, S(3)/2, 5] for b in bvals: X = Weibull('x', a, b) assert simplify(E(X)) == expand_func(a * gamma(1 + 1/S(b))) assert simplify(variance(X)) == simplify( a**2 * gamma(1 + 2/S(b)) - E(X)**2)
def test_gamma(): assert gamma(nan) == nan assert gamma(oo) == oo assert gamma(-100) == zoo assert gamma(0) == zoo assert gamma(1) == 1 assert gamma(2) == 1 assert gamma(3) == 2 assert gamma(102) == factorial(101) assert gamma(Rational(1,2)) == sqrt(pi) assert gamma(Rational(3, 2)) == Rational(1, 2)*sqrt(pi) assert gamma(Rational(5, 2)) == Rational(3, 4)*sqrt(pi) assert gamma(Rational(7, 2)) == Rational(15, 8)*sqrt(pi) assert gamma(Rational(-1, 2)) == -2*sqrt(pi) assert gamma(Rational(-3, 2)) == Rational(4, 3)*sqrt(pi) assert gamma(Rational(-5, 2)) == -Rational(8, 15)*sqrt(pi) assert gamma(Rational(-15, 2)) == Rational(256, 2027025)*sqrt(pi) assert gamma(Rational(-11, 8)).expand(func=True) == Rational(64, 33)*gamma(Rational(5, 8)) assert gamma(Rational(-10, 3)).expand(func=True) == Rational(81, 280)*gamma(Rational(2, 3)) assert gamma(Rational(14, 3)).expand(func=True) == Rational(880, 81)*gamma(Rational(2, 3)) assert gamma(Rational(17, 7)).expand(func=True) == Rational(30, 49)*gamma(Rational(3, 7)) assert gamma(Rational(19, 8)).expand(func=True) == Rational(33, 64)*gamma(Rational(3, 8)) assert gamma(x).diff(x) == gamma(x)*polygamma(0, x) assert gamma(x - 1).expand(func=True) == gamma(x)/(x-1) assert gamma(x + 2).expand(func=True, mul=False) == x*(x+1)*gamma(x) assert expand_func(gamma(x + Rational(3, 2))) == \ (x + Rational(1, 2))*gamma(x + Rational(1, 2)) assert expand_func(gamma(x - Rational(1, 2))) == \ gamma(Rational(1, 2) + x)/(x - Rational(1, 2)) # Test a bug: assert expand_func(gamma(x + Rational(3, 4))) == gamma(x + Rational(3, 4))
def test_harmonic_rewrite_sum_fail(): n = Symbol("n") m = Symbol("m") assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma assert harmonic(n).rewrite(trigamma) == polygamma(0, n + 1) + EulerGamma assert harmonic(n).rewrite(polygamma) == polygamma(0, n + 1) + EulerGamma assert harmonic(n,3).rewrite(polygamma) == polygamma(2, n + 1)/2 - polygamma(2, 1)/2 assert harmonic(n,m).rewrite(polygamma) == (-1)**m*(polygamma(m - 1, 1) - polygamma(m - 1, n + 1))/factorial(m - 1) assert expand_func(harmonic(n+4)) == harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1) assert expand_func(harmonic(n-4)) == harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n assert harmonic(n, m).rewrite("tractable") == harmonic(n, m).rewrite(polygamma) _k = Dummy("k") assert harmonic(n).rewrite(Sum) == Sum(1/_k, (_k, 1, n)) assert harmonic(n, m).rewrite(Sum) == Sum(_k**(-m), (_k, 1, n))
def test_beta(): a, b = symbols('alpha beta', positive=True) B = Beta('x', a, b) assert pspace(B).domain.set == Interval(0, 1) dens = density(B) x = Symbol('x') assert dens(x) == x**(a - 1)*(1 - x)**(b - 1) / beta(a, b) assert simplify(E(B)) == a / (a + b) assert simplify(variance(B)) == a*b / (a**3 + 3*a**2*b + a**2 + 3*a*b**2 + 2*a*b + b**3 + b**2) # Full symbolic solution is too much, test with numeric version a, b = 1, 2 B = Beta('x', a, b) assert expand_func(E(B)) == a / S(a + b) assert expand_func(variance(B)) == (a*b) / S((a + b)**2 * (a + b + 1))
def test_beta(): a, b = symbols("alpha beta", positive=True) B = Beta("x", a, b) assert pspace(B).domain.set == Interval(0, 1) dens = density(B) x = Symbol("x") assert dens(x) == x ** (a - 1) * (1 - x) ** (b - 1) / beta(a, b) # This is too slow # assert E(B) == a / (a + b) # assert variance(B) == (a*b) / ((a+b)**2 * (a+b+1)) # Full symbolic solution is too much, test with numeric version a, b = 1, 2 B = Beta("x", a, b) assert expand_func(E(B)) == a / S(a + b) assert expand_func(variance(B)) == (a * b) / S((a + b) ** 2 * (a + b + 1))
def myexpand(func, target): expanded = expand_func(func) if target is not None: return expanded == target if expanded == func: # it didn't expand return False # check to see that the expanded and original evaluate to the same value subs = {} for a in func.free_symbols: subs[a] = randcplx() return abs(func.subs(subs).n() - expanded.replace(exp_polar, exp).subs(subs).n()) < 1e-10
def test_yn(): z = symbols("z") assert myn(0, z) == -cos(z)/z assert myn(1, z) == -cos(z)/z**2 - sin(z)/z assert myn(2, z) == -((3/z**3 - 1/z)*cos(z) + (3/z**2)*sin(z)) assert expand_func(yn(n, z)) == yn(n, z) # SBFs not defined for complex-valued orders assert yn(2+3j, 5.2+0.3j).evalf() == yn(2+3j, 5.2+0.3j) assert eq([yn(2, 5.2+0.3j).evalf(10)], [0.185250342 + 0.01489557397*I])
def test_polylog_expansion(): from sympy import log assert polylog(s, 0) == 0 assert polylog(s, 1) == zeta(s) assert polylog(s, -1) == -dirichlet_eta(s) assert polylog(s, exp_polar(4*I*pi/3)) == polylog(s, exp(4*I*pi/3)) assert polylog(s, exp_polar(I*pi)/3) == polylog(s, exp(I*pi)/3) assert myexpand(polylog(1, z), -log(1 - z)) assert myexpand(polylog(0, z), z/(1 - z)) assert myexpand(polylog(-1, z), z/(1 - z)**2) assert ((1-z)**3 * expand_func(polylog(-2, z))).simplify() == z*(1 + z) assert myexpand(polylog(-5, z), None)
def test_erf2(): assert erf2(0, 0) == S.Zero assert erf2(x, x) == S.Zero assert erf2(nan, 0) == nan assert erf2(-oo, y) == erf(y) + 1 assert erf2(oo, y) == erf(y) - 1 assert erf2(x, oo) == 1 - erf(x) assert erf2(x, -oo) == -1 - erf(x) assert erf2(x, erf2inv(x, y)) == y assert erf2(-x, -y) == -erf2(x, y) assert erf2(-x, y) == erf(y) + erf(x) assert erf2(x, -y) == -erf(y) - erf(x) assert erf2(x, y).rewrite( 'fresnels') == erf(y).rewrite(fresnels) - erf(x).rewrite(fresnels) assert erf2(x, y).rewrite( 'fresnelc') == erf(y).rewrite(fresnelc) - erf(x).rewrite(fresnelc) assert erf2( x, y).rewrite('hyper') == erf(y).rewrite(hyper) - erf(x).rewrite(hyper) assert erf2(x, y).rewrite( 'meijerg') == erf(y).rewrite(meijerg) - erf(x).rewrite(meijerg) assert erf2( x, y).rewrite('uppergamma' ) == erf(y).rewrite(uppergamma) - erf(x).rewrite(uppergamma) assert erf2( x, y).rewrite('expint') == erf(y).rewrite(expint) - erf(x).rewrite(expint) assert erf2(I, 0).is_real is False assert erf2(0, 0).is_real is True assert expand_func(erf(x) + erf2(x, y)) == erf(y) assert conjugate(erf2(x, y)) == erf2(conjugate(x), conjugate(y)) assert erf2(x, y).rewrite('erf') == erf(y) - erf(x) assert erf2(x, y).rewrite('erfc') == erfc(x) - erfc(y) assert erf2(x, y).rewrite('erfi') == I * (erfi(I * x) - erfi(I * y)) assert erf2(x, y).diff(x) == erf2(x, y).fdiff(1) assert erf2(x, y).diff(y) == erf2(x, y).fdiff(2) assert erf2(x, y).diff(x) == -2 * exp(-x**2) / sqrt(pi) assert erf2(x, y).diff(y) == 2 * exp(-y**2) / sqrt(pi) raises(ArgumentIndexError, lambda: erf2(x, y).fdiff(3)) assert erf2(x, y).is_extended_real is None xr, yr = symbols('xr yr', extended_real=True) assert erf2(xr, yr).is_extended_real is True
def test_gamma(): assert gamma(nan) == nan assert gamma(oo) == oo assert gamma(-100) == zoo assert gamma(0) == zoo assert gamma(1) == 1 assert gamma(2) == 1 assert gamma(3) == 2 assert gamma(102) == factorial(101) assert gamma(Rational(1, 2)) == sqrt(pi) assert gamma(Rational(3, 2)) == Rational(1, 2) * sqrt(pi) assert gamma(Rational(5, 2)) == Rational(3, 4) * sqrt(pi) assert gamma(Rational(7, 2)) == Rational(15, 8) * sqrt(pi) assert gamma(Rational(-1, 2)) == -2 * sqrt(pi) assert gamma(Rational(-3, 2)) == Rational(4, 3) * sqrt(pi) assert gamma(Rational(-5, 2)) == -Rational(8, 15) * sqrt(pi) assert gamma(Rational(-15, 2)) == Rational(256, 2027025) * sqrt(pi) assert gamma(x).diff(x) == gamma(x) * polygamma(0, x) assert gamma(x - 1).expand(func=True) == gamma(x) / (x - 1) assert gamma(x + 2).expand(func=True, mul=False) == x * (x + 1) * gamma(x) assert expand_func(gamma(x + Rational(3, 2))) == \ (x + Rational(1, 2))*gamma(x + Rational(1, 2)) assert expand_func(gamma(x - Rational(1, 2))) == \ gamma(Rational(1, 2) + x)/(x - Rational(1, 2)) # Test a bug: assert expand_func(gamma(x + Rational(3, 4))) == gamma(x + Rational(3, 4))
def test_polylog_expansion(): from sympy import log assert polylog(s, 0) == 0 assert polylog(s, 1) == zeta(s) assert polylog(s, -1) == -dirichlet_eta(s) assert polylog(s, exp_polar(4 * I * pi / 3)) == polylog( s, exp(4 * I * pi / 3)) assert polylog(s, exp_polar(I * pi) / 3) == polylog(s, exp(I * pi) / 3) assert myexpand(polylog(1, z), -log(1 - z)) assert myexpand(polylog(0, z), z / (1 - z)) assert myexpand(polylog(-1, z), z / (1 - z)**2) assert ((1 - z)**3 * expand_func(polylog(-2, z))).simplify() == z * (1 + z) assert myexpand(polylog(-5, z), None)
def myexpand(func, target): expanded = expand_func(func) if target is not None: return expanded == target if expanded == func: # it didn't expand return False # check to see that the expanded and original evaluate to the same value subs = {} for a in func.free_symbols: subs[a] = randcplx() return abs( func.subs(subs).n() - expanded.replace(exp_polar, exp).subs(subs).n()) < 1e-10
def test_beta(): x, y = symbols('x y') t = Dummy('t') assert unchanged(beta, x, y) assert beta(5, -3).is_real == True assert beta(3, y).is_real is None assert expand_func(beta(x, y)) == gamma(x)*gamma(y)/gamma(x + y) assert expand_func(beta(x, y) - beta(y, x)) == 0 # Symmetric assert expand_func(beta(x, y)) == expand_func(beta(x, y + 1) + beta(x + 1, y)).simplify() assert diff(beta(x, y), x) == beta(x, y)*(polygamma(0, x) - polygamma(0, x + y)) assert diff(beta(x, y), y) == beta(x, y)*(polygamma(0, y) - polygamma(0, x + y)) assert conjugate(beta(x, y)) == beta(conjugate(x), conjugate(y)) raises(ArgumentIndexError, lambda: beta(x, y).fdiff(3)) assert beta(x, y).rewrite(gamma) == gamma(x)*gamma(y)/gamma(x + y) assert beta(x).rewrite(gamma) == gamma(x)**2/gamma(2*x) assert beta(x, y).rewrite(Integral).dummy_eq(Integral(t**(x - 1) * (1 - t)**(y - 1), (t, 0, 1)))
def test_jn(): z = symbols("z") assert mjn(0, z) == sin(z)/z assert mjn(1, z) == sin(z)/z**2 - cos(z)/z assert mjn(2, z) == (3/z**3 - 1/z)*sin(z) - (3/z**2) * cos(z) assert mjn(3, z) == (15/z**4 - 6/z**2)*sin(z) + (1/z - 15/z**3)*cos(z) assert mjn(4, z) == (1/z + 105/z**5 - 45/z**3)*sin(z) + \ (-105/z**4 + 10/z**2)*cos(z) assert mjn(5, z) == (945/z**6 - 420/z**4 + 15/z**2)*sin(z) + \ (-1/z - 945/z**5 + 105/z**3)*cos(z) assert mjn(6, z) == (-1/z + 10395/z**7 - 4725/z**5 + 210/z**3)*sin(z) + \ (-10395/z**6 + 1260/z**4 - 21/z**2)*cos(z) assert expand_func(jn(n, z)) == jn(n, z)
def test_erfc(): assert erfc(nan) is nan assert erfc(oo) == 0 assert erfc(-oo) == 2 assert erfc(0) == 1 assert erfc(I*oo) == -oo*I assert erfc(-I*oo) == oo*I assert erfc(-x) == S(2) - erfc(x) assert erfc(erfcinv(x)) == x assert erfc(I).is_real is False assert erfc(0).is_real is True assert erfc(erfinv(x)) == 1 - x assert conjugate(erfc(z)) == erfc(conjugate(z)) assert erfc(x).as_leading_term(x) is S.One assert erfc(1/x).as_leading_term(x) == erfc(1/x) assert erfc(z).rewrite('erf') == 1 - erf(z) assert erfc(z).rewrite('erfi') == 1 + I*erfi(I*z) assert erfc(z).rewrite('fresnels') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erfc(z).rewrite('fresnelc') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erfc(z).rewrite('hyper') == 1 - 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi) assert erfc(z).rewrite('meijerg') == 1 - z*meijerg([S.Half], [], [0], [Rational(-1, 2)], z**2)/sqrt(pi) assert erfc(z).rewrite('uppergamma') == 1 - sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z assert erfc(z).rewrite('expint') == S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(S.Pi) assert erfc(z).rewrite('tractable') == _erfs(z)*exp(-z**2) assert expand_func(erf(x) + erfc(x)) is S.One assert erfc(x).as_real_imag() == \ (erfc(re(x) - I*im(x))/2 + erfc(re(x) + I*im(x))/2, -I*(-erfc(re(x) - I*im(x)) + erfc(re(x) + I*im(x)))/2) assert erfc(x).as_real_imag(deep=False) == \ (erfc(re(x) - I*im(x))/2 + erfc(re(x) + I*im(x))/2, -I*(-erfc(re(x) - I*im(x)) + erfc(re(x) + I*im(x)))/2) assert erfc(w).as_real_imag() == (erfc(w), 0) assert erfc(w).as_real_imag(deep=False) == (erfc(w), 0) raises(ArgumentIndexError, lambda: erfc(x).fdiff(2)) assert erfc(x).inverse() == erfcinv
def test_harmonic_rewrite_polygamma(): n = Symbol("n") m = Symbol("m") assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma assert harmonic(n).rewrite(trigamma) == polygamma(0, n + 1) + EulerGamma assert harmonic(n).rewrite(polygamma) == polygamma(0, n + 1) + EulerGamma assert harmonic( n, 3).rewrite(polygamma) == polygamma(2, n + 1) / 2 - polygamma(2, 1) / 2 assert harmonic(n, m).rewrite(polygamma) == (-1)**m * ( polygamma(m - 1, 1) - polygamma(m - 1, n + 1)) / factorial(m - 1) assert expand_func( harmonic(n + 4) ) == harmonic(n) + 1 / (n + 4) + 1 / (n + 3) + 1 / (n + 2) + 1 / (n + 1) assert expand_func(harmonic( n - 4)) == harmonic(n) - 1 / (n - 1) - 1 / (n - 2) - 1 / (n - 3) - 1 / n assert harmonic(n, m).rewrite("tractable") == harmonic(n, m).rewrite(polygamma)
def test_erfc(): assert erfc(nan) == nan assert erfc(oo) == 0 assert erfc(-oo) == 2 assert erfc(0) == 1 assert erfc(I * oo) == -oo * I assert erfc(-I * oo) == oo * I assert erfc(-x) == S(2) - erfc(x) assert erfc(erfcinv(x)) == x assert erfc(I).is_real is False assert erfc(0).is_real is True assert conjugate(erfc(z)) == erfc(conjugate(z)) assert erfc(x).as_leading_term(x) == S.One assert erfc(1 / x).as_leading_term(x) == erfc(1 / x) assert erfc(z).rewrite('erf') == 1 - erf(z) assert erfc(z).rewrite('erfi') == 1 + I * erfi(I * z) assert erfc(z).rewrite('fresnels') == 1 - (1 + I) * ( fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z * (1 - I) / sqrt(pi))) assert erfc(z).rewrite('fresnelc') == 1 - (1 + I) * ( fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z * (1 - I) / sqrt(pi))) assert erfc(z).rewrite( 'hyper') == 1 - 2 * z * hyper([S.Half], [3 * S.Half], -z**2) / sqrt(pi) assert erfc(z).rewrite('meijerg') == 1 - z * meijerg( [S.Half], [], [0], [-S.Half], z**2) / sqrt(pi) assert erfc(z).rewrite( 'uppergamma') == 1 - sqrt(z**2) * (1 - erfc(sqrt(z**2))) / z assert erfc(z).rewrite('expint') == S.One - sqrt(z**2) / z + z * expint( S.Half, z**2) / sqrt(S.Pi) assert expand_func(erf(x) + erfc(x)) == S.One assert erfc(x).as_real_imag() == \ ((erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 + erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2, I*(erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) - erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) * re(x)*Abs(im(x))/(2*im(x)*Abs(re(x))))) raises(ArgumentIndexError, lambda: erfc(x).fdiff(2))
def test_erfi(): assert erfi(nan) == nan assert erfi(oo) == S.Infinity assert erfi(-oo) == S.NegativeInfinity assert erfi(0) == S.Zero assert erfi(I * oo) == I assert erfi(-I * oo) == -I assert erfi(-x) == -erfi(x) assert erfi(I * erfinv(x)) == I * x assert erfi(I * erfcinv(x)) == I * (1 - x) assert erfi(I * erf2inv(0, x)) == I * x assert erfi(I).is_real is False assert erfi(0).is_real is True assert conjugate(erfi(z)) == erfi(conjugate(z)) assert erfi(z).rewrite('erf') == -I * erf(I * z) assert erfi(z).rewrite('erfc') == I * erfc(I * z) - I assert erfi(z).rewrite('fresnels') == (1 - I) * ( fresnelc(z * (1 + I) / sqrt(pi)) - I * fresnels(z * (1 + I) / sqrt(pi))) assert erfi(z).rewrite('fresnelc') == (1 - I) * ( fresnelc(z * (1 + I) / sqrt(pi)) - I * fresnels(z * (1 + I) / sqrt(pi))) assert erfi(z).rewrite('hyper') == 2 * z * hyper([S.Half], [3 * S.Half], z **2) / sqrt(pi) assert erfi(z).rewrite('meijerg') == z * meijerg( [S.Half], [], [0], [-S.Half], -z**2) / sqrt(pi) assert erfi(z).rewrite('uppergamma') == ( sqrt(-z**2) / z * (uppergamma(S.Half, -z**2) / sqrt(S.Pi) - S.One)) assert erfi(z).rewrite( 'expint') == sqrt(-z**2) / z - z * expint(S.Half, -z**2) / sqrt(S.Pi) assert expand_func(erfi(I * z)) == I * erf(z) assert erfi(x).as_real_imag() == \ ((erfi(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 + erfi(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2, I*(erfi(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) - erfi(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) * re(x)*Abs(im(x))/(2*im(x)*Abs(re(x))))) raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))
def test_erfi(): assert erfi(nan) is nan assert erfi(oo) is S.Infinity assert erfi(-oo) is S.NegativeInfinity assert erfi(0) is S.Zero assert erfi(I*oo) == I assert erfi(-I*oo) == -I assert erfi(-x) == -erfi(x) assert erfi(I*erfinv(x)) == I*x assert erfi(I*erfcinv(x)) == I*(1 - x) assert erfi(I*erf2inv(0, x)) == I*x assert erfi(I*erf2inv(0, x, evaluate=False)) == I*x # To cover code in erfi assert erfi(I).is_real is False assert erfi(0).is_real is True assert conjugate(erfi(z)) == erfi(conjugate(z)) assert erfi(z).rewrite('erf') == -I*erf(I*z) assert erfi(z).rewrite('erfc') == I*erfc(I*z) - I assert erfi(z).rewrite('fresnels') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) - I*fresnels(z*(1 + I)/sqrt(pi))) assert erfi(z).rewrite('fresnelc') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) - I*fresnels(z*(1 + I)/sqrt(pi))) assert erfi(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], z**2)/sqrt(pi) assert erfi(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [Rational(-1, 2)], -z**2)/sqrt(pi) assert erfi(z).rewrite('uppergamma') == (sqrt(-z**2)/z*(uppergamma(S.Half, -z**2)/sqrt(S.Pi) - S.One)) assert erfi(z).rewrite('expint') == sqrt(-z**2)/z - z*expint(S.Half, -z**2)/sqrt(S.Pi) assert erfi(z).rewrite('tractable') == -I*(-_erfs(I*z)*exp(z**2) + 1) assert expand_func(erfi(I*z)) == I*erf(z) assert erfi(x).as_real_imag() == \ (erfi(re(x) - I*im(x))/2 + erfi(re(x) + I*im(x))/2, -I*(-erfi(re(x) - I*im(x)) + erfi(re(x) + I*im(x)))/2) assert erfi(x).as_real_imag(deep=False) == \ (erfi(re(x) - I*im(x))/2 + erfi(re(x) + I*im(x))/2, -I*(-erfi(re(x) - I*im(x)) + erfi(re(x) + I*im(x)))/2) assert erfi(w).as_real_imag() == (erfi(w), 0) assert erfi(w).as_real_imag(deep=False) == (erfi(w), 0) raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))
def solve_eq(pad: Expr) -> dict: # expand left term left = cancel(expand_func(pad).subs(sin(theta)**2, 1 - cos(theta)**2)) terms_lft = tuple(expend_cos(left, theta)) # expand right term b0, b1, b2, b3, b4 = symbols('b_0 b_1 b_2 b_3 b_4', real=True) right = cancel(b0 + b1 * legendre(1, cos(theta)) + b2 * legendre(2, cos(theta)) + b3 * legendre(3, cos(theta)) + b4 * legendre(4, cos(theta))) terms_rgt = tuple(expend_cos(right, theta)) # solve equations b4_cmpx = simplify(cancel(solve((terms_lft[4] - terms_rgt[4]), b4)[0])) b4_real = simplify(re(expand(b4_cmpx))) b3_cmpx = simplify(cancel(solve((terms_lft[3] - terms_rgt[3]), b3)[0])) b3_real = simplify(re(expand(b3_cmpx))) b3_amp, b3_shift = amp_and_shift(b3_real, phi) b2_cmpx = simplify( cancel(solve((terms_lft[2] - terms_rgt[2]).subs(b4, b4_cmpx), b2)[0])) b2_real = simplify(re(expand(b2_cmpx))) b1_cmpx = simplify( cancel(solve((terms_lft[1] - terms_rgt[1]).subs(b3, b3_cmpx), b1)[0])) b1_real = simplify(re(expand(b1_cmpx))) b1_amp, b1_shift = amp_and_shift(b1_real, phi) b0_cmpx = simplify( cancel( solve( (terms_lft[0] - terms_rgt[0]).subs(b4, b4_cmpx).subs(b2, b2_cmpx), b0)[0])) b0_real = simplify(re(expand(b0_cmpx))) b1m3_real = simplify(cancel(b1_real - b3_real * 2 / 3)) b1m3_amp, b1m3_shift = amp_and_shift(b1m3_real, phi) return { 'beta1': b1_real / b0_real, 'beta1_amp': b1_amp / b0_real, 'beta1_shift': b1_shift, 'beta2': b2_real / b0_real, 'beta3': b3_real / b0_real, 'beta3_amp': b3_amp / b0_real, 'beta3_shift': b3_shift, 'beta4': b4_real / b0_real, 'beta1m3': b1m3_real / b0_real, 'beta1m3_amp': b1m3_amp / b0_real, 'beta1m3_shift': b1m3_shift, }
def test_binomial(): x = Symbol('x') n = Symbol('n', integer=True) nz = Symbol('nz', integer=True, nonzero=True) k = Symbol('k', integer=True) kp = Symbol('kp', integer=True, positive=True) u = Symbol('u', negative=True) p = Symbol('p', positive=True) z = Symbol('z', zero=True) nt = Symbol('nt', integer=False) a = Symbol('a', integer=True, nonnegative=True) b = Symbol('b', integer=True, nonnegative=True) assert binomial(0, 0) == 1 assert binomial(1, 1) == 1 assert binomial(10, 10) == 1 assert binomial(n, z) == 1 assert binomial(1, 2) == 0 assert binomial(1, -1) == 0 assert binomial(-1, 1) == -1 assert binomial(-1, -1) == 1 assert binomial(S.Half, S.Half) == 1 assert binomial(-10, 1) == -10 assert binomial(-10, 7) == -11440 assert binomial(n, -1).func == binomial assert binomial(kp, -1) == 0 assert binomial(nz, 0) == 1 assert expand_func(binomial(n, 1)) == n assert expand_func(binomial(n, 2)) == n*(n - 1)/2 assert expand_func(binomial(n, n - 2)) == n*(n - 1)/2 assert expand_func(binomial(n, n - 1)) == n assert binomial(n, 3).func == binomial assert binomial(n, 3).expand(func=True) == n**3/6 - n**2/2 + n/3 assert expand_func(binomial(n, 3)) == n*(n - 2)*(n - 1)/6 assert binomial(n, n) == 1 assert binomial(n, n + 1).func == binomial # e.g. (-1, 0) == 1 assert binomial(kp, kp + 1) == 0 assert binomial(n, u).func == binomial assert binomial(kp, u) == 0 assert binomial(n, p).func == binomial assert binomial(n, k).func == binomial assert binomial(n, n + p).func == binomial assert binomial(kp, kp + p) == 0 assert expand_func(binomial(n, n - 3)) == n*(n - 2)*(n - 1)/6 assert binomial(n, k).is_integer assert binomial(nt, k).is_integer is None assert binomial(x, nt).is_integer is False assert binomial(gamma(25), 6) == 79232165267303928292058750056084441948572511312165380965440075720159859792344339983120618959044048198214221915637090855535036339620413440000 assert binomial(a, b).is_nonnegative is True
def test_airybi(): z = Symbol('z', real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airybi(z), airybi) assert airybi(0) == 3**Rational(5, 6) / (3 * gamma(Rational(2, 3))) assert airybi(oo) is oo assert airybi(-oo) == 0 assert diff(airybi(z), z) == airybiprime(z) assert series(airybi(z), z, 0, 3) == (3**Rational(1, 3) * gamma(Rational(1, 3)) / (2 * pi) + 3**Rational(2, 3) * z * gamma(Rational(2, 3)) / (2 * pi) + O(z**3)) assert airybi(z).rewrite(hyper) == (3**Rational(1, 6) * z * hyper( (), (Rational(4, 3), ), z**3 / 9) / gamma(Rational(1, 3)) + 3**Rational(5, 6) * hyper( (), (Rational(2, 3), ), z**3 / 9) / (3 * gamma(Rational(2, 3)))) assert isinstance(airybi(z).rewrite(besselj), airybi) assert airyai(t).rewrite(besselj) == ( sqrt(-t) * (besselj(Rational(-1, 3), 2 * (-t)**Rational(3, 2) / 3) + besselj(Rational(1, 3), 2 * (-t)**Rational(3, 2) / 3)) / 3) assert airybi(z).rewrite(besseli) == ( sqrt(3) * (z * besseli(Rational(1, 3), 2 * z**Rational(3, 2) / 3) / (z**Rational(3, 2))**Rational(1, 3) + (z**Rational(3, 2))**Rational(1, 3) * besseli(Rational(-1, 3), 2 * z**Rational(3, 2) / 3)) / 3) assert airybi(p).rewrite(besseli) == ( sqrt(3) * sqrt(p) * (besseli(Rational(-1, 3), 2 * p**Rational(3, 2) / 3) + besseli(Rational(1, 3), 2 * p**Rational(3, 2) / 3)) / 3) assert expand_func(airybi(2 * (3 * z**5)**Rational(1, 3))) == ( sqrt(3) * (1 - (z**5)**Rational(1, 3) / z**Rational(5, 3)) * airyai(2 * 3**Rational(1, 3) * z**Rational(5, 3)) / 2 + (1 + (z**5)**Rational(1, 3) / z**Rational(5, 3)) * airybi(2 * 3**Rational(1, 3) * z**Rational(5, 3)) / 2)
def test_erf2(): assert erf2(0, 0) == S.Zero assert erf2(x, x) == S.Zero assert erf2(nan, 0) == nan assert erf2(-oo, y) == erf(y) + 1 assert erf2(oo, y) == erf(y) - 1 assert erf2(x, oo) == 1 - erf(x) assert erf2(x, -oo) == -1 - erf(x) assert erf2(x, erf2inv(x, y)) == y assert erf2(-x, -y) == -erf2(x, y) assert erf2(-x, y) == erf(y) + erf(x) assert erf2(x, -y) == -erf(y) - erf(x) assert erf2(x, y).rewrite( 'fresnels') == erf(y).rewrite(fresnels) - erf(x).rewrite(fresnels) assert erf2(x, y).rewrite( 'fresnelc') == erf(y).rewrite(fresnelc) - erf(x).rewrite(fresnelc) assert erf2( x, y).rewrite('hyper') == erf(y).rewrite(hyper) - erf(x).rewrite(hyper) assert erf2(x, y).rewrite( 'meijerg') == erf(y).rewrite(meijerg) - erf(x).rewrite(meijerg) assert erf2( x, y).rewrite('uppergamma' ) == erf(y).rewrite(uppergamma) - erf(x).rewrite(uppergamma) assert erf2( x, y).rewrite('expint') == erf(y).rewrite(expint) - erf(x).rewrite(expint) assert erf2(I, 0).is_real is False assert erf2(0, 0).is_real is True assert expand_func(erf(x) + erf2(x, y)) == erf(y) assert conjugate(erf2(x, y)) == erf2(conjugate(x), conjugate(y)) assert erf2(x, y).rewrite('erf') == erf(y) - erf(x) assert erf2(x, y).rewrite('erfc') == erfc(x) - erfc(y) assert erf2(x, y).rewrite('erfi') == I * (erfi(I * x) - erfi(I * y)) raises(ArgumentIndexError, lambda: erfi(x).fdiff(3))
def test_erfi(): assert erfi(nan) == nan assert erfi(oo) == S.Infinity assert erfi(-oo) == S.NegativeInfinity assert erfi(0) == S.Zero assert erfi(I*oo) == I assert erfi(-I*oo) == -I assert erfi(-x) == -erfi(x) assert erfi(I*erfinv(x)) == I*x assert erfi(I*erfcinv(x)) == I*(1 - x) assert erfi(I*erf2inv(0, x)) == I*x assert erfi(I).is_real is False assert erfi(0).is_real is True assert conjugate(erfi(z)) == erfi(conjugate(z)) assert erfi(z).rewrite('erf') == -I*erf(I*z) assert erfi(z).rewrite('erfc') == I*erfc(I*z) - I assert erfi(z).rewrite('fresnels') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) - I*fresnels(z*(1 + I)/sqrt(pi))) assert erfi(z).rewrite('fresnelc') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) - I*fresnels(z*(1 + I)/sqrt(pi))) assert erfi(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], z**2)/sqrt(pi) assert erfi(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [-S.Half], -z**2)/sqrt(pi) assert erfi(z).rewrite('uppergamma') == (sqrt(-z**2)/z*(uppergamma(S.Half, -z**2)/sqrt(S.Pi) - S.One)) assert erfi(z).rewrite('expint') == sqrt(-z**2)/z - z*expint(S.Half, -z**2)/sqrt(S.Pi) assert expand_func(erfi(I*z)) == I*erf(z) assert erfi(x).as_real_imag() == \ ((erfi(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 + erfi(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2, I*(erfi(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) - erfi(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) * re(x)*Abs(im(x))/(2*im(x)*Abs(re(x))))) raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))
def test_expand_func(): # evaluation at 1 of Gauss' hypergeometric function: from sympy.abc import a, b, c from sympy import gamma, expand_func a1, b1, c1 = randcplx(), randcplx(), randcplx() + 5 assert expand_func(hyper([a, b], [c], 1)) == gamma(c) * gamma(-a - b + c) / (gamma(-a + c) * gamma(-b + c)) assert abs(expand_func(hyper([a1, b1], [c1], 1)).n() - hyper([a1, b1], [c1], 1).n()) < 1e-10 # hyperexpand wrapper for hyper: assert expand_func(hyper([], [], z)) == exp(z) assert expand_func(hyper([1, 2, 3], [], z)) == hyper([1, 2, 3], [], z) assert expand_func(meijerg([[1, 1], []], [[1], [0]], z)) == log(z + 1) assert expand_func(meijerg([[1, 1], []], [[], []], z)) == meijerg([[1, 1], []], [[], []], z)
def test_erfc(): assert erfc(nan) == nan assert erfc(oo) == 0 assert erfc(-oo) == 2 assert erfc(0) == 1 assert erfc(I*oo) == -oo*I assert erfc(-I*oo) == oo*I assert erfc(-x) == S(2) - erfc(x) assert erfc(erfcinv(x)) == x assert erfc(I).is_real is False assert erfc(0).is_real is True assert conjugate(erfc(z)) == erfc(conjugate(z)) assert erfc(x).as_leading_term(x) == S.One assert erfc(1/x).as_leading_term(x) == erfc(1/x) assert erfc(z).rewrite('erf') == 1 - erf(z) assert erfc(z).rewrite('erfi') == 1 + I*erfi(I*z) assert erfc(z).rewrite('fresnels') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erfc(z).rewrite('fresnelc') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erfc(z).rewrite('hyper') == 1 - 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi) assert erfc(z).rewrite('meijerg') == 1 - z*meijerg([S.Half], [], [0], [-S.Half], z**2)/sqrt(pi) assert erfc(z).rewrite('uppergamma') == 1 - sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z assert erfc(z).rewrite('expint') == S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(S.Pi) assert expand_func(erf(x) + erfc(x)) == S.One assert erfc(x).as_real_imag() == \ ((erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 + erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2, I*(erfc(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) - erfc(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) * re(x)*Abs(im(x))/(2*im(x)*Abs(re(x))))) raises(ArgumentIndexError, lambda: erfc(x).fdiff(2))
def test_binomial(): x = Symbol('x') n = Symbol('n', integer=True) nz = Symbol('nz', integer=True, nonzero=True) k = Symbol('k', integer=True) kp = Symbol('kp', integer=True, positive=True) u = Symbol('u', negative=True) p = Symbol('p', positive=True) z = Symbol('z', zero=True) nt = Symbol('nt', integer=False) assert binomial(0, 0) == 1 assert binomial(1, 1) == 1 assert binomial(10, 10) == 1 assert binomial(n, z) == 1 assert binomial(1, 2) == 0 assert binomial(1, -1) == 0 assert binomial(-1, 1) == -1 assert binomial(-1, -1) == 1 assert binomial(S.Half, S.Half) == 1 assert binomial(-10, 1) == -10 assert binomial(-10, 7) == -11440 assert binomial(n, -1).func == binomial assert binomial(kp, -1) == 0 assert binomial(nz, 0) == 1 assert expand_func(binomial(n, 1)) == n assert expand_func(binomial(n, 2)) == n*(n - 1)/2 assert expand_func(binomial(n, n - 2)) == n*(n - 1)/2 assert expand_func(binomial(n, n - 1)) == n assert binomial(n, 3).func == binomial assert binomial(n, 3).expand(func=True) == n**3/6 - n**2/2 + n/3 assert expand_func(binomial(n, 3)) == n*(n - 2)*(n - 1)/6 assert binomial(n, n) == 1 assert binomial(n, n + 1).func == binomial # e.g. (-1, 0) == 1 assert binomial(kp, kp + 1) == 0 assert binomial(n, u).func == binomial assert binomial(kp, u) == 0 assert binomial(n, p).func == binomial assert binomial(n, k).func == binomial assert binomial(n, n + p).func == binomial assert binomial(kp, kp + p) == 0 assert expand_func(binomial(n, n - 3)) == n*(n - 2)*(n - 1)/6 assert binomial(n, k).is_integer assert binomial(nt, k).is_integer is None assert binomial(x, nt).is_integer is False assert binomial(gamma(25), 6) == 79232165267303928292058750056084441948572511312165380965440075720159859792344339983120618959044048198214221915637090855535036339620413440000
def test_jn(): z = symbols("z") assert mjn(0, z) == sin(z)/z assert mjn(1, z) == sin(z)/z**2 - cos(z)/z assert mjn(2, z) == (3/z**3 - 1/z)*sin(z) - (3/z**2) * cos(z) assert mjn(3, z) == (15/z**4 - 6/z**2)*sin(z) + (1/z - 15/z**3)*cos(z) assert mjn(4, z) == (1/z + 105/z**5 - 45/z**3)*sin(z) + \ (-105/z**4 + 10/z**2)*cos(z) assert mjn(5, z) == (945/z**6 - 420/z**4 + 15/z**2)*sin(z) + \ (-1/z - 945/z**5 + 105/z**3)*cos(z) assert mjn(6, z) == (-1/z + 10395/z**7 - 4725/z**5 + 210/z**3)*sin(z) + \ (-10395/z**6 + 1260/z**4 - 21/z**2)*cos(z) assert expand_func(jn(n, z)) == jn(n, z) # SBFs not defined for complex-valued orders assert jn(2+3j, 5.2+0.3j).evalf() == jn(2+3j, 5.2+0.3j) assert eq([jn(2, 5.2+0.3j).evalf(10)], [0.09941975672 - 0.05452508024*I])
def test_airyai(): z = Symbol('z', real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airyai(z), airyai) assert airyai(0) == 3**(S(1) / 3) / (3 * gamma(S(2) / 3)) assert airyai(oo) == 0 assert airyai(-oo) == 0 assert diff(airyai(z), z) == airyaiprime(z) assert series(airyai(z), z, 0, 3) == (3**(S(5) / 6) * gamma(S(1) / 3) / (6 * pi) - 3**(S(1) / 6) * z * gamma(S(2) / 3) / (2 * pi) + O(z**3)) assert airyai(z).rewrite(hyper) == (-3**(S(2) / 3) * z * hyper( (), (S(4) / 3, ), z**S(3) / 9) / (3 * gamma(S(1) / 3)) + 3**(S(1) / 3) * hyper( (), (S(2) / 3, ), z**S(3) / 9) / (3 * gamma(S(2) / 3))) assert isinstance(airyai(z).rewrite(besselj), airyai) assert airyai(t).rewrite(besselj) == ( sqrt(-t) * (besselj(-S(1) / 3, 2 * (-t)**(S(3) / 2) / 3) + besselj(S(1) / 3, 2 * (-t)**(S(3) / 2) / 3)) / 3) assert airyai(z).rewrite(besseli) == ( -z * besseli(S(1) / 3, 2 * z**(S(3) / 2) / 3) / (3 * (z**(S(3) / 2))**(S(1) / 3)) + (z**(S(3) / 2))**(S(1) / 3) * besseli(-S(1) / 3, 2 * z**(S(3) / 2) / 3) / 3) assert airyai(p).rewrite(besseli) == ( sqrt(p) * (besseli(-S(1) / 3, 2 * p**(S(3) / 2) / 3) - besseli(S(1) / 3, 2 * p**(S(3) / 2) / 3)) / 3) assert expand_func(airyai(2 * (3 * z**5)**(S(1) / 3))) == ( -sqrt(3) * (-1 + (z**5)**(S(1) / 3) / z**(S(5) / 3)) * airybi(2 * 3**(S(1) / 3) * z**(S(5) / 3)) / 6 + (1 + (z**5)**(S(1) / 3) / z**(S(5) / 3)) * airyai(2 * 3**(S(1) / 3) * z**(S(5) / 3)) / 2)
def test_airybiprime(): z = Symbol("z", real=False) t = Symbol("t", negative=True) p = Symbol("p", positive=True) assert isinstance(airybiprime(z), airybiprime) assert airybiprime(0) == 3**Rational(1, 6) / gamma(Rational(1, 3)) assert airybiprime(oo) is oo assert airybiprime(-oo) == 0 assert diff(airybiprime(z), z) == z * airybi(z) assert series(airybiprime(z), z, 0, 3) == (3**Rational(1, 6) / gamma(Rational(1, 3)) + 3**Rational(5, 6) * z**2 / (6 * gamma(Rational(2, 3))) + O(z**3)) assert airybiprime(z).rewrite(hyper) == ( 3**Rational(5, 6) * z**2 * hyper((), (Rational(5, 3), ), z**3 / 9) / (6 * gamma(Rational(2, 3))) + 3**Rational(1, 6) * hyper( (), (Rational(1, 3), ), z**3 / 9) / gamma(Rational(1, 3))) assert isinstance(airybiprime(z).rewrite(besselj), airybiprime) assert airyai(t).rewrite(besselj) == ( sqrt(-t) * (besselj(Rational(-1, 3), 2 * (-t)**Rational(3, 2) / 3) + besselj(Rational(1, 3), 2 * (-t)**Rational(3, 2) / 3)) / 3) assert airybiprime(z).rewrite(besseli) == ( sqrt(3) * (z**2 * besseli(Rational(2, 3), 2 * z**Rational(3, 2) / 3) / (z**Rational(3, 2))**Rational(2, 3) + (z**Rational(3, 2))**Rational(2, 3) * besseli(Rational(-2, 3), 2 * z**Rational(3, 2) / 3)) / 3) assert airybiprime(p).rewrite(besseli) == ( sqrt(3) * p * (besseli(Rational(-2, 3), 2 * p**Rational(3, 2) / 3) + besseli(Rational(2, 3), 2 * p**Rational(3, 2) / 3)) / 3) assert expand_func(airybiprime(2 * (3 * z**5)**Rational(1, 3))) == ( sqrt(3) * (z**Rational(5, 3) / (z**5)**Rational(1, 3) - 1) * airyaiprime(2 * 3**Rational(1, 3) * z**Rational(5, 3)) / 2 + (z**Rational(5, 3) / (z**5)**Rational(1, 3) + 1) * airybiprime(2 * 3**Rational(1, 3) * z**Rational(5, 3)) / 2)
def test_airybiprime(): z = Symbol('z', real=False) t = Symbol('t', negative=True) p = Symbol('p', positive=True) assert isinstance(airybiprime(z), airybiprime) assert airybiprime(0) == 3**(S(1) / 6) / gamma(S(1) / 3) assert airybiprime(oo) == oo assert airybiprime(-oo) == 0 assert diff(airybiprime(z), z) == z * airybi(z) assert series(airybiprime(z), z, 0, 3) == (3**(S(1) / 6) / gamma(S(1) / 3) + 3**(S(5) / 6) * z**2 / (6 * gamma(S(2) / 3)) + O(z**3)) assert airybiprime(z).rewrite(hyper) == (3**(S(5) / 6) * z**2 * hyper( (), (S(5) / 3, ), z**S(3) / 9) / (6 * gamma(S(2) / 3)) + 3**(S(1) / 6) * hyper( (), (S(1) / 3, ), z**S(3) / 9) / gamma(S(1) / 3)) assert isinstance(airybiprime(z).rewrite(besselj), airybiprime) assert airyai(t).rewrite(besselj) == ( sqrt(-t) * (besselj(-S(1) / 3, 2 * (-t)**(S(3) / 2) / 3) + besselj(S(1) / 3, 2 * (-t)**(S(3) / 2) / 3)) / 3) assert airybiprime(z).rewrite(besseli) == ( sqrt(3) * (z**2 * besseli(S(2) / 3, 2 * z**(S(3) / 2) / 3) / (z**(S(3) / 2))**(S(2) / 3) + (z**(S(3) / 2))** (S(2) / 3) * besseli(-S(2) / 3, 2 * z**(S(3) / 2) / 3)) / 3) assert airybiprime(p).rewrite(besseli) == ( sqrt(3) * p * (besseli(-S(2) / 3, 2 * p**(S(3) / 2) / 3) + besseli(S(2) / 3, 2 * p**(S(3) / 2) / 3)) / 3) assert expand_func(airybiprime(2 * (3 * z**5)**(S(1) / 3))) == ( sqrt(3) * (z**(S(5) / 3) / (z**5)**(S(1) / 3) - 1) * airyaiprime(2 * 3**(S(1) / 3) * z**(S(5) / 3)) / 2 + (z**(S(5) / 3) / (z**5)**(S(1) / 3) + 1) * airybiprime(2 * 3**(S(1) / 3) * z**(S(5) / 3)) / 2)
def test_expand_func(): # evaluation at 1 of Gauss' hypergeometric function: from sympy.abc import a, b, c from sympy import gamma, expand_func a1, b1, c1 = randcplx(), randcplx(), randcplx() + 5 assert expand_func(hyper([a, b], [c], 1)) == \ gamma(c)*gamma(-a - b + c)/(gamma(-a + c)*gamma(-b + c)) assert abs(expand_func(hyper([a1, b1], [c1], 1)).n() - hyper([a1, b1], [c1], 1).n()) < 1e-10 # hyperexpand wrapper for hyper: assert expand_func(hyper([], [], z)) == exp(z) assert expand_func(hyper([1, 2, 3], [], z)) == hyper([1, 2, 3], [], z) assert expand_func(meijerg([[1, 1], []], [[1], [0]], z)) == log(z + 1) assert expand_func(meijerg([[1, 1], []], [[], []], z)) == \ meijerg([[1, 1], []], [[], []], z)
def test_binomial(): x = Symbol('x') n = Symbol('n', integer=True) nz = Symbol('nz', integer=True, nonzero=True) k = Symbol('k', integer=True) kp = Symbol('kp', integer=True, positive=True) u = Symbol('u', negative=True) p = Symbol('p', positive=True) z = Symbol('z', zero=True) nt = Symbol('nt', integer=False) assert binomial(0, 0) == 1 assert binomial(1, 1) == 1 assert binomial(10, 10) == 1 assert binomial(n, z) == 1 assert binomial(1, 2) == 0 assert binomial(1, -1) == 0 assert binomial(-1, 1) == -1 assert binomial(-1, -1) == 1 assert binomial(S.Half, S.Half) == 1 assert binomial(-10, 1) == -10 assert binomial(-10, 7) == -11440 assert binomial(n, -1).func == binomial assert binomial(kp, -1) == 0 assert binomial(nz, 0) == 1 assert expand_func(binomial(n, 1)) == n assert expand_func(binomial(n, 2)) == n * (n - 1) / 2 assert expand_func(binomial(n, n - 2)) == n * (n - 1) / 2 assert expand_func(binomial(n, n - 1)) == n assert binomial(n, 3).func == binomial assert binomial(n, 3).expand(func=True) == n**3 / 6 - n**2 / 2 + n / 3 assert expand_func(binomial(n, 3)) == n * (n - 2) * (n - 1) / 6 assert binomial(n, n) == 1 assert binomial(n, n + 1).func == binomial # e.g. (-1, 0) == 1 assert binomial(kp, kp + 1) == 0 assert binomial(n, u).func == binomial assert binomial(kp, u) == 0 assert binomial(n, p).func == binomial assert binomial(n, k).func == binomial assert binomial(n, n + p).func == binomial assert binomial(kp, kp + p) == 0 assert expand_func(binomial(n, n - 3)) == n * (n - 2) * (n - 1) / 6 assert binomial(n, k).is_integer assert binomial(nt, k).is_integer is None assert binomial(x, nt).is_integer is False
def test_binomial(): x = Symbol('x') n = Symbol('n', integer=True) nz = Symbol('nz', integer=True, nonzero=True) k = Symbol('k', integer=True) kp = Symbol('kp', integer=True, positive=True) u = Symbol('u', negative=True) p = Symbol('p', positive=True) z = Symbol('z', zero=True) nt = Symbol('nt', integer=False) assert binomial(0, 0) == 1 assert binomial(1, 1) == 1 assert binomial(10, 10) == 1 assert binomial(n, z) == 1 assert binomial(1, 2) == 0 assert binomial(1, -1) == 0 assert binomial(-1, 1) == -1 assert binomial(-1, -1) == 1 assert binomial(S.Half, S.Half) == 1 assert binomial(-10, 1) == -10 assert binomial(-10, 7) == -11440 assert binomial(n, -1).func == binomial assert binomial(kp, -1) == 0 assert binomial(nz, 0) == 1 assert expand_func(binomial(n, 1)) == n assert expand_func(binomial(n, 2)) == n*(n - 1)/2 assert expand_func(binomial(n, n - 2)) == n*(n - 1)/2 assert expand_func(binomial(n, n - 1)) == n assert binomial(n, 3).func == binomial assert binomial(n, 3).expand(func=True) == n**3/6 - n**2/2 + n/3 assert expand_func(binomial(n, 3)) == n*(n - 2)*(n - 1)/6 assert binomial(n, n) == 1 assert binomial(n, n + 1).func == binomial # e.g. (-1, 0) == 1 assert binomial(kp, kp + 1) == 0 assert binomial(n, u).func == binomial assert binomial(kp, u) == 0 assert binomial(n, p).func == binomial assert binomial(n, k).func == binomial assert binomial(n, n + p).func == binomial assert binomial(kp, kp + p) == 0 assert expand_func(binomial(n, n - 3)) == n*(n - 2)*(n - 1)/6 assert binomial(n, k).is_integer assert binomial(nt, k).is_integer is None assert binomial(x, nt).is_integer is False
def test_issue1893(): from sympy import simplify, expand_func, polygamma, gamma a = Symbol('a', positive=True) assert simplify(expand_func(integrate(exp(-x)*log(x)*x**a, (x, 0, oo)))) == \ (a*polygamma(0, a) + 1)*gamma(a)
def test_yn(): z = symbols("z") assert myn(0, z) == -cos(z)/z assert myn(1, z) == -cos(z)/z**2 - sin(z)/z assert myn(2, z) == -((3/z**3 - 1/z)*cos(z) + (3/z**2)*sin(z)) assert expand_func(yn(n, z)) == yn(n, z)
def myn(n, z): return expand_func(yn(n, z))
def test_issue_3686(): # remove this when fresnel itegrals are implemented from sympy import expand_func, fresnels assert expand_func(integrate(sin(x**2), x)) == \ sqrt(2)*sqrt(pi)*fresnels(sqrt(2)*x/sqrt(pi))/2
def test_binomial(): x = Symbol('x') n = Symbol('n', integer=True) nz = Symbol('nz', integer=True, nonzero=True) k = Symbol('k', integer=True) kp = Symbol('kp', integer=True, positive=True) kn = Symbol('kn', integer=True, negative=True) u = Symbol('u', negative=True) v = Symbol('v', nonnegative=True) p = Symbol('p', positive=True) z = Symbol('z', zero=True) nt = Symbol('nt', integer=False) kt = Symbol('kt', integer=False) a = Symbol('a', integer=True, nonnegative=True) b = Symbol('b', integer=True, nonnegative=True) assert binomial(0, 0) == 1 assert binomial(1, 1) == 1 assert binomial(10, 10) == 1 assert binomial(n, z) == 1 assert binomial(1, 2) == 0 assert binomial(-1, 2) == 1 assert binomial(1, -1) == 0 assert binomial(-1, 1) == -1 assert binomial(-1, -1) == 0 assert binomial(S.Half, S.Half) == 1 assert binomial(-10, 1) == -10 assert binomial(-10, 7) == -11440 assert binomial( n, -1) == 0 # holds for all integers (negative, zero, positive) assert binomial(kp, -1) == 0 assert binomial(nz, 0) == 1 assert expand_func(binomial(n, 1)) == n assert expand_func(binomial(n, 2)) == n * (n - 1) / 2 assert expand_func(binomial(n, n - 2)) == n * (n - 1) / 2 assert expand_func(binomial(n, n - 1)) == n assert binomial(n, 3).func == binomial assert binomial(n, 3).expand(func=True) == n**3 / 6 - n**2 / 2 + n / 3 assert expand_func(binomial(n, 3)) == n * (n - 2) * (n - 1) / 6 assert binomial(n, n).func == binomial # e.g. (-1, -1) == 0, (2, 2) == 1 assert binomial(n, n + 1).func == binomial # e.g. (-1, 0) == 1 assert binomial(kp, kp + 1) == 0 assert binomial(kn, kn) == 0 # issue #14529 assert binomial(n, u).func == binomial assert binomial(kp, u).func == binomial assert binomial(n, p).func == binomial assert binomial(n, k).func == binomial assert binomial(n, n + p).func == binomial assert binomial(kp, kp + p).func == binomial assert expand_func(binomial(n, n - 3)) == n * (n - 2) * (n - 1) / 6 assert binomial(n, k).is_integer assert binomial(nt, k).is_integer is None assert binomial(x, nt).is_integer is False assert binomial( gamma(25), 6 ) == 79232165267303928292058750056084441948572511312165380965440075720159859792344339983120618959044048198214221915637090855535036339620413440000 assert binomial( 1324, 47 ) == 906266255662694632984994480774946083064699457235920708992926525848438478406790323869952 assert binomial( 1735, 43 ) == 190910140420204130794758005450919715396159959034348676124678207874195064798202216379800 assert binomial( 2512, 53 ) == 213894469313832631145798303740098720367984955243020898718979538096223399813295457822575338958939834177325304000 assert binomial( 3383, 52 ) == 27922807788818096863529701501764372757272890613101645521813434902890007725667814813832027795881839396839287659777235 assert binomial( 4321, 51 ) == 124595639629264868916081001263541480185227731958274383287107643816863897851139048158022599533438936036467601690983780576 assert binomial(a, b).is_nonnegative is True assert binomial(-1, 2, evaluate=False).is_nonnegative is True assert binomial(10, 5, evaluate=False).is_nonnegative is True assert binomial(10, -3, evaluate=False).is_nonnegative is True assert binomial(-10, -3, evaluate=False).is_nonnegative is True assert binomial(-10, 2, evaluate=False).is_nonnegative is True assert binomial(-10, 1, evaluate=False).is_nonnegative is False assert binomial(-10, 7, evaluate=False).is_nonnegative is False # issue #14625 for _ in (pi, -pi, nt, v, a): assert binomial(_, _) == 1 assert binomial(_, _ - 1) == _ assert isinstance(binomial(u, u), binomial) assert isinstance(binomial(u, u - 1), binomial) assert isinstance(binomial(x, x), binomial) assert isinstance(binomial(x, x - 1), binomial) # issue #13980 and #13981 assert binomial(-7, -5) == 0 assert binomial(-23, -12) == 0 assert binomial(Rational(13, 2), -10) == 0 assert binomial(-49, -51) == 0 assert binomial(19, Rational(-7, 2)) == S(-68719476736) / (911337863661225 * pi) assert binomial(0, Rational(3, 2)) == S(-2) / (3 * pi) assert binomial(-3, Rational(-7, 2)) is zoo assert binomial(kn, kt) is zoo assert binomial(nt, kt).func == binomial assert binomial(nt, Rational( 15, 6)) == 8 * gamma(nt + 1) / (15 * sqrt(pi) * gamma(nt - Rational(3, 2))) assert binomial(Rational(20, 3), Rational(-10, 8)) == gamma(Rational( 23, 3)) / (gamma(Rational(-1, 4)) * gamma(Rational(107, 12))) assert binomial(Rational(19, 2), Rational(-7, 2)) == Rational(-1615, 8388608) assert binomial(Rational(-13, 5), Rational(-7, 8)) == gamma(Rational( -8, 5)) / (gamma(Rational(-29, 40)) * gamma(Rational(1, 8))) assert binomial(Rational(-19, 8), Rational(-13, 5)) == gamma( Rational(-11, 8)) / (gamma(Rational(-8, 5)) * gamma(Rational(49, 40))) # binomial for complexes from sympy import I assert binomial(I, Rational(-89, 8)) == gamma(1 + I) / (gamma(Rational(-81, 8)) * gamma(Rational(97, 8) + I)) assert binomial(I, 2 * I) == gamma(1 + I) / (gamma(1 - I) * gamma(1 + 2 * I)) assert binomial(-7, I) is zoo assert binomial(Rational(-7, 6), I) == gamma(Rational( -1, 6)) / (gamma(Rational(-1, 6) - I) * gamma(1 + I)) assert binomial( (1 + 2 * I), (1 + 3 * I)) == gamma(2 + 2 * I) / (gamma(1 - I) * gamma(2 + 3 * I)) assert binomial(I, 5) == Rational(1, 3) - I / S(12) assert binomial((2 * I + 3), 7) == -13 * I / S(63) assert isinstance(binomial(I, n), binomial) assert expand_func(binomial(3, 2, evaluate=False)) == 3 assert expand_func(binomial(n, 0, evaluate=False)) == 1 assert expand_func(binomial(n, -2, evaluate=False)) == 0 assert expand_func(binomial(n, k)) == binomial(n, k)
def test_loggamma(): raises(TypeError, lambda: loggamma(2, 3)) raises(ArgumentIndexError, lambda: loggamma(x).fdiff(2)) assert loggamma(-1) == oo assert loggamma(-2) == oo assert loggamma(0) == oo assert loggamma(1) == 0 assert loggamma(2) == 0 assert loggamma(3) == log(2) assert loggamma(4) == log(6) n = Symbol("n", integer=True, positive=True) assert loggamma(n) == log(gamma(n)) assert loggamma(-n) == oo assert loggamma(n / 2) == log(2**(-n + 1) * sqrt(pi) * gamma(n) / gamma(n / 2 + S.Half)) from sympy import I assert loggamma(oo) == oo assert loggamma(-oo) == zoo assert loggamma(I * oo) == zoo assert loggamma(-I * oo) == zoo assert loggamma(zoo) == zoo assert loggamma(nan) == nan L = loggamma(S(16) / 3) E = -5 * log(3) + loggamma(S(1) / 3) + log(4) + log(7) + log(10) + log(13) assert expand_func(L).doit() == E assert L.n() == E.n() L = loggamma(19 / S(4)) E = -4 * log(4) + loggamma(S(3) / 4) + log(3) + log(7) + log(11) + log(15) assert expand_func(L).doit() == E assert L.n() == E.n() L = loggamma(S(23) / 7) E = -3 * log(7) + log(2) + loggamma(S(2) / 7) + log(9) + log(16) assert expand_func(L).doit() == E assert L.n() == E.n() L = loggamma(19 / S(4) - 7) E = -log(9) - log(5) + loggamma(S(3) / 4) + 3 * log(4) - 3 * I * pi assert expand_func(L).doit() == E assert L.n() == E.n() L = loggamma(23 / S(7) - 6) E = -log(19) - log(12) - log(5) + loggamma( S(2) / 7) + 3 * log(7) - 3 * I * pi assert expand_func(L).doit() == E assert L.n() == E.n() assert loggamma(x).diff(x) == polygamma(0, x) s1 = loggamma(1 / (x + sin(x)) + cos(x)).nseries(x, n=4) s2 = (-log(2*x) - 1)/(2*x) - log(x/pi)/2 + (4 - log(2*x))*x/24 + O(x**2) + \ log(x)*x**2/2 assert (s1 - s2).expand(force=True).removeO() == 0 s1 = loggamma(1 / x).series(x) s2 = (1/x - S(1)/2)*log(1/x) - 1/x + log(2*pi)/2 + \ x/12 - x**3/360 + x**5/1260 + O(x**7) assert ((s1 - s2).expand(force=True)).removeO() == 0 assert loggamma(x).rewrite('intractable') == log(gamma(x)) s1 = loggamma(x).series(x) assert s1 == -log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + \ pi**4*x**4/360 + x**5*polygamma(4, 1)/120 + O(x**6) assert s1 == loggamma(x).rewrite('intractable').series(x) assert conjugate(loggamma(x)) == loggamma(conjugate(x)) assert conjugate(loggamma(0)) == conjugate(loggamma(0)) assert conjugate(loggamma(1)) == loggamma(conjugate(1)) assert conjugate(loggamma(-oo)) == conjugate(loggamma(-oo)) assert loggamma(x).is_real is None y, z = Symbol('y', real=True), Symbol('z', imaginary=True) assert loggamma(y).is_real assert loggamma(z).is_real is False def tN(N, M): assert loggamma(1 / x)._eval_nseries(x, n=N).getn() == M tN(0, 0) tN(1, 1) tN(2, 3) tN(3, 3) tN(4, 5) tN(5, 5)
def test_fresnel(): assert fresnels(0) == 0 assert fresnels(oo) == S.Half assert fresnels(-oo) == -S.Half assert fresnels(z) == fresnels(z) assert fresnels(-z) == -fresnels(z) assert fresnels(I*z) == -I*fresnels(z) assert fresnels(-I*z) == I*fresnels(z) assert conjugate(fresnels(z)) == fresnels(conjugate(z)) assert fresnels(z).diff(z) == sin(pi*z**2/2) assert fresnels(z).rewrite(erf) == (S.One + I)/4 * ( erf((S.One + I)/2*sqrt(pi)*z) - I*erf((S.One - I)/2*sqrt(pi)*z)) assert fresnels(z).rewrite(hyper) == \ pi*z**3/6 * hyper([S(3)/4], [S(3)/2, S(7)/4], -pi**2*z**4/16) assert fresnels(z).series(z, n=15) == \ pi*z**3/6 - pi**3*z**7/336 + pi**5*z**11/42240 + O(z**15) assert fresnels(w).is_real is True assert fresnels(z).as_real_imag() == \ ((fresnels(re(z) - I*re(z)*Abs(im(z))/Abs(re(z)))/2 + fresnels(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))/2, I*(fresnels(re(z) - I*re(z)*Abs(im(z))/Abs(re(z))) - fresnels(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))) * re(z)*Abs(im(z))/(2*im(z)*Abs(re(z))))) assert fresnels(2 + 3*I).as_real_imag() == ( fresnels(2 + 3*I)/2 + fresnels(2 - 3*I)/2, I*(fresnels(2 - 3*I) - fresnels(2 + 3*I))/2 ) assert expand_func(integrate(fresnels(z), z)) == \ z*fresnels(z) + cos(pi*z**2/2)/pi assert fresnels(z).rewrite(meijerg) == sqrt(2)*pi*z**(S(9)/4) * \ meijerg(((), (1,)), ((S(3)/4,), (S(1)/4, 0)), -pi**2*z**4/16)/(2*(-z)**(S(3)/4)*(z**2)**(S(3)/4)) assert fresnelc(0) == 0 assert fresnelc(oo) == S.Half assert fresnelc(-oo) == -S.Half assert fresnelc(z) == fresnelc(z) assert fresnelc(-z) == -fresnelc(z) assert fresnelc(I*z) == I*fresnelc(z) assert fresnelc(-I*z) == -I*fresnelc(z) assert conjugate(fresnelc(z)) == fresnelc(conjugate(z)) assert fresnelc(z).diff(z) == cos(pi*z**2/2) assert fresnelc(z).rewrite(erf) == (S.One - I)/4 * ( erf((S.One + I)/2*sqrt(pi)*z) + I*erf((S.One - I)/2*sqrt(pi)*z)) assert fresnelc(z).rewrite(hyper) == \ z * hyper([S.One/4], [S.One/2, S(5)/4], -pi**2*z**4/16) assert fresnelc(z).series(z, n=15) == \ z - pi**2*z**5/40 + pi**4*z**9/3456 - pi**6*z**13/599040 + O(z**15) # issue 6510 assert fresnels(z).series(z, S.Infinity) == \ (-1/(pi**2*z**3) + O(z**(-6), (z, oo)))*sin(pi*z**2/2) + \ (3/(pi**3*z**5) - 1/(pi*z) + O(z**(-6), (z, oo)))*cos(pi*z**2/2) + S.Half assert fresnelc(z).series(z, S.Infinity) == \ (-1/(pi**2*z**3) + O(z**(-6), (z, oo)))*cos(pi*z**2/2) + \ (-3/(pi**3*z**5) + 1/(pi*z) + O(z**(-6), (z, oo)))*sin(pi*z**2/2) + S.Half assert fresnels(1/z).series(z) == \ (-z**3/pi**2 + O(z**6))*sin(pi/(2*z**2)) + (-z/pi + 3*z**5/pi**3 + \ O(z**6))*cos(pi/(2*z**2)) + S.Half assert fresnelc(1/z).series(z) == \ (-z**3/pi**2 + O(z**6))*cos(pi/(2*z**2)) + (z/pi - 3*z**5/pi**3 + \ O(z**6))*sin(pi/(2*z**2)) + S.Half assert fresnelc(w).is_real is True assert fresnelc(z).as_real_imag() == \ ((fresnelc(re(z) - I*re(z)*Abs(im(z))/Abs(re(z)))/2 + fresnelc(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))/2, I*(fresnelc(re(z) - I*re(z)*Abs(im(z))/Abs(re(z))) - fresnelc(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))) * re(z)*Abs(im(z))/(2*im(z)*Abs(re(z))))) assert fresnelc(2 + 3*I).as_real_imag() == ( fresnelc(2 - 3*I)/2 + fresnelc(2 + 3*I)/2, I*(fresnelc(2 - 3*I) - fresnelc(2 + 3*I))/2 ) assert expand_func(integrate(fresnelc(z), z)) == \ z*fresnelc(z) - sin(pi*z**2/2)/pi assert fresnelc(z).rewrite(meijerg) == sqrt(2)*pi*z**(S(3)/4) * \ meijerg(((), (1,)), ((S(1)/4,), (S(3)/4, 0)), -pi**2*z**4/16)/(2*(-z)**(S(1)/4)*(z**2)**(S(1)/4)) from sympy.utilities.randtest import verify_numerically verify_numerically(re(fresnels(z)), fresnels(z).as_real_imag()[0], z) verify_numerically(im(fresnels(z)), fresnels(z).as_real_imag()[1], z) verify_numerically(fresnels(z), fresnels(z).rewrite(hyper), z) verify_numerically(fresnels(z), fresnels(z).rewrite(meijerg), z) verify_numerically(re(fresnelc(z)), fresnelc(z).as_real_imag()[0], z) verify_numerically(im(fresnelc(z)), fresnelc(z).as_real_imag()[1], z) verify_numerically(fresnelc(z), fresnelc(z).rewrite(hyper), z) verify_numerically(fresnelc(z), fresnelc(z).rewrite(meijerg), z)
def test_harmonic_rational(): ne = S(6) no = S(5) pe = S(8) po = S(9) qe = S(10) qo = S(13) Heee = harmonic(ne + pe / qe) Aeee = (-log(10) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + pi * sqrt(2 * sqrt(5) / 5 + 1) / 2 + Rational(13944145, 4720968)) Heeo = harmonic(ne + pe / qo) Aeeo = (-log(26) + 2 * log(sin(pi * Rational(3, 13))) * cos(pi * Rational(4, 13)) + 2 * log(sin(pi * Rational(2, 13))) * cos(pi * Rational(32, 13)) + 2 * log(sin(pi * Rational(5, 13))) * cos(pi * Rational(80, 13)) - 2 * log(sin(pi * Rational(6, 13))) * cos(pi * Rational(5, 13)) - 2 * log(sin(pi * Rational(4, 13))) * cos(pi / 13) + pi * cot(pi * Rational(5, 13)) / 2 - 2 * log(sin(pi / 13)) * cos(pi * Rational(3, 13)) + Rational(2422020029, 702257080)) Heoe = harmonic(ne + po / qe) Aeoe = ( -log(20) + 2 * (Rational(1, 4) + sqrt(5) / 4) * log(Rational(-1, 4) + sqrt(5) / 4) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 + Rational(1, 4)) * log(Rational(1, 4) + sqrt(5) / 4) + Rational(11818877030, 4286604231) + pi * sqrt(2 * sqrt(5) + 5) / 2) Heoo = harmonic(ne + po / qo) Aeoo = (-log(26) + 2 * log(sin(pi * Rational(3, 13))) * cos(pi * Rational(54, 13)) + 2 * log(sin(pi * Rational(4, 13))) * cos(pi * Rational(6, 13)) + 2 * log(sin(pi * Rational(6, 13))) * cos(pi * Rational(108, 13)) - 2 * log(sin(pi * Rational(5, 13))) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(pi * Rational(5, 13)) + pi * cot(pi * Rational(4, 13)) / 2 - 2 * log(sin(pi * Rational(2, 13))) * cos(pi * Rational(3, 13)) + Rational(11669332571, 3628714320)) Hoee = harmonic(no + pe / qe) Aoee = (-log(10) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + pi * sqrt(2 * sqrt(5) / 5 + 1) / 2 + Rational(779405, 277704)) Hoeo = harmonic(no + pe / qo) Aoeo = (-log(26) + 2 * log(sin(pi * Rational(3, 13))) * cos(pi * Rational(4, 13)) + 2 * log(sin(pi * Rational(2, 13))) * cos(pi * Rational(32, 13)) + 2 * log(sin(pi * Rational(5, 13))) * cos(pi * Rational(80, 13)) - 2 * log(sin(pi * Rational(6, 13))) * cos(pi * Rational(5, 13)) - 2 * log(sin(pi * Rational(4, 13))) * cos(pi / 13) + pi * cot(pi * Rational(5, 13)) / 2 - 2 * log(sin(pi / 13)) * cos(pi * Rational(3, 13)) + Rational(53857323, 16331560)) Hooe = harmonic(no + po / qe) Aooe = ( -log(20) + 2 * (Rational(1, 4) + sqrt(5) / 4) * log(Rational(-1, 4) + sqrt(5) / 4) + 2 * (Rational(-1, 4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 - Rational(1, 4)) * log(sqrt(sqrt(5) / 8 + Rational(5, 8))) + 2 * (-sqrt(5) / 4 + Rational(1, 4)) * log(Rational(1, 4) + sqrt(5) / 4) + Rational(486853480, 186374097) + pi * sqrt(2 * sqrt(5) + 5) / 2) Hooo = harmonic(no + po / qo) Aooo = (-log(26) + 2 * log(sin(pi * Rational(3, 13))) * cos(pi * Rational(54, 13)) + 2 * log(sin(pi * Rational(4, 13))) * cos(pi * Rational(6, 13)) + 2 * log(sin(pi * Rational(6, 13))) * cos(pi * Rational(108, 13)) - 2 * log(sin(pi * Rational(5, 13))) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(pi * Rational(5, 13)) + pi * cot(pi * Rational(4, 13)) / 2 - 2 * log(sin(pi * Rational(2, 13))) * cos(3 * pi / 13) + Rational(383693479, 125128080)) H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo] A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo] for h, a in zip(H, A): e = expand_func(h).doit() assert cancel(e / a) == 1 assert abs(h.n() - a.n()) < 1e-12
def test_harmonic_rational(): ne = S(6) no = S(5) pe = S(8) po = S(9) qe = S(10) qo = S(13) Heee = harmonic(ne + pe / qe) Aeee = (-log(10) + 2 * (-1 / S(4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + 5 / S(8))) + 2 * (-sqrt(5) / 4 - 1 / S(4)) * log(sqrt(sqrt(5) / 8 + 5 / S(8))) + pi * (1 / S(4) + sqrt(5) / 4) / (2 * sqrt(-sqrt(5) / 8 + 5 / S(8))) + 13944145 / S(4720968)) Heeo = harmonic(ne + pe / qo) Aeeo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(4 * pi / 13) + 2 * log(sin(2 * pi / 13)) * cos(32 * pi / 13) + 2 * log(sin(5 * pi / 13)) * cos(80 * pi / 13) - 2 * log(sin(6 * pi / 13)) * cos(5 * pi / 13) - 2 * log(sin(4 * pi / 13)) * cos(pi / 13) + pi * cot(5 * pi / 13) / 2 - 2 * log(sin(pi / 13)) * cos(3 * pi / 13) + 2422020029 / S(702257080)) Heoe = harmonic(ne + po / qe) Aeoe = (-log(20) + 2 * (1 / S(4) + sqrt(5) / 4) * log(-1 / S(4) + sqrt(5) / 4) + 2 * (-1 / S(4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + 5 / S(8))) + 2 * (-sqrt(5) / 4 - 1 / S(4)) * log(sqrt(sqrt(5) / 8 + 5 / S(8))) + 2 * (-sqrt(5) / 4 + 1 / S(4)) * log(1 / S(4) + sqrt(5) / 4) + 11818877030 / S(4286604231) + pi * (sqrt(5) / 8 + 5 / S(8)) / sqrt(-sqrt(5) / 8 + 5 / S(8))) Heoo = harmonic(ne + po / qo) Aeoo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(54 * pi / 13) + 2 * log(sin(4 * pi / 13)) * cos(6 * pi / 13) + 2 * log(sin(6 * pi / 13)) * cos(108 * pi / 13) - 2 * log(sin(5 * pi / 13)) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(5 * pi / 13) + pi * cot(4 * pi / 13) / 2 - 2 * log(sin(2 * pi / 13)) * cos(3 * pi / 13) + 11669332571 / S(3628714320)) Hoee = harmonic(no + pe / qe) Aoee = (-log(10) + 2 * (-1 / S(4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + 5 / S(8))) + 2 * (-sqrt(5) / 4 - 1 / S(4)) * log(sqrt(sqrt(5) / 8 + 5 / S(8))) + pi * (1 / S(4) + sqrt(5) / 4) / (2 * sqrt(-sqrt(5) / 8 + 5 / S(8))) + 779405 / S(277704)) Hoeo = harmonic(no + pe / qo) Aoeo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(4 * pi / 13) + 2 * log(sin(2 * pi / 13)) * cos(32 * pi / 13) + 2 * log(sin(5 * pi / 13)) * cos(80 * pi / 13) - 2 * log(sin(6 * pi / 13)) * cos(5 * pi / 13) - 2 * log(sin(4 * pi / 13)) * cos(pi / 13) + pi * cot(5 * pi / 13) / 2 - 2 * log(sin(pi / 13)) * cos(3 * pi / 13) + 53857323 / S(16331560)) Hooe = harmonic(no + po / qe) Aooe = (-log(20) + 2 * (1 / S(4) + sqrt(5) / 4) * log(-1 / S(4) + sqrt(5) / 4) + 2 * (-1 / S(4) + sqrt(5) / 4) * log(sqrt(-sqrt(5) / 8 + 5 / S(8))) + 2 * (-sqrt(5) / 4 - 1 / S(4)) * log(sqrt(sqrt(5) / 8 + 5 / S(8))) + 2 * (-sqrt(5) / 4 + 1 / S(4)) * log(1 / S(4) + sqrt(5) / 4) + 486853480 / S(186374097) + pi * (sqrt(5) / 8 + 5 / S(8)) / sqrt(-sqrt(5) / 8 + 5 / S(8))) Hooo = harmonic(no + po / qo) Aooo = (-log(26) + 2 * log(sin(3 * pi / 13)) * cos(54 * pi / 13) + 2 * log(sin(4 * pi / 13)) * cos(6 * pi / 13) + 2 * log(sin(6 * pi / 13)) * cos(108 * pi / 13) - 2 * log(sin(5 * pi / 13)) * cos(pi / 13) - 2 * log(sin(pi / 13)) * cos(5 * pi / 13) + pi * cot(4 * pi / 13) / 2 - 2 * log(sin(2 * pi / 13)) * cos(3 * pi / 13) + 383693479 / S(125128080)) H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo] A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo] for h, a in zip(H, A): e = expand_func(h).doit() assert cancel(e / a) == 1 assert abs(h.n() - a.n()) < 1e-12
def test_binomial(): x = Symbol('x') n = Symbol('n', integer=True) nz = Symbol('nz', integer=True, nonzero=True) k = Symbol('k', integer=True) kp = Symbol('kp', integer=True, positive=True) kn = Symbol('kn', integer=True, negative=True) u = Symbol('u', negative=True) v = Symbol('v', nonnegative=True) p = Symbol('p', positive=True) z = Symbol('z', zero=True) nt = Symbol('nt', integer=False) kt = Symbol('kt', integer=False) a = Symbol('a', integer=True, nonnegative=True) b = Symbol('b', integer=True, nonnegative=True) assert binomial(0, 0) == 1 assert binomial(1, 1) == 1 assert binomial(10, 10) == 1 assert binomial(n, z) == 1 assert binomial(1, 2) == 0 assert binomial(-1, 2) == 1 assert binomial(1, -1) == 0 assert binomial(-1, 1) == -1 assert binomial(-1, -1) == 0 assert binomial(S.Half, S.Half) == 1 assert binomial(-10, 1) == -10 assert binomial(-10, 7) == -11440 assert binomial(n, -1) == 0 # holds for all integers (negative, zero, positive) assert binomial(kp, -1) == 0 assert binomial(nz, 0) == 1 assert expand_func(binomial(n, 1)) == n assert expand_func(binomial(n, 2)) == n*(n - 1)/2 assert expand_func(binomial(n, n - 2)) == n*(n - 1)/2 assert expand_func(binomial(n, n - 1)) == n assert binomial(n, 3).func == binomial assert binomial(n, 3).expand(func=True) == n**3/6 - n**2/2 + n/3 assert expand_func(binomial(n, 3)) == n*(n - 2)*(n - 1)/6 assert binomial(n, n).func == binomial # e.g. (-1, -1) == 0, (2, 2) == 1 assert binomial(n, n + 1).func == binomial # e.g. (-1, 0) == 1 assert binomial(kp, kp + 1) == 0 assert binomial(kn, kn) == 0 # issue #14529 assert binomial(n, u).func == binomial assert binomial(kp, u).func == binomial assert binomial(n, p).func == binomial assert binomial(n, k).func == binomial assert binomial(n, n + p).func == binomial assert binomial(kp, kp + p).func == binomial assert expand_func(binomial(n, n - 3)) == n*(n - 2)*(n - 1)/6 assert binomial(n, k).is_integer assert binomial(nt, k).is_integer is None assert binomial(x, nt).is_integer is False assert binomial(gamma(25), 6) == 79232165267303928292058750056084441948572511312165380965440075720159859792344339983120618959044048198214221915637090855535036339620413440000 assert binomial(1324, 47) == 906266255662694632984994480774946083064699457235920708992926525848438478406790323869952 assert binomial(1735, 43) == 190910140420204130794758005450919715396159959034348676124678207874195064798202216379800 assert binomial(2512, 53) == 213894469313832631145798303740098720367984955243020898718979538096223399813295457822575338958939834177325304000 assert binomial(3383, 52) == 27922807788818096863529701501764372757272890613101645521813434902890007725667814813832027795881839396839287659777235 assert binomial(4321, 51) == 124595639629264868916081001263541480185227731958274383287107643816863897851139048158022599533438936036467601690983780576 assert binomial(a, b).is_nonnegative is True assert binomial(-1, 2, evaluate=False).is_nonnegative is True assert binomial(10, 5, evaluate=False).is_nonnegative is True assert binomial(10, -3, evaluate=False).is_nonnegative is True assert binomial(-10, -3, evaluate=False).is_nonnegative is True assert binomial(-10, 2, evaluate=False).is_nonnegative is True assert binomial(-10, 1, evaluate=False).is_nonnegative is False assert binomial(-10, 7, evaluate=False).is_nonnegative is False # issue #14625 for _ in (pi, -pi, nt, v, a): assert binomial(_, _) == 1 assert binomial(_, _ - 1) == _ assert isinstance(binomial(u, u), binomial) assert isinstance(binomial(u, u - 1), binomial) assert isinstance(binomial(x, x), binomial) assert isinstance(binomial(x, x - 1), binomial) # issue #13980 and #13981 assert binomial(-7, -5) == 0 assert binomial(-23, -12) == 0 assert binomial(S(13)/2, -10) == 0 assert binomial(-49, -51) == 0 assert binomial(19, S(-7)/2) == S(-68719476736)/(911337863661225*pi) assert binomial(0, S(3)/2) == S(-2)/(3*pi) assert binomial(-3, S(-7)/2) == zoo assert binomial(kn, kt) == zoo assert binomial(nt, kt).func == binomial assert binomial(nt, S(15)/6) == 8*gamma(nt + 1)/(15*sqrt(pi)*gamma(nt - S(3)/2)) assert binomial(S(20)/3, S(-10)/8) == gamma(S(23)/3)/(gamma(S(-1)/4)*gamma(S(107)/12)) assert binomial(S(19)/2, S(-7)/2) == S(-1615)/8388608 assert binomial(S(-13)/5, S(-7)/8) == gamma(S(-8)/5)/(gamma(S(-29)/40)*gamma(S(1)/8)) assert binomial(S(-19)/8, S(-13)/5) == gamma(S(-11)/8)/(gamma(S(-8)/5)*gamma(S(49)/40)) # binomial for complexes from sympy import I assert binomial(I, S(-89)/8) == gamma(1 + I)/(gamma(S(-81)/8)*gamma(S(97)/8 + I)) assert binomial(I, 2*I) == gamma(1 + I)/(gamma(1 - I)*gamma(1 + 2*I)) assert binomial(-7, I) == zoo assert binomial(-7/S(6), I) == gamma(-1/S(6))/(gamma(-1/S(6) - I)*gamma(1 + I)) assert binomial((1+2*I), (1+3*I)) == gamma(2 + 2*I)/(gamma(1 - I)*gamma(2 + 3*I)) assert binomial(I, 5) == S(1)/3 - I/S(12) assert binomial((2*I + 3), 7) == -13*I/S(63) assert isinstance(binomial(I, n), binomial)
def test_issue_4992(): # Note: psi in _check_antecedents becomes NaN. from sympy import simplify, expand_func, polygamma, gamma a = Symbol('a', positive=True) assert simplify(expand_func(integrate(exp(-x)*log(x)*x**a, (x, 0, oo)))) == \ (a*polygamma(0, a) + 1)*gamma(a)
def test_expand(): assert expand_func(besselj(S(1)/2, z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z)) assert expand_func( bessely(S(1)/2, z)) == -sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))
def test_fresnel(): from sympy import fresnels, fresnelc assert expand_func(integrate(sin(pi * x**2 / 2), x)) == fresnels(x) assert expand_func(integrate(cos(pi * x**2 / 2), x)) == fresnelc(x)
def mjn(n, z): return expand_func(jn(n, z))