Module Expr Str ImportFrom alias alias ImportFrom alias alias alias ImportFrom alias alias ImportFrom alias ImportFrom alias alias ImportFrom alias ImportFrom alias ImportFrom alias alias ImportFrom alias ClassDef bases NameLoad body Expr Str FunctionDef arguments args NameParam NameParam defaults body If CompareIs NameLoad AttributeLoad NameLoad attr body Return BinOpDiv NameLoad Num orelse If CompareIs NameLoad AttributeLoad NameLoad attr body Return BinOpDiv BinOpMult Num BinOpPow NameLoad BinOpDiv Call NameLoad Num Num BinOpPow Call NameLoad BinOpDiv UnaryOpUSub Call NameLoad Num Num Num orelse If CompareIs NameLoad AttributeLoad NameLoad attr body Return AttributeLoad NameLoad attr orelse If CompareIs NameLoad AttributeLoad NameLoad attr body Return BinOpDiv BinOpPow Call NameLoad BinOpDiv Call NameLoad Num Num Num BinOpMult Num Call NameLoad BinOpMult Num NameLoad orelse If CompareIn NameLoad TupleLoad AttributeLoad NameLoad attr AttributeLoad NameLoad attr BinOpMult NameLoad 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Return NameLoad orelse If AttributeLoad NameLoad attr body Return AttributeLoad NameLoad attr orelse If AttributeLoad NameLoad attr body Return BinOpMult NameLoad Call NameLoad NameLoad orelse If CompareIn NameLoad TupleLoad AttributeLoad NameLoad attr AttributeLoad NameLoad attr body Return AttributeLoad NameLoad attr orelse If Call AttributeLoad NameLoad attr body Return UnaryOpUSub Call NameLoad UnaryOpUSub NameLoad NameLoad decorator_list NameLoad FunctionDef arguments args NameParam NameParam defaults Num body Assign TupleStore NameStore NameStore AttributeLoad NameLoad attr Assign NameStore Call NameLoad BinOpSub Num BinOpMult NameLoad BinOpPow Call NameLoad NameLoad Num If CompareEq NameLoad Num body Return BinOpDiv Num NameLoad orelse If CompareEq NameLoad Num body Return BinOpSub BinOpSub BinOpDiv Call NameLoad NameLoad NameLoad BinOpMult BinOpMult Num NameLoad BinOpSub Num NameLoad BinOpDiv Call NameLoad NameLoad NameLoad BinOpMult Num NameLoad BinOpDiv Call NameLoad BinOpMult Num NameLoad BinOpMult BinOpMult Num BinOpSub Num NameLoad NameLoad Raise Call NameLoad NameLoad NameLoad decorator_list FunctionDef arguments args NameParam defaults body Assign TupleStore NameStore NameStore AttributeLoad NameLoad attr If CompareIs BoolOpAnd AttributeLoad NameLoad attr AttributeLoad BinOpSub NameLoad Num attr NameLoad body Return Call AttributeLoad NameLoad attr Call AttributeLoad NameLoad attr Call AttributeLoad NameLoad attr decorator_list decorator_list ClassDef bases NameLoad body Expr Str FunctionDef arguments args NameParam NameParam NameParam defaults NameLoad body If CompareIsNot NameLoad NameLoad body Assign NameStore BinOpDiv BinOpMult Num NameLoad NameLoad If AttributeLoad NameLoad attr body Return NameLoad If AttributeLoad NameLoad attr body Return AttributeLoad NameLoad attr orelse If AttributeLoad NameLoad attr body Return BinOpMult NameLoad Call NameLoad NameLoad orelse If CompareIn NameLoad TupleLoad AttributeLoad NameLoad attr AttributeLoad NameLoad attr body Return AttributeLoad NameLoad attr orelse If Call AttributeLoad NameLoad attr body Return UnaryOpUSub Call NameLoad UnaryOpUSub NameLoad NameLoad orelse If AttributeLoad NameLoad attr body Return BinOpDiv NameLoad Num orelse If CompareIs NameLoad AttributeLoad NameLoad attr body Return AttributeLoad NameLoad attr orelse If CompareIs NameLoad AttributeLoad NameLoad attr body Return BinOpMult NameLoad AttributeLoad NameLoad attr orelse If CompareIs NameLoad AttributeLoad NameLoad attr body Return AttributeLoad NameLoad attr orelse If CompareIs NameLoad AttributeLoad NameLoad attr body Return AttributeLoad NameLoad attr decorator_list NameLoad FunctionDef arguments args NameParam NameParam defaults Num body If CompareEq Call NameLoad AttributeLoad NameLoad attr Num body Assign TupleStore NameStore NameStore AttributeLoad NameLoad attr If CompareEq NameLoad Num body Return Call NameLoad BinOpSub Num BinOpMult NameLoad BinOpPow Call NameLoad NameLoad Num orelse If CompareEq NameLoad Num body Return BinOpDiv BinOpSub Call NameLoad NameLoad NameLoad Call NameLoad NameLoad NameLoad BinOpMult Num NameLoad orelse Assign NameStore SubscriptLoad AttributeLoad NameLoad attr Index Num If CompareEq NameLoad Num body Return BinOpDiv BinOpSub Call NameLoad NameLoad Call NameLoad NameLoad BinOpMult Num NameLoad Raise Call NameLoad NameLoad NameLoad decorator_list FunctionDef arguments args NameParam defaults body If CompareEq Call NameLoad AttributeLoad NameLoad attr Num body Assign TupleStore NameStore NameStore AttributeLoad NameLoad attr If CompareIs BoolOpAnd AttributeLoad NameLoad attr AttributeLoad BinOpSub NameLoad Num attr NameLoad body Return Call AttributeLoad NameLoad attr Call AttributeLoad NameLoad attr Call AttributeLoad NameLoad attr orelse Assign NameStore SubscriptLoad AttributeLoad NameLoad attr Index Num If CompareIs BoolOpAnd AttributeLoad NameLoad attr AttributeLoad BinOpSub NameLoad Num attr NameLoad body Return Call AttributeLoad NameLoad attr Call AttributeLoad NameLoad attr decorator_list FunctionDef arguments args NameParam NameParam NameParam NameParam defaults body ImportFrom alias If CompareEq Call NameLoad AttributeLoad NameLoad attr Num body Return Call NameLoad Call AttributeLoad Call AttributeLoad NameLoad attr NameLoad attr NameLoad keyword NameLoad keyword NameLoad Return Call AttributeLoad Call NameLoad NameLoad NameLoad attr NameLoad keyword NameLoad keyword NameLoad decorator_list FunctionDef arguments args NameParam defaults vararg body If CompareEq Call NameLoad NameLoad Num body Assign NameStore SubscriptLoad NameLoad Index Num Return BinOpMult BinOpDiv NameLoad Num Call NameLoad TupleLoad UnaryOpUSub AttributeLoad NameLoad attr AttributeLoad NameLoad attr TupleLoad AttributeLoad NameLoad attr NameLoad decorator_list FunctionDef arguments args NameParam defaults vararg body If CompareEq Call NameLoad NameLoad Num body Assign NameStore SubscriptLoad NameLoad Index Num Return BinOpDiv 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NameLoad NameLoad NameLoad BinOpDiv BinOpMult BinOpSub NameLoad NameLoad Call NameLoad NameLoad NameLoad NameLoad BinOpDiv BinOpMult BinOpSub BinOpPow NameLoad Num NameLoad Call NameLoad NameLoad NameLoad NameLoad NameLoad BinOpDiv BinOpMult BinOpMult NameLoad NameLoad Call NameLoad BinOpMult Num NameLoad BinOpMult Num NameLoad BinOpMult BinOpMult Num BinOpSub NameLoad NameLoad BinOpSub NameLoad Num orelse If CompareEq NameLoad Num body Return BinOpDiv Num BinOpMult NameLoad NameLoad orelse If CompareEq NameLoad Num body Return BinOpDiv BinOpSub BinOpAdd BinOpDiv Call NameLoad NameLoad NameLoad BinOpSub NameLoad Num Call NameLoad NameLoad NameLoad NameLoad BinOpDiv BinOpMult NameLoad Call NameLoad BinOpMult Num NameLoad BinOpMult BinOpMult Num BinOpSub NameLoad Num NameLoad BinOpMult Num BinOpSub NameLoad NameLoad orelse Assign TupleStore NameStore NameStore AttributeLoad NameLoad attr If CompareEq NameLoad Num body Return BinOpDiv BinOpAdd BinOpAdd Call NameLoad NameLoad BinOpDiv BinOpMult BinOpSub NameLoad NameLoad Call NameLoad NameLoad NameLoad BinOpDiv BinOpMult BinOpSub BinOpPow NameLoad Num NameLoad Call NameLoad NameLoad NameLoad NameLoad BinOpMult BinOpMult Num BinOpSub NameLoad NameLoad BinOpSub NameLoad Num orelse If CompareEq NameLoad Num body Return BinOpDiv BinOpAdd BinOpDiv Call NameLoad NameLoad BinOpSub NameLoad Num Call NameLoad NameLoad NameLoad BinOpMult Num BinOpSub NameLoad NameLoad Raise Call NameLoad NameLoad NameLoad decorator_list decorator_list
N/A N/A  Elliptic integrals.  __future__ print_function division sympy.core S pi I sympy.core.function Function ArgumentIndexError sympy.functions.elementary.hyperbolic atanh sympy.functions.elementary.trigonometric sin tan sympy.functions.elementary.miscellaneous sqrt sympy.functions.elementary.complexes sign sympy.functions.special.hyper hyper meijerg sympy.functions.special.gamma_functions gamma elliptic_k N/A Function N/A N/A 
    The complete elliptic integral of the first kind, defined by

    .. math:: K(z) = F\left(\tfrac{\pi}{2}\middle| z\right)

    where `F\left(z\middle| m\right)` is the Legendre incomplete
    elliptic integral of the first kind.

    The function `K(z)` is a single-valued function on the complex
    plane with branch cut along the interval `(1, \infty)`.

    Examples
    ========

    >>> from sympy import elliptic_k, I, pi
    >>> from sympy.abc import z
    >>> elliptic_k(0)
    pi/2
    >>> elliptic_k(1.0 + I)
    1.50923695405127 + 0.625146415202697*I
    >>> elliptic_k(z).series(z, n=3)
    pi/2 + pi*z/8 + 9*pi*z**2/128 + O(z**3)

    References
    ==========

    .. [1] http://en.wikipedia.org/wiki/Elliptic_integrals
    .. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticK

    See Also
    ========

    elliptic_f
     eval N/A N/A cls z N/A N/A N/A N/A z N/A S Zero N/A N/A N/A pi 2 N/A N/A N/A z N/A S Half N/A N/A N/A N/A 8 N/A pi N/A N/A S 3 2 N/A N/A gamma N/A N/A N/A S 1 4 2 N/A N/A N/A z N/A S One N/A N/A N/A S ComplexInfinity N/A N/A N/A z N/A S NegativeOne N/A N/A N/A N/A N/A gamma N/A N/A S 1 4 2 N/A 4 N/A sqrt N/A 2 pi N/A N/A N/A z N/A N/A S Infinity N/A S NegativeInfinity N/A I N/A S Infinity N/A I N/A S NegativeInfinity N/A S ComplexInfinity N/A N/A N/A S Zero N/A classmethod fdiff N/A N/A self argindex N/A 1 N/A N/A z N/A N/A self args N/A 0 N/A N/A N/A N/A elliptic_e z N/A N/A 1 z N/A elliptic_k z N/A N/A 2 z N/A 1 z N/A _eval_conjugate N/A N/A self N/A N/A N/A z N/A N/A self args N/A 0 N/A N/A N/A N/A z is_real N/A N/A z 1 is_positive False N/A N/A N/A N/A self func N/A N/A z conjugate N/A _eval_nseries N/A N/A self x n logx N/A N/A sympy.simplify hyperexpand N/A N/A hyperexpand N/A N/A N/A N/A self rewrite hyper _eval_nseries x n n logx logx N/A _eval_rewrite_as_hyper N/A N/A self z N/A N/A N/A N/A N/A pi 2 N/A hyper N/A N/A S Half N/A S Half N/A N/A S One z N/A _eval_rewrite_as_meijerg N/A N/A self z N/A N/A N/A N/A N/A meijerg N/A N/A N/A S Half N/A S Half N/A N/A N/A N/A S Zero N/A N/A S Zero N/A z 2 N/A _sage_ N/A N/A self N/A N/A N/A sage.all sage N/A N/A N/A sage elliptic_kc N/A N/A N/A N/A self args N/A 0 _sage_ N/A N/A elliptic_f N/A Function N/A N/A 
    The Legendre incomplete elliptic integral of the first
    kind, defined by

    .. math:: F\left(z\middle| m\right) =
              \int_0^z \frac{dt}{\sqrt{1 - m \sin^2 t}}

    This function reduces to a complete elliptic integral of
    the first kind, `K(m)`, when `z = \pi/2`.

    Examples
    ========

    >>> from sympy import elliptic_f, I, O
    >>> from sympy.abc import z, m
    >>> elliptic_f(z, m).series(z)
    z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6)
    >>> elliptic_f(3.0 + I/2, 1.0 + I)
    2.909449841483 + 1.74720545502474*I

    References
    ==========

    .. [1] http://en.wikipedia.org/wiki/Elliptic_integrals
    .. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticF

    See Also
    ========

    elliptic_k
     eval N/A N/A cls z m N/A N/A N/A k N/A N/A 2 z pi N/A N/A m is_zero N/A N/A z N/A N/A N/A z is_zero N/A N/A N/A S Zero N/A N/A N/A k is_integer N/A N/A N/A k N/A elliptic_k m N/A N/A N/A m N/A N/A S Infinity N/A S NegativeInfinity N/A N/A N/A S Zero N/A N/A N/A N/A z could_extract_minus_sign N/A N/A N/A N/A elliptic_f N/A z m N/A classmethod fdiff N/A N/A self argindex N/A 1 N/A N/A N/A z m N/A self args N/A fm N/A sqrt N/A 1 N/A m N/A N/A sin z 2 N/A N/A argindex 1 N/A N/A N/A 1 fm N/A N/A N/A argindex 2 N/A N/A N/A N/A N/A N/A elliptic_e z m N/A N/A 2 m N/A 1 m N/A N/A elliptic_f z m N/A 2 m N/A N/A sin N/A 2 z N/A N/A 4 N/A 1 m fm N/A N/A ArgumentIndexError self argindex N/A _eval_conjugate N/A N/A self N/A N/A N/A N/A z m N/A self args N/A N/A N/A N/A m is_real N/A N/A m 1 is_positive False N/A N/A N/A N/A self func N/A N/A z conjugate N/A N/A m conjugate N/A N/A elliptic_e N/A Function N/A N/A 
    Called with two arguments `z` and `m`, evaluates the
    incomplete elliptic integral of the second kind, defined by

    .. math:: E\left(z\middle| m\right) = \int_0^z \sqrt{1 - m \sin^2 t} dt

    Called with a single argument `z`, evaluates the Legendre complete
    elliptic integral of the second kind

    .. math:: E(z) = E\left(\tfrac{\pi}{2}\middle| z\right)

    The function `E(z)` is a single-valued function on the complex
    plane with branch cut along the interval `(1, \infty)`.

    Examples
    ========

    >>> from sympy import elliptic_e, I, pi, O
    >>> from sympy.abc import z, m
    >>> elliptic_e(z, m).series(z)
    z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)
    >>> elliptic_e(z).series(z, n=4)
    pi/2 - pi*z/8 - 3*pi*z**2/128 - 5*pi*z**3/512 + O(z**4)
    >>> elliptic_e(1 + I, 2 - I/2).n()
    1.55203744279187 + 0.290764986058437*I
    >>> elliptic_e(0)
    pi/2
    >>> elliptic_e(2.0 - I)
    0.991052601328069 + 0.81879421395609*I

    References
    ==========

    .. [1] http://en.wikipedia.org/wiki/Elliptic_integrals
    .. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticE2
    .. [3] http://functions.wolfram.com/EllipticIntegrals/EllipticE
     eval N/A N/A cls z m N/A None N/A N/A N/A m None N/A N/A k N/A N/A 2 z pi N/A N/A m is_zero N/A N/A z N/A N/A z is_zero N/A N/A N/A S Zero N/A N/A N/A k is_integer N/A N/A N/A k N/A elliptic_e m N/A N/A N/A m N/A N/A S Infinity N/A S NegativeInfinity N/A N/A N/A S ComplexInfinity N/A N/A N/A N/A z could_extract_minus_sign N/A N/A N/A N/A elliptic_e N/A z m N/A N/A N/A z is_zero N/A N/A N/A pi 2 N/A N/A N/A z N/A S One N/A N/A N/A S One N/A N/A N/A z N/A S Infinity N/A N/A N/A I N/A S Infinity N/A N/A N/A z N/A S NegativeInfinity N/A N/A N/A S Infinity N/A N/A N/A z N/A S ComplexInfinity N/A N/A N/A S ComplexInfinity N/A classmethod fdiff N/A N/A self argindex N/A 1 N/A N/A N/A N/A len N/A self args 2 N/A N/A N/A z m N/A self args N/A N/A argindex 1 N/A N/A N/A sqrt N/A 1 N/A m N/A N/A sin z 2 N/A N/A N/A argindex 2 N/A N/A N/A N/A N/A elliptic_e z m N/A elliptic_f z m N/A 2 m N/A N/A z N/A N/A self args N/A 0 N/A N/A argindex 1 N/A N/A N/A N/A N/A elliptic_e z N/A elliptic_k z N/A 2 z N/A N/A ArgumentIndexError self argindex N/A _eval_conjugate N/A N/A self N/A N/A N/A N/A N/A len N/A self args 2 N/A N/A N/A z m N/A self args N/A N/A N/A N/A m is_real N/A N/A m 1 is_positive False N/A N/A N/A N/A self func N/A N/A z conjugate N/A N/A m conjugate N/A N/A z N/A N/A self args N/A 0 N/A N/A N/A N/A z is_real N/A N/A z 1 is_positive False N/A N/A N/A N/A self func N/A N/A z conjugate N/A _eval_nseries N/A N/A self x n logx N/A N/A sympy.simplify hyperexpand N/A N/A N/A len N/A self args 1 N/A N/A N/A hyperexpand N/A N/A N/A N/A self rewrite hyper _eval_nseries x n n logx logx N/A N/A N/A N/A super elliptic_e self _eval_nseries x n n logx logx N/A _eval_rewrite_as_hyper N/A N/A self N/A args N/A N/A N/A N/A len args 1 N/A N/A z N/A args N/A 0 N/A N/A N/A pi 2 N/A hyper N/A N/A N/A S Half N/A S Half N/A N/A S One z N/A _eval_rewrite_as_meijerg N/A N/A self N/A args N/A N/A N/A N/A len args 1 N/A N/A z N/A args N/A 0 N/A N/A N/A N/A meijerg N/A N/A N/A S Half N/A N/A S 3 2 N/A N/A N/A N/A S Zero N/A N/A S Zero N/A z 4 N/A N/A elliptic_pi N/A Function N/A N/A 
    Called with three arguments `n`, `z` and `m`, evaluates the
    Legendre incomplete elliptic integral of the third kind, defined by

    .. math:: \Pi\left(n; z\middle| m\right) = \int_0^z \frac{dt}
              {\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}}

    Called with two arguments `n` and `m`, evaluates the complete
    elliptic integral of the third kind:

    .. math:: \Pi\left(n\middle| m\right) =
              \Pi\left(n; \tfrac{\pi}{2}\middle| m\right)

    Examples
    ========

    >>> from sympy import elliptic_pi, I, pi, O, S
    >>> from sympy.abc import z, n, m
    >>> elliptic_pi(n, z, m).series(z, n=4)
    z + z**3*(m/6 + n/3) + O(z**4)
    >>> elliptic_pi(0.5 + I, 1.0 - I, 1.2)
    2.50232379629182 - 0.760939574180767*I
    >>> elliptic_pi(0, 0)
    pi/2
    >>> elliptic_pi(1.0 - I/3, 2.0 + I)
    3.29136443417283 + 0.32555634906645*I

    References
    ==========

    .. [1] http://en.wikipedia.org/wiki/Elliptic_integrals
    .. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticPi3
    .. [3] http://functions.wolfram.com/EllipticIntegrals/EllipticPi
     eval N/A N/A cls n m z N/A None N/A N/A N/A z None N/A N/A N/A n z m N/A n m z N/A k N/A N/A 2 z pi N/A N/A n N/A S Zero N/A N/A N/A elliptic_f z m N/A N/A N/A n N/A S One N/A N/A N/A N/A elliptic_f z m N/A N/A N/A N/A sqrt N/A 1 N/A m N/A N/A sin z 2 N/A tan z N/A elliptic_e z m N/A 1 m N/A N/A N/A k is_integer N/A N/A N/A k N/A elliptic_pi n m N/A N/A N/A m N/A S Zero N/A N/A N/A N/A atanh N/A N/A sqrt N/A n 1 N/A tan z N/A sqrt N/A n 1 N/A N/A N/A n m N/A N/A N/A N/A N/A elliptic_f z n N/A elliptic_pi 1 z n N/A N/A tan z N/A sqrt N/A 1 N/A n N/A N/A sin z 2 N/A N/A N/A n N/A N/A S Infinity N/A S NegativeInfinity N/A N/A N/A S Zero N/A N/A N/A m N/A N/A S Infinity N/A S NegativeInfinity N/A N/A N/A S Zero N/A N/A N/A N/A z could_extract_minus_sign N/A N/A N/A N/A elliptic_pi n N/A z m N/A N/A N/A n N/A S Zero N/A N/A N/A elliptic_k m N/A N/A N/A n N/A S One N/A N/A N/A S ComplexInfinity N/A N/A N/A m N/A S Zero N/A N/A N/A pi N/A 2 N/A sqrt N/A 1 n N/A N/A N/A m N/A S One N/A N/A N/A N/A N/A S Infinity N/A sign N/A n 1 N/A N/A N/A n m N/A N/A N/A N/A elliptic_e n N/A 1 n N/A N/A N/A n N/A N/A S Infinity N/A S NegativeInfinity N/A N/A N/A S Zero N/A N/A N/A m N/A N/A S Infinity N/A S NegativeInfinity N/A N/A N/A S Zero N/A classmethod _eval_conjugate N/A N/A self N/A N/A N/A N/A N/A len N/A self args 3 N/A N/A N/A n z m N/A self args N/A N/A N/A N/A N/A n is_real N/A N/A n 1 is_positive False N/A N/A N/A m is_real N/A N/A m 1 is_positive False N/A N/A N/A N/A self func N/A N/A n conjugate N/A N/A z conjugate N/A N/A m conjugate N/A N/A N/A n m N/A self args N/A N/A N/A self func N/A N/A n conjugate N/A N/A m conjugate N/A fdiff N/A N/A self argindex N/A 1 N/A N/A N/A N/A len N/A self args 3 N/A N/A N/A n z m N/A self args N/A N/A fm fn N/A N/A sqrt N/A 1 N/A m N/A N/A sin z 2 N/A 1 N/A n N/A N/A sin z 2 N/A N/A argindex 1 N/A N/A N/A N/A N/A N/A N/A elliptic_e z m N/A N/A N/A m n N/A elliptic_f z m n N/A N/A N/A N/A n 2 m N/A elliptic_pi n z m n N/A N/A N/A n fm N/A sin N/A 2 z N/A 2 fn N/A N/A 2 N/A m n N/A n 1 N/A N/A N/A argindex 2 N/A N/A N/A 1 N/A fm fn N/A N/A N/A argindex 3 N/A N/A N/A N/A N/A N/A N/A elliptic_e z m N/A m 1 N/A elliptic_pi n z m N/A N/A m N/A sin N/A 2 z N/A N/A 2 N/A m 1 fm N/A 2 N/A n m N/A N/A N/A n m N/A self args N/A N/A argindex 1 N/A N/A N/A N/A N/A N/A elliptic_e m N/A N/A N/A m n N/A elliptic_k m n N/A N/A N/A N/A n 2 m N/A elliptic_pi n m n N/A N/A 2 N/A m n N/A n 1 N/A N/A N/A argindex 2 N/A N/A N/A N/A N/A N/A elliptic_e m N/A m 1 N/A elliptic_pi n m N/A 2 N/A n m N/A N/A ArgumentIndexError self argindex N/A N/A