def _sympysage_derivative(self):
"""
EXAMPLES::
sage: from sympy import Derivative
sage: f = function('f')
sage: sympy_diff = Derivative(f(x)._sympy_(), x._sympy_())
sage: assert diff(f(x),x)._sympy_() == sympy_diff
sage: assert diff(f(x),x) == sympy_diff._sage_()
TESTS:
Check that :trac:`28964` is fixed::
sage: f = function('f')
sage: _ = var('x,t')
sage: assert diff(f(x, t), t)._sympy_()._sage_() == diff(f(x, t), t)
sage: assert diff(f(x, t), x, 2, t)._sympy_()._sage_() == diff(f(x, t), x, 2, t)
sage: diff(f(x, t), x).integrate(x)
f(x, t)
sage: diff(f(x, t), x).integrate(t, algorithm='maxima')
integrate(diff(f(x, t), x), t)
sage: diff(f(x, t), x).integrate(t, algorithm='sympy')
integrate(diff(f(x, t), x), t)
sage: integrate(f(x, t), x).diff(t)
integrate(diff(f(x, t), t), x)
"""
from sage.calculus.functional import derivative
from sympy.core.containers import Tuple
f = self.args[0]._sage_()
args = [a._sage_() for arg in self.args[1:]
for a in (arg if isinstance(arg, (tuple, Tuple)) else [arg])]
return derivative(f, *args)
def special_curve(j, Phi):
field = j.parent()
if j == field(0) or j == field(1728):
return True
x = PolynomialRing(field, 'x').gen()
f = Phi(x, j)
der = derivative(f, x)
return f.gcd(der) != 1
def _sympysage_derivative(self):
"""
EXAMPLES::
sage: from sympy import Derivative
sage: f = function('f')
sage: sympy_diff = Derivative(f(x)._sympy_(), x._sympy_())
sage: assert diff(f(x),x)._sympy_() == sympy_diff
sage: assert diff(f(x),x) == sympy_diff._sage_()
"""
from sage.calculus.functional import derivative
args = [arg._sage_() for arg in self.args]
return derivative(*args)
def _sympysage_derivative(self):
"""
EXAMPLES::
sage: from sympy import Derivative
sage: f = function('f')
sage: sympy_diff = Derivative(f(x)._sympy_(), x._sympy_())
sage: assert diff(f(x),x)._sympy_() == sympy_diff
sage: assert diff(f(x),x) == sympy_diff._sage_()
"""
from sage.calculus.functional import derivative
args = [arg._sage_() for arg in self.args]
return derivative(*args)
def _sympysage_derivative(self):
"""
EXAMPLES::
sage: from sympy import Derivative
sage: f = function('f')
sage: sympy_diff = Derivative(f(x)._sympy_(), x._sympy_())
sage: assert diff(f(x),x)._sympy_() == sympy_diff
sage: assert diff(f(x),x) == sympy_diff._sage_()
"""
from sage.calculus.functional import derivative
f = self.args[0]._sage_()
args = [[a._sage_()
for a in arg] if isinstance(arg, tuple) else arg._sage_()
for arg in self.args[2:]]
return derivative(f, *args)
def _tderivative_(self, a, b, z, *args, **kwargs):
"""
EXAMPLES::
sage: hypergeometric([1/3, 2/3], [5], x^2).diff(x)
4/45*x*hypergeometric((4/3, 5/3), (6,), x^2)
sage: hypergeometric([1, 2], [x], 2).diff(x)
Traceback (most recent call last):
...
NotImplementedError: derivative of hypergeometric function with...
respect to parameters. Try calling .simplify_hypergeometric()...
first.
sage: hypergeometric([1/3, 2/3], [5], 2).diff(x)
0
"""
diff_param = kwargs['diff_param']
if diff_param in hypergeometric(a, b, 1).variables(): # ignore z
raise NotImplementedError("derivative of hypergeometric function "
"with respect to parameters. Try calling"
" .simplify_hypergeometric() first.")
t = (reduce(lambda x, y: x * y, a, 1) *
reduce(lambda x, y: x / y, b, Integer(1)))
return (t * derivative(z, diff_param) *
hypergeometric([c + 1 for c in a], [c + 1 for c in b], z))
def _tderivative_(self, a, b, z, *args, **kwargs):
"""
EXAMPLES::
sage: hypergeometric([1/3, 2/3], [5], x^2).diff(x)
4/45*x*hypergeometric((4/3, 5/3), (6,), x^2)
sage: hypergeometric([1, 2], [x], 2).diff(x)
Traceback (most recent call last):
...
NotImplementedError: derivative of hypergeometric function with...
respect to parameters. Try calling .simplify_hypergeometric()...
first.
sage: hypergeometric([1/3, 2/3], [5], 2).diff(x)
0
"""
diff_param = kwargs['diff_param']
if diff_param in hypergeometric(a, b, 1).variables(): # ignore z
raise NotImplementedError("derivative of hypergeometric function "
"with respect to parameters. Try calling"
" .simplify_hypergeometric() first.")
t = (reduce(lambda x, y: x * y, a, 1) *
reduce(lambda x, y: x / y, b, Integer(1)))
return (t * derivative(z, diff_param) *
hypergeometric([c + 1 for c in a], [c + 1 for c in b], z))
def airy_bi(alpha, x=None, hold_derivative=True, **kwds):
r"""
The Airy Bi function
The Airy Bi function `\operatorname{Bi}(x)` is (along with
`\operatorname{Ai}(x)`) one of the two linearly independent standard
solutions to the Airy differential equation `f''(x) - x f(x) = 0`. It is
defined by the initial conditions:
.. MATH::
\operatorname{Bi}(0)=\frac{1}{3^{1/6} \Gamma\left(\frac{2}{3}\right)},
\operatorname{Bi}'(0)=\frac{3^{1/6}}{ \Gamma\left(\frac{1}{3}\right)}.
Another way to define the Airy Bi function is:
.. MATH::
\operatorname{Bi}(x)=\frac{1}{\pi}\int_0^\infty
\left[ \exp\left( xt -\frac{t^3}{3} \right)
+\sin\left(xt + \frac{1}{3}t^3\right) \right ] dt.
INPUT:
- ``alpha`` -- Return the `\alpha`-th order fractional derivative with
respect to `z`.
For `\alpha = n = 1,2,3,\ldots` this gives the derivative
`\operatorname{Bi}^{(n)}(z)`, and for `\alpha = -n = -1,-2,-3,\ldots`
this gives the `n`-fold iterated integral.
.. MATH::
f_0(z) = \operatorname{Bi}(z)
f_n(z) = \int_0^z f_{n-1}(t) dt
- ``x`` -- The argument of the function
- ``hold_derivative`` -- Whether or not to stop from returning higher
derivatives in terms of `\operatorname{Bi}(x)` and
`\operatorname{Bi}'(x)`
.. SEEALSO:: :func:`airy_ai`
EXAMPLES::
sage: n, x = var('n x')
sage: airy_bi(x)
airy_bi(x)
It can return derivatives or integrals::
sage: airy_bi(2, x)
airy_bi(2, x)
sage: airy_bi(1, x, hold_derivative=False)
airy_bi_prime(x)
sage: airy_bi(2, x, hold_derivative=False)
x*airy_bi(x)
sage: airy_bi(-2, x, hold_derivative=False)
airy_bi(-2, x)
sage: airy_bi(n, x)
airy_bi(n, x)
It can be evaluated symbolically or numerically for real or complex
values::
sage: airy_bi(0)
1/3*3^(5/6)/gamma(2/3)
sage: airy_bi(0.0)
0.614926627446001
sage: airy_bi(I)
airy_bi(I)
sage: airy_bi(1.0*I)
0.648858208330395 + 0.344958634768048*I
The functions can be evaluated numerically using mpmath,
which can compute the values to arbitrary precision, and scipy::
sage: airy_bi(2).n(prec=100)
3.2980949999782147102806044252
sage: airy_bi(2).n(algorithm='mpmath', prec=100)
3.2980949999782147102806044252
sage: airy_bi(2).n(algorithm='scipy') # rel tol 1e-10
3.2980949999782134
And the derivatives can be evaluated::
sage: airy_bi(1, 0)
3^(1/6)/gamma(1/3)
sage: airy_bi(1, 0.0)
0.448288357353826
Plots::
sage: plot(airy_bi(x), (x, -10, 5)) + plot(airy_bi_prime(x),
....: (x, -10, 5), color='red')
Graphics object consisting of 2 graphics primitives
**References**
- Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 10"
- :wikipedia:`Airy_function`
"""
# We catch the case with no alpha
if x is None:
x = alpha
return airy_bi_simple(x, **kwds)
# We take care of all other cases.
if not alpha in ZZ and not isinstance(alpha, Expression):
return airy_bi_general(alpha, x, **kwds)
if hold_derivative:
return airy_bi_general(alpha, x, **kwds)
elif alpha == 0:
return airy_bi_simple(x, **kwds)
elif alpha == 1:
return airy_bi_prime(x, **kwds)
elif alpha > 1:
# We use a different variable here because if x is a
# particular value, we would be differentiating a constant
# which would return 0. What we want is the value of
# the derivative at the value and not the derivative of
# a particular value of the function.
v = SR.symbol()
return derivative(airy_bi_simple(v, **kwds), v, alpha).subs({v: x})
else:
return airy_bi_general(alpha, x, **kwds)
def airy_bi(alpha, x=None, hold_derivative=True, **kwds):
r"""
The Airy Bi function
The Airy Bi function `\operatorname{Bi}(x)` is (along with
`\operatorname{Ai}(x)`) one of the two linearly independent standard
solutions to the Airy differential equation `f''(x) - x f(x) = 0`. It is
defined by the initial conditions:
.. MATH::
\operatorname{Bi}(0)=\frac{1}{3^{1/6} \Gamma\left(\frac{2}{3}\right)},
\operatorname{Bi}'(0)=\frac{3^{1/6}}{ \Gamma\left(\frac{1}{3}\right)}.
Another way to define the Airy Bi function is:
.. MATH::
\operatorname{Bi}(x)=\frac{1}{\pi}\int_0^\infty
\left[ \exp\left( xt -\frac{t^3}{3} \right)
+\sin\left(xt + \frac{1}{3}t^3\right) \right ] dt.
INPUT:
- ``alpha`` -- Return the `\alpha`-th order fractional derivative with
respect to `z`.
For `\alpha = n = 1,2,3,\ldots` this gives the derivative
`\operatorname{Bi}^{(n)}(z)`, and for `\alpha = -n = -1,-2,-3,\ldots`
this gives the `n`-fold iterated integral.
.. MATH::
f_0(z) = \operatorname{Bi}(z)
f_n(z) = \int_0^z f_{n-1}(t) dt
- ``x`` -- The argument of the function
- ``hold_derivative`` -- Whether or not to stop from returning higher
derivatives in terms of `\operatorname{Bi}(x)` and
`\operatorname{Bi}'(x)`
.. SEEALSO:: :func:`airy_ai`
EXAMPLES::
sage: n, x = var('n x')
sage: airy_bi(x)
airy_bi(x)
It can return derivatives or integrals::
sage: airy_bi(2, x)
airy_bi(2, x)
sage: airy_bi(1, x, hold_derivative=False)
airy_bi_prime(x)
sage: airy_bi(2, x, hold_derivative=False)
x*airy_bi(x)
sage: airy_bi(-2, x, hold_derivative=False)
airy_bi(-2, x)
sage: airy_bi(n, x)
airy_bi(n, x)
It can be evaluated symbolically or numerically for real or complex
values::
sage: airy_bi(0)
1/3*3^(5/6)/gamma(2/3)
sage: airy_bi(0.0)
0.614926627446001
sage: airy_bi(I)
airy_bi(I)
sage: airy_bi(1.0*I)
0.648858208330395 + 0.344958634768048*I
The functions can be evaluated numerically using mpmath,
which can compute the values to arbitrary precision, and scipy::
sage: airy_bi(2).n(prec=100)
3.2980949999782147102806044252
sage: airy_bi(2).n(algorithm='mpmath', prec=100)
3.2980949999782147102806044252
sage: airy_bi(2).n(algorithm='scipy') # rel tol 1e-10
3.2980949999782134
And the derivatives can be evaluated::
sage: airy_bi(1, 0)
3^(1/6)/gamma(1/3)
sage: airy_bi(1, 0.0)
0.448288357353826
Plots::
sage: plot(airy_bi(x), (x, -10, 5)) + plot(airy_bi_prime(x),
....: (x, -10, 5), color='red')
Graphics object consisting of 2 graphics primitives
**References**
- Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 10"
- :wikipedia:`Airy_function`
"""
# We catch the case with no alpha
if x is None:
x = alpha
return airy_bi_simple(x, **kwds)
# We take care of all other cases.
if not alpha in ZZ and not isinstance(alpha, Expression):
return airy_bi_general(alpha, x, **kwds)
if hold_derivative:
return airy_bi_general(alpha, x, **kwds)
elif alpha == 0:
return airy_bi_simple(x, **kwds)
elif alpha == 1:
return airy_bi_prime(x, **kwds)
elif alpha > 1:
# We use a different variable here because if x is a
# particular value, we would be differentiating a constant
# which would return 0. What we want is the value of
# the derivative at the value and not the derivative of
# a particular value of the function.
v = SR.symbol()
return derivative(airy_bi_simple(v, **kwds), v, alpha).subs({v: x})
else:
return airy_bi_general(alpha, x, **kwds)
def airy_ai(alpha, x=None, hold_derivative=True, **kwds):
r"""
The Airy Ai function
The Airy Ai function `\operatorname{Ai}(x)` is (along with
`\operatorname{Bi}(x)`) one of the two linearly independent standard
solutions to the Airy differential equation `f''(x) - x f(x) = 0`. It is
defined by the initial conditions:
.. MATH::
\operatorname{Ai}(0)=\frac{1}{2^{2/3} \Gamma\left(\frac{2}{3}\right)},
\operatorname{Ai}'(0)=-\frac{1}{2^{1/3}\Gamma\left(\frac{1}{3}\right)}.
Another way to define the Airy Ai function is:
.. MATH::
\operatorname{Ai}(x)=\frac{1}{\pi}\int_0^\infty
\cos\left(\frac{1}{3}t^3+xt\right) dt.
INPUT:
- ``alpha`` -- Return the `\alpha`-th order fractional derivative with
respect to `z`.
For `\alpha = n = 1,2,3,\ldots` this gives the derivative
`\operatorname{Ai}^{(n)}(z)`, and for `\alpha = -n = -1,-2,-3,\ldots`
this gives the `n`-fold iterated integral.
.. MATH::
f_0(z) = \operatorname{Ai}(z)
f_n(z) = \int_0^z f_{n-1}(t) dt
- ``x`` -- The argument of the function
- ``hold_derivative`` -- Whether or not to stop from returning higher
derivatives in terms of `\operatorname{Ai}(x)` and
`\operatorname{Ai}'(x)`
.. SEEALSO:: :func:`airy_bi`
EXAMPLES::
sage: n, x = var('n x')
sage: airy_ai(x)
airy_ai(x)
It can return derivatives or integrals::
sage: airy_ai(2, x)
airy_ai(2, x)
sage: airy_ai(1, x, hold_derivative=False)
airy_ai_prime(x)
sage: airy_ai(2, x, hold_derivative=False)
x*airy_ai(x)
sage: airy_ai(-2, x, hold_derivative=False)
airy_ai(-2, x)
sage: airy_ai(n, x)
airy_ai(n, x)
It can be evaluated symbolically or numerically for real or complex
values::
sage: airy_ai(0)
1/3*3^(1/3)/gamma(2/3)
sage: airy_ai(0.0)
0.355028053887817
sage: airy_ai(I)
airy_ai(I)
sage: airy_ai(1.0*I)
0.331493305432141 - 0.317449858968444*I
The functions can be evaluated numerically either using mpmath. which
can compute the values to arbitrary precision, and scipy::
sage: airy_ai(2).n(prec=100)
0.034924130423274379135322080792
sage: airy_ai(2).n(algorithm='mpmath', prec=100)
0.034924130423274379135322080792
sage: airy_ai(2).n(algorithm='scipy') # rel tol 1e-10
0.03492413042327323
And the derivatives can be evaluated::
sage: airy_ai(1, 0)
-1/3*3^(2/3)/gamma(1/3)
sage: airy_ai(1, 0.0)
-0.258819403792807
Plots::
sage: plot(airy_ai(x), (x, -10, 5)) + plot(airy_ai_prime(x),
....: (x, -10, 5), color='red')
Graphics object consisting of 2 graphics primitives
REFERENCES:
- Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 10"
- :wikipedia:`Airy_function`
"""
# We catch the case with no alpha
if x is None:
x = alpha
return airy_ai_simple(x, **kwds)
# We take care of all other cases.
if alpha not in ZZ and not isinstance(alpha, Expression):
return airy_ai_general(alpha, x, **kwds)
if hold_derivative:
return airy_ai_general(alpha, x, **kwds)
elif alpha == 0:
return airy_ai_simple(x, **kwds)
elif alpha == 1:
return airy_ai_prime(x, **kwds)
elif alpha > 1:
# We use a different variable here because if x is a
# particular value, we would be differentiating a constant
# which would return 0. What we want is the value of
# the derivative at the value and not the derivative of
# a particular value of the function.
v = SR.symbol()
return derivative(airy_ai_simple(v, **kwds), v, alpha).subs({v: x})
else:
return airy_ai_general(alpha, x, **kwds)
def f(p):
return apply_young_idempotent(derivative(p, x), mu)
