def test_erf(): assert erf(nan) == nan assert erf(oo) == 1 assert erf(-oo) == -1 assert erf(0) == 0 assert erf(I * oo) == oo * I assert erf(-I * oo) == -oo * I assert erf(-2) == -erf(2) assert erf(-x * y) == -erf(x * y) assert erf(-x - y) == -erf(x + y) assert erf(erfinv(x)) == x assert erf(erfcinv(x)) == 1 - x assert erf(erf2inv(0, x)) == x assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x assert erf(I).is_extended_real is False assert erf(w).is_extended_real is True assert erf(z).is_extended_real is None assert conjugate(erf(z)) == erf(conjugate(z)) assert erf(x).as_leading_term(x) == 2 * x / sqrt(pi) assert erf(1 / x).as_leading_term(x) == erf(1 / x) assert erf(z).rewrite('uppergamma') == sqrt(z**2) * erf(sqrt(z**2)) / z assert erf(z).rewrite('erfc') == S.One - erfc(z) assert erf(z).rewrite('erfi') == -I * erfi(I * z) assert erf(z).rewrite('fresnels') == (1 + I) * ( fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z * (1 - I) / sqrt(pi))) assert erf(z).rewrite('fresnelc') == (1 + I) * ( fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z * (1 - I) / sqrt(pi))) assert erf(z).rewrite('hyper') == 2 * z * hyper([S.Half], [3 * S.Half], -z**2) / sqrt(pi) assert erf(z).rewrite('meijerg') == z * meijerg([S.Half], [], [0], [-S.Half], z**2) / sqrt(pi) assert erf(z).rewrite( 'expint') == sqrt(z**2) / z - z * expint(S.Half, z**2) / sqrt(S.Pi) assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \ 2/sqrt(pi) assert limit((1 - erf(z)) * exp(z**2) * z, z, oo) == 1 / sqrt(pi) assert limit((1 - erf(x)) * exp(x**2) * sqrt(pi) * x, x, oo) == 1 assert limit(((1 - erf(x)) * exp(x**2) * sqrt(pi) * x - 1) * 2 * x**2, x, oo) == -1 assert erf(x).as_real_imag() == \ ((erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 + erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2, I*(erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) - erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) * re(x)*Abs(im(x))/(2*im(x)*Abs(re(x))))) pytest.raises(ArgumentIndexError, lambda: erf(x).fdiff(2))
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) == Integer(2) - erfc(x) assert erfc(erfcinv(x)) == x assert erfc(erfinv(x)) == 1 - x assert erfc(I).is_extended_real is False assert erfc(w).is_extended_real is True assert erfc(z).is_extended_real is None assert conjugate(erfc(z)) == erfc(conjugate(z)) assert erfc(x).as_leading_term(x) == 1 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( [Rational(1, 2)], [Rational(3, 2)], -z**2) / sqrt(pi) assert erfc(z).rewrite('meijerg') == 1 - z * meijerg( [Rational(1, 2)], [], [0], [Rational(-1, 2)], z**2) / sqrt(pi) assert erfc(z).rewrite( 'uppergamma') == 1 - sqrt(z**2) * erf(sqrt(z**2)) / z assert erfc(z).rewrite('expint') == 1 - sqrt(z**2) / z + z * expint( Rational(1, 2), z**2) / sqrt(pi) 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))))) assert erfc(x).as_real_imag(deep=False) == erfc(x).as_real_imag() assert erfc(w).as_real_imag() == (erfc(w), 0) assert erfc(w).as_real_imag(deep=False) == erfc(w).as_real_imag() assert erfc(I).as_real_imag() == (1, -erfi(1)) pytest.raises(ArgumentIndexError, lambda: erfc(x).fdiff(2)) assert erfc(x).taylor_term(3, x, *(-2 * x / sqrt(pi), 0)) == 2 * x**3 / 3 / sqrt(pi) assert erfc(x).limit(x, oo) == 0 assert erfc(x).diff(x) == -2 * exp(-x**2) / sqrt(pi)
def test_erfi(): assert erfi(nan) == nan assert erfi(+oo) == +oo assert erfi(-oo) == -oo assert erfi(0) == 0 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_extended_real is False assert erfi(w).is_extended_real is True assert erfi(z).is_extended_real is None 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( [Rational(1, 2)], [Rational(3, 2)], z**2) / sqrt(pi) assert erfi(z).rewrite('meijerg') == z * meijerg( [Rational(1, 2)], [], [0], [Rational(-1, 2)], -z**2) / sqrt(pi) assert erfi(z).rewrite('uppergamma') == ( sqrt(-z**2) / z * (uppergamma(Rational(1, 2), -z**2) / sqrt(pi) - 1)) assert erfi(z).rewrite('expint') == sqrt(-z**2) / z - z * expint( Rational(1, 2), -z**2) / sqrt(pi) 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))))) assert erfi(x).as_real_imag(deep=False) == erfi(x).as_real_imag() assert erfi(w).as_real_imag() == (erfi(w), 0) assert erfi(w).as_real_imag(deep=False) == erfi(w).as_real_imag() assert erfi(I).as_real_imag() == (0, erf(1)) pytest.raises(ArgumentIndexError, lambda: erfi(x).fdiff(2)) assert erfi(x).taylor_term(3, x, *(2 * x / sqrt(pi), 0)) == 2 * x**3 / 3 / sqrt(pi) assert erfi(x).limit(x, oo) == oo
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) == Integer(2) - erfc(x) assert erfc(erfcinv(x)) == x assert erfc(erfinv(x)) == 1 - x assert erfc(I).is_extended_real is False assert erfc(w).is_extended_real is True assert erfc(z).is_extended_real is None assert conjugate(erfc(z)) == erfc(conjugate(z)) assert erfc(x).as_leading_term(x) == 1 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([Rational(1, 2)], [Rational(3, 2)], -z**2)/sqrt(pi) assert erfc(z).rewrite('meijerg') == 1 - z*meijerg([Rational(1, 2)], [], [0], [Rational(-1, 2)], z**2)/sqrt(pi) assert erfc(z).rewrite('uppergamma') == 1 - sqrt(z**2)*erf(sqrt(z**2))/z assert erfc(z).rewrite('expint') == 1 - sqrt(z**2)/z + z*expint(Rational(1, 2), z**2)/sqrt(pi) 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))))) assert erfc(x).as_real_imag(deep=False) == erfc(x).as_real_imag() assert erfc(w).as_real_imag() == (erfc(w), 0) assert erfc(w).as_real_imag(deep=False) == erfc(w).as_real_imag() assert erfc(I).as_real_imag() == (1, -erfi(1)) pytest.raises(ArgumentIndexError, lambda: erfc(x).fdiff(2)) assert erfc(x).taylor_term(3, x, *(-2*x/sqrt(pi), 0)) == 2*x**3/3/sqrt(pi) assert erfc(x).limit(x, oo) == 0 assert erfc(x).diff(x) == -2*exp(-x**2)/sqrt(pi)
def test_erfi(): assert erfi(nan) == nan assert erfi(+oo) == +oo assert erfi(-oo) == -oo assert erfi(0) == 0 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_extended_real is False assert erfi(w).is_extended_real is True assert erfi(z).is_extended_real is None 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([Rational(1, 2)], [Rational(3, 2)], z**2)/sqrt(pi) assert erfi(z).rewrite('meijerg') == z*meijerg([Rational(1, 2)], [], [0], [Rational(-1, 2)], -z**2)/sqrt(pi) assert erfi(z).rewrite('uppergamma') == (sqrt(-z**2)/z*(uppergamma(Rational(1, 2), -z**2)/sqrt(pi) - 1)) assert erfi(z).rewrite('expint') == sqrt(-z**2)/z - z*expint(Rational(1, 2), -z**2)/sqrt(pi) 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))))) assert erfi(x).as_real_imag(deep=False) == erfi(x).as_real_imag() assert erfi(w).as_real_imag() == (erfi(w), 0) assert erfi(w).as_real_imag(deep=False) == erfi(w).as_real_imag() assert erfi(I).as_real_imag() == (0, erf(1)) pytest.raises(ArgumentIndexError, lambda: erfi(x).fdiff(2)) assert erfi(x).taylor_term(3, x, *(2*x/sqrt(pi), 0)) == 2*x**3/3/sqrt(pi) assert erfi(x).limit(x, oo) == oo
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_extended_real is False assert erfi(w).is_extended_real is True assert erfi(z).is_extended_real is None 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 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))))) pytest.raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))
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) == Integer(2) - erfc(x) assert erfc(erfcinv(x)) == x assert erfc(I).is_extended_real is False assert erfc(w).is_extended_real is True assert erfc(z).is_extended_real is None 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) * erf(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(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))))) pytest.raises(ArgumentIndexError, lambda: erfc(x).fdiff(2))
def test_fresnel(): 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 test_sympyissue_3686( ): # remove this when fresnel itegrals are implemented assert expand_func(integrate(sin(x**2), x)) == \ sqrt(2)*sqrt(pi)*fresnels(sqrt(2)*x/sqrt(pi))/2
def test_fresnel(): 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 test_fresnel(): from diofant 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 test_fresnel(): assert fresnels(0) == 0 assert fresnels(+oo) == Rational(+1, 2) assert fresnels(-oo) == Rational(-1, 2) 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) == (1 + I) / 4 * (erf( (1 + I) / 2 * sqrt(pi) * z) - I * erf((1 - I) / 2 * sqrt(pi) * z)) assert fresnels(z).rewrite(hyper) == \ pi*z**3/6 * hyper([Rational(3, 4)], [Rational(3, 2), Rational(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(y / z).limit(z, 0) == fresnels(oo * sign(y)) assert fresnels(x).taylor_term(-1, z) == 0 assert fresnels(x).taylor_term(1, z, *(pi * z**3 / 6, )) == -pi**3 * z**7 / 336 assert fresnels(x).taylor_term(1, z) == -pi**3 * z**7 / 336 assert fresnels(w).is_extended_real is True assert fresnels(z).is_extended_real is None 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(z).as_real_imag(deep=False) == fresnels(z).as_real_imag() assert fresnels(w).as_real_imag() == (fresnels(w), 0) assert fresnels(w).as_real_imag(deep=False) == fresnels(w).as_real_imag() assert (fresnels(I, evaluate=False).as_real_imag() == ( 0, -erf(sqrt(pi) / 2 + I * sqrt(pi) / 2) / 4 + I * (-erf(sqrt(pi) / 2 + I * sqrt(pi) / 2) + erf(sqrt(pi) / 2 - I * sqrt(pi) / 2)) / 4 - erf(sqrt(pi) / 2 - I * sqrt(pi) / 2) / 4)) 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**Rational(9, 4) * \ meijerg(((), (1,)), ((Rational(3, 4),), (Rational(1, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(3, 4)*(z**2)**Rational(3, 4)) assert fresnelc(0) == 0 assert fresnelc(+oo) == Rational(+1, 2) assert fresnelc(-oo) == Rational(-1, 2) 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) pytest.raises(ArgumentIndexError, lambda: fresnels(z).fdiff(2)) pytest.raises(ArgumentIndexError, lambda: fresnelc(z).fdiff(2)) assert fresnelc(z).rewrite(erf) == (1 - I) / 4 * (erf( (1 + I) / 2 * sqrt(pi) * z) + I * erf((1 - I) / 2 * sqrt(pi) * z)) assert fresnelc(z).rewrite(hyper) == \ z * hyper([Rational(1, 4)], [Rational(1, 2), Rational(5, 4)], -pi**2*z**4/16) assert fresnelc(x).taylor_term(-1, z) == 0 assert fresnelc(x).taylor_term(1, z, *(z, )) == -pi**2 * z**5 / 40 assert fresnelc(x).taylor_term(1, z) == -pi**2 * z**5 / 40 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) assert fresnelc(y / z).limit(z, 0) == fresnelc(oo * sign(y)) # issue sympy/sympy#6510 assert fresnels(z).series(z, oo) == \ (-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) + Rational(1, 2) assert fresnelc(z).series(z, oo) == \ (-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) + Rational(1, 2) 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)) + Rational(1, 2) 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)) + Rational(1, 2) assert fresnelc(w).is_extended_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**Rational(3, 4) * \ meijerg(((), (1,)), ((Rational(1, 4),), (Rational(3, 4), 0)), -pi**2*z**4/16)/(2*root(-z, 4)*root(z**2, 4)) 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_erf(): assert erf(nan) == nan assert erf(oo) == 1 assert erf(-oo) == -1 assert erf(0) == 0 assert erf(I*oo) == oo*I assert erf(-I*oo) == -oo*I assert erf(-2) == -erf(2) assert erf(-x*y) == -erf(x*y) assert erf(-x - y) == -erf(x + y) assert erf(erfinv(x)) == x assert erf(erfcinv(x)) == 1 - x assert erf(erf2inv(0, x)) == x assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x assert erf(I).is_extended_real is False assert erf(w).is_extended_real is True assert erf(z).is_extended_real is None assert conjugate(erf(z)) == erf(conjugate(z)) assert erf(x).as_leading_term(x) == 2*x/sqrt(pi) assert erf(1/x).as_leading_term(x) == erf(1/x) assert erf(z).rewrite('uppergamma') == sqrt(z**2)*erf(sqrt(z**2))/z assert erf(z).rewrite('erfc') == 1 - erfc(z) assert erf(z).rewrite('erfi') == -I*erfi(I*z) assert erf(z).rewrite('fresnels') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erf(z).rewrite('fresnelc') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erf(z).rewrite('hyper') == 2*z*hyper([Rational(1, 2)], [Rational(3, 2)], -z**2)/sqrt(pi) assert erf(z).rewrite('meijerg') == z*meijerg([Rational(1, 2)], [], [0], [Rational(-1, 2)], z**2)/sqrt(pi) assert erf(z).rewrite('expint') == sqrt(z**2)/z - z*expint(Rational(1, 2), z**2)/sqrt(pi) assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \ 2/sqrt(pi) assert limit((1 - erf(z))*exp(z**2)*z, z, oo) == 1/sqrt(pi) assert limit((1 - erf(x))*exp(x**2)*sqrt(pi)*x, x, oo) == 1 assert limit(((1 - erf(x))*exp(x**2)*sqrt(pi)*x - 1)*2*x**2, x, oo) == -1 l = Limit((1 - erf(y/x))*exp(y**2/x**2), x, 0) assert l.doit() == l # cover _erfs._eval_aseries assert erf(x).as_real_imag() == \ ((erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x)))/2 + erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))/2, I*(erf(re(x) - I*re(x)*Abs(im(x))/Abs(re(x))) - erf(re(x) + I*re(x)*Abs(im(x))/Abs(re(x)))) * re(x)*Abs(im(x))/(2*im(x)*Abs(re(x))))) assert erf(x).as_real_imag() == erf(x).as_real_imag(deep=False) assert erf(w).as_real_imag() == (erf(w), 0) assert erf(w).as_real_imag() == erf(w).as_real_imag(deep=False) assert erf(I).as_real_imag() == (0, erfi(1)) pytest.raises(ArgumentIndexError, lambda: erf(x).fdiff(2)) assert erf(x).taylor_term(3, x, *(2*x/sqrt(pi), 0)) == -2*x**3/3/sqrt(pi)
def test_sympyissue_3686(): # remove this when fresnel itegrals are implemented assert expand_func(integrate(sin(x**2), x)) == \ sqrt(2)*sqrt(pi)*fresnels(sqrt(2)*x/sqrt(pi))/2
def test_fresnel(): assert fresnels(0) == 0 assert fresnels(+oo) == Rational(+1, 2) assert fresnels(-oo) == Rational(-1, 2) 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) == (1 + I)/4 * ( erf((1 + I)/2*sqrt(pi)*z) - I*erf((1 - I)/2*sqrt(pi)*z)) assert fresnels(z).rewrite(hyper) == \ pi*z**3/6 * hyper([Rational(3, 4)], [Rational(3, 2), Rational(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(y/z).limit(z, 0) == fresnels(oo*sign(y)) assert fresnels(x).taylor_term(-1, z) == 0 assert fresnels(x).taylor_term(1, z, *(pi*z**3/6,)) == -pi**3*z**7/336 assert fresnels(x).taylor_term(1, z) == -pi**3*z**7/336 assert fresnels(w).is_extended_real is True assert fresnels(z).is_extended_real is None 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(z).as_real_imag(deep=False) == fresnels(z).as_real_imag() assert fresnels(w).as_real_imag() == (fresnels(w), 0) assert fresnels(w).as_real_imag(deep=False) == fresnels(w).as_real_imag() assert (fresnels(I, evaluate=False).as_real_imag() == (0, -erf(sqrt(pi)/2 + I*sqrt(pi)/2)/4 + I*(-erf(sqrt(pi)/2 + I*sqrt(pi)/2) + erf(sqrt(pi)/2 - I*sqrt(pi)/2))/4 - erf(sqrt(pi)/2 - I*sqrt(pi)/2)/4)) 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**Rational(9, 4) * \ meijerg(((), (1,)), ((Rational(3, 4),), (Rational(1, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(3, 4)*(z**2)**Rational(3, 4)) assert fresnelc(0) == 0 assert fresnelc(+oo) == Rational(+1, 2) assert fresnelc(-oo) == Rational(-1, 2) 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) pytest.raises(ArgumentIndexError, lambda: fresnels(z).fdiff(2)) pytest.raises(ArgumentIndexError, lambda: fresnelc(z).fdiff(2)) assert fresnelc(z).rewrite(erf) == (1 - I)/4 * ( erf((1 + I)/2*sqrt(pi)*z) + I*erf((1 - I)/2*sqrt(pi)*z)) assert fresnelc(z).rewrite(hyper) == \ z * hyper([Rational(1, 4)], [Rational(1, 2), Rational(5, 4)], -pi**2*z**4/16) assert fresnelc(x).taylor_term(-1, z) == 0 assert fresnelc(x).taylor_term(1, z, *(z,)) == -pi**2*z**5/40 assert fresnelc(x).taylor_term(1, z) == -pi**2*z**5/40 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) assert fresnelc(y/z).limit(z, 0) == fresnelc(oo*sign(y)) # issue sympy/sympy#6510 assert fresnels(z).series(z, oo) == \ (-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) + Rational(1, 2) assert fresnelc(z).series(z, oo) == \ (-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) + Rational(1, 2) 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)) + Rational(1, 2) 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)) + Rational(1, 2) assert fresnelc(w).is_extended_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**Rational(3, 4) * \ meijerg(((), (1,)), ((Rational(1, 4),), (Rational(3, 4), 0)), -pi**2*z**4/16)/(2*root(-z, 4)*root(z**2, 4)) 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_erf(): assert erf(nan) == nan assert erf(oo) == 1 assert erf(-oo) == -1 assert erf(0) == 0 assert erf(I * oo) == oo * I assert erf(-I * oo) == -oo * I assert erf(-2) == -erf(2) assert erf(-x * y) == -erf(x * y) assert erf(-x - y) == -erf(x + y) assert erf(erfinv(x)) == x assert erf(erfcinv(x)) == 1 - x assert erf(erf2inv(0, x)) == x assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x assert erf(I).is_extended_real is False assert erf(w).is_extended_real is True assert erf(z).is_extended_real is None assert conjugate(erf(z)) == erf(conjugate(z)) assert erf(x).as_leading_term(x) == 2 * x / sqrt(pi) assert erf(1 / x).as_leading_term(x) == erf(1 / x) assert erf(z).rewrite('uppergamma') == sqrt(z**2) * erf(sqrt(z**2)) / z assert erf(z).rewrite('erfc') == 1 - erfc(z) assert erf(z).rewrite('erfi') == -I * erfi(I * z) assert erf(z).rewrite('fresnels') == (1 + I) * ( fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z * (1 - I) / sqrt(pi))) assert erf(z).rewrite('fresnelc') == (1 + I) * ( fresnelc(z * (1 - I) / sqrt(pi)) - I * fresnels(z * (1 - I) / sqrt(pi))) assert erf(z).rewrite('hyper') == 2 * z * hyper( [Rational(1, 2)], [Rational(3, 2)], -z**2) / sqrt(pi) assert erf(z).rewrite('meijerg') == z * meijerg( [Rational(1, 2)], [], [0], [Rational(-1, 2)], z**2) / sqrt(pi) assert erf(z).rewrite('expint') == sqrt(z**2) / z - z * expint( Rational(1, 2), z**2) / sqrt(pi) assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \ 2/sqrt(pi) assert limit((1 - erf(z)) * exp(z**2) * z, z, oo) == 1 / sqrt(pi) assert limit((1 - erf(x)) * exp(x**2) * sqrt(pi) * x, x, oo) == 1 assert limit(((1 - erf(x)) * exp(x**2) * sqrt(pi) * x - 1) * 2 * x**2, x, oo) == -1 l = Limit((1 - erf(y / x)) * exp(y**2 / x**2), x, 0) assert l.doit() == l # cover _erfs._eval_aseries assert erf(x).as_real_imag() == \ ((erf(re(x) - I*re(x)*abs(im(x))/abs(re(x)))/2 + erf(re(x) + I*re(x)*abs(im(x))/abs(re(x)))/2, I*(erf(re(x) - I*re(x)*abs(im(x))/abs(re(x))) - erf(re(x) + I*re(x)*abs(im(x))/abs(re(x)))) * re(x)*abs(im(x))/(2*im(x)*abs(re(x))))) assert erf(x).as_real_imag() == erf(x).as_real_imag(deep=False) assert erf(w).as_real_imag() == (erf(w), 0) assert erf(w).as_real_imag() == erf(w).as_real_imag(deep=False) assert erf(I).as_real_imag() == (0, erfi(1)) pytest.raises(ArgumentIndexError, lambda: erf(x).fdiff(2)) assert erf(x).taylor_term(3, x, *(2 * x / sqrt(pi), 0)) == -2 * x**3 / 3 / sqrt(pi)
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([Rational(3, 4)], [Rational(3, 2), Rational(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_extended_real is True assert fresnels(z).is_extended_real is None 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**Rational(9, 4) * \ meijerg(((), (1,)), ((Rational(3, 4),), (Rational(1, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(3, 4)*(z**2)**Rational(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, Rational(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_extended_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**Rational(3, 4) * \ meijerg(((), (1,)), ((Rational(1, 4),), (Rational(3, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(1, 4)*(z**2)**Rational(1, 4)) from diofant.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_laplace_transform(): from diofant import fresnels, fresnelc LT = laplace_transform a, b, c, = symbols('a b c', positive=True) t = symbols('t') w = Symbol("w") f = Function("f") # Test unevaluated form assert laplace_transform(f(t), t, w) == LaplaceTransform(f(t), t, w) assert inverse_laplace_transform(f(w), w, t, plane=0) == InverseLaplaceTransform( f(w), w, t, 0) # test a bug spos = symbols('s', positive=True) assert LT(exp(t), t, spos)[:2] == (1 / (spos - 1), True) # basic tests from wikipedia assert LT((t - a)**b*exp(-c*(t - a))*Heaviside(t - a), t, s) == \ ((s + c)**(-b - 1)*exp(-a*s)*gamma(b + 1), -c, True) assert LT(t**a, t, s) == (s**(-a - 1) * gamma(a + 1), 0, True) assert LT(Heaviside(t), t, s) == (1 / s, 0, True) assert LT(Heaviside(t - a), t, s) == (exp(-a * s) / s, 0, True) assert LT(1 - exp(-a * t), t, s) == (a / (s * (a + s)), 0, True) assert LT((exp(2*t) - 1)*exp(-b - t)*Heaviside(t)/2, t, s, noconds=True) \ == exp(-b)/(s**2 - 1) assert LT(exp(t), t, s)[:2] == (1 / (s - 1), 1) assert LT(exp(2 * t), t, s)[:2] == (1 / (s - 2), 2) assert LT(exp(a * t), t, s)[:2] == (1 / (s - a), a) assert LT(log(t / a), t, s) == ((log(a * s) + EulerGamma) / s / -1, 0, True) assert LT(erf(t), t, s) == ((-erf(s / 2) + 1) * exp(s**2 / 4) / s, 0, True) assert LT(sin(a * t), t, s) == (a / (a**2 + s**2), 0, True) assert LT(cos(a * t), t, s) == (s / (a**2 + s**2), 0, True) # TODO would be nice to have these come out better assert LT(exp(-a * t) * sin(b * t), t, s) == (b / (b**2 + (a + s)**2), -a, True) assert LT(exp(-a*t)*cos(b*t), t, s) == \ ((a + s)/(b**2 + (a + s)**2), -a, True) assert LT(besselj(0, t), t, s) == (1 / sqrt(1 + s**2), 0, True) assert LT(besselj(1, t), t, s) == (1 - 1 / sqrt(1 + 1 / s**2), 0, True) # TODO general order works, but is a *mess* # TODO besseli also works, but is an even greater mess # test a bug in conditions processing # TODO the auxiliary condition should be recognised/simplified assert LT(exp(t) * cos(t), t, s)[:-1] in [ ((s - 1) / (s**2 - 2 * s + 2), -oo), ((s - 1) / ((s - 1)**2 + 1), -oo), ] # Fresnel functions assert laplace_transform(fresnels(t), t, s) == \ ((-sin(s**2/(2*pi))*fresnels(s/pi) + sin(s**2/(2*pi))/2 - cos(s**2/(2*pi))*fresnelc(s/pi) + cos(s**2/(2*pi))/2)/s, 0, True) assert laplace_transform( fresnelc(t), t, s) == ((sin(s**2 / (2 * pi)) * fresnelc(s / pi) / s - cos(s**2 / (2 * pi)) * fresnels(s / pi) / s + sqrt(2) * cos(s**2 / (2 * pi) + pi / 4) / (2 * s), 0, True)) assert LT(Matrix([[exp(t), t*exp(-t)], [t*exp(-t), exp(t)]]), t, s) ==\ Matrix([ [(1/(s - 1), 1, True), ((s + 1)**(-2), 0, True)], [((s + 1)**(-2), 0, True), (1/(s - 1), 1, True)] ])
def test_laplace_transform(): LT = laplace_transform a, b, c, = symbols('a b c', positive=True) f = Function("f") # Test unevaluated form assert laplace_transform(f(t), t, w) == LaplaceTransform(f(t), t, w) assert inverse_laplace_transform( f(w), w, t, plane=0) == InverseLaplaceTransform(f(w), w, t, 0) # test a bug spos = symbols('s', positive=True) assert LT(exp(t), t, spos)[:2] == (1/(spos - 1), True) # basic tests from wikipedia assert LT((t - a)**b*exp(-c*(t - a))*Heaviside(t - a), t, s) == \ ((s + c)**(-b - 1)*exp(-a*s)*gamma(b + 1), -c, True) assert LT(t**a, t, s) == (s**(-a - 1)*gamma(a + 1), 0, True) assert LT(Heaviside(t), t, s) == (1/s, 0, True) assert LT(Heaviside(t - a), t, s) == (exp(-a*s)/s, 0, True) assert LT(1 - exp(-a*t), t, s) == (a/(s*(a + s)), 0, True) assert LT((exp(2*t) - 1)*exp(-b - t)*Heaviside(t)/2, t, s, noconds=True) \ == exp(-b)/(s**2 - 1) assert LT(exp(t), t, s)[:2] == (1/(s - 1), 1) assert LT(exp(2*t), t, s)[:2] == (1/(s - 2), 2) assert LT(exp(a*t), t, s)[:2] == (1/(s - a), a) assert LT(log(t/a), t, s) == ((log(a*s) + EulerGamma)/s/-1, 0, True) assert LT(erf(t), t, s) == ((-erf(s/2) + 1)*exp(s**2/4)/s, 0, True) assert LT(sin(a*t), t, s) == (a/(a**2 + s**2), 0, True) assert LT(cos(a*t), t, s) == (s/(a**2 + s**2), 0, True) # TODO would be nice to have these come out better assert LT( exp(-a*t)*sin(b*t), t, s) == (b/(b**2 + (a + s)**2), -a, True) assert LT(exp(-a*t)*cos(b*t), t, s) == \ ((a + s)/(b**2 + (a + s)**2), -a, True) assert LT(besselj(0, t), t, s) == (1/sqrt(1 + s**2), 0, True) assert LT(besselj(1, t), t, s) == (1 - 1/sqrt(1 + 1/s**2), 0, True) # TODO general order works, but is a *mess* # TODO besseli also works, but is an even greater mess # test a bug in conditions processing # TODO the auxiliary condition should be recognised/simplified assert LT(exp(t)*cos(t), t, s)[:-1] in [ ((s - 1)/(s**2 - 2*s + 2), -oo), ((s - 1)/((s - 1)**2 + 1), -oo), ] # Fresnel functions assert laplace_transform(fresnels(t), t, s) == \ ((-sin(s**2/(2*pi))*fresnels(s/pi) + sin(s**2/(2*pi))/2 - cos(s**2/(2*pi))*fresnelc(s/pi) + cos(s**2/(2*pi))/2)/s, 0, True) assert laplace_transform(fresnelc(t), t, s) == ( (sin(s**2/(2*pi))*fresnelc(s/pi)/s - cos(s**2/(2*pi))*fresnels(s/pi)/s + sqrt(2)*cos(s**2/(2*pi) + pi/4)/(2*s), 0, True)) assert LT(Matrix([[exp(t), t*exp(-t)], [t*exp(-t), exp(t)]]), t, s) ==\ Matrix([ [(1/(s - 1), 1, True), ((s + 1)**(-2), 0, True)], [((s + 1)**(-2), 0, True), (1/(s - 1), 1, True)] ])