(ctx, *args)
return ctx.sum_accurately(terms) + P
r"""
    Called with three arguments `n, \phi, m`, evaluates the Legendre
    incomplete elliptic integral of the third kind

    .. math ::

        \Pi(n; \phi, m) = \int_0^{\phi}
            \frac{dt}{(1-n \sin^2 t) \sqrt{1-m \sin^2 t}} =
            \int_0^{\sin \phi}
            \frac{dt}{(1-nt^2) \sqrt{1-t^2} \sqrt{1-mt^2}}.

    Called with two arguments `n, m`, evaluates the complete
    elliptic integral of the third kind
    `\Pi(n,m) = \Pi(n; \frac{\pi}{2},m)`.

    In the defining integral, it is assumed that the principal branch
    of the square root is taken and that the path of integration avoids
    crossing any branch cuts. Outside `-\pi/2 \le \Re(\phi) \le \pi/2`,
    the function extends quasi-periodically as

    .. math ::

        \Pi(n,\phi+k\pi,m) = 2k\Pi(n,m) + \Pi(n,\phi,m), k \in \mathbb{Z}.

    **Plots**

    .. literalinclude :: /modules/mpmath/plots/ellippi.py
    .. image :: /modules/mpmath/plots/ellippi.png

    **Examples for the complete integral**

    Some basic values and limits::

        >>> from sympy.mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> ellippi(0,-5); ellipk(-5)
        0.9555039270640439337379334
        0.9555039270640439337379334
        >>> ellippi(inf,2)
        0.0
        >>> ellippi(2,inf)
        0.0
        >>> abs(ellippi(1,5))
        +inf
        >>> abs(ellippi(0.25,1))
        +inf

    Evaluation in terms of simpler functions::

        >>> ellippi(0.25,0.25); ellipe(0.25)/(1-0.25)
        1.956616279119236207279727
        1.956616279119236207279727
        >>> ellippi(3,0); pi/(2*sqrt(-2))
        (0.0 - 1.11072073453959156175397j)
        (0.0 - 1.11072073453959156175397j)
        >>> ellippi(-3,0); pi/(2*sqrt(4))
        0.7853981633974483096156609
        0.7853981633974483096156609

    **Examples for the incomplete integral**

    Basic values and limits::

        >>> ellippi(0.25,-0.5); ellippi(0.25,pi/2,-0.5)
        1.622944760954741603710555
        1.622944760954741603710555
        >>> ellippi(1,0,1)
        0.0
        >>> ellippi(inf,0,1)
        0.0
        >>> ellippi(0,0.25,0.5); ellipf(0.25,0.5)
        0.2513040086544925794134591
        0.2513040086544925794134591
        >>> ellippi(1,1,1); (log(sec(1)+tan(1))+sec(1)*tan(1))/2
        2.054332933256248668692452
        2.054332933256248668692452
        >>> ellippi(0.25, 53*pi/2, 0.75); 53*ellippi(0.25,0.75)
        135.240868757890840755058
        135.240868757890840755058
        >>> ellippi(0.5,pi/4,0.5); 2*ellipe(pi/4,0.5)-1/sqrt(3)
        0.9190227391656969903987269
        0.9190227391656969903987269

    Complex arguments are supported::

        >>> ellippi(0.5, 5+6j-2*pi, -7-8j)
        (-0.3612856620076747660410167 + 0.5217735339984807829755815j)

    Some degenerate cases::

        >>> ellippi(1,1)
        +inf
        >>> ellippi(1,0)
        +inf
        >>> ellippi(1,2,0)
        +inf
        >>> ellippi(1,2,1)
        +inf
        >>> ellippi(1,0,1)
        0.0
r"""
    Called with three arguments `n, \phi, m`, evaluates the Legendre
    incomplete elliptic integral of the third kind
r Called with three arguments n \ phi m evaluates the Legendre incomplete elliptic integral of the third kind
def ellippi(ctx, *args):
    r"""
    Called with three arguments `n, \phi, m`, evaluates the Legendre
    incomplete elliptic integral of the third kind

    .. math ::

        \Pi(n; \phi, m) = \int_0^{\phi}
            \frac{dt}{(1-n \sin^2 t) \sqrt{1-m \sin^2 t}} =
            \int_0^{\sin \phi}
            \frac{dt}{(1-nt^2) \sqrt{1-t^2} \sqrt{1-mt^2}}.

    Called with two arguments `n, m`, evaluates the complete
    elliptic integral of the third kind
    `\Pi(n,m) = \Pi(n; \frac{\pi}{2},m)`.

    In the defining integral, it is assumed that the principal branch
    of the square root is taken and that the path of integration avoids
    crossing any branch cuts. Outside `-\pi/2 \le \Re(\phi) \le \pi/2`,
    the function extends quasi-periodically as

    .. math ::

        \Pi(n,\phi+k\pi,m) = 2k\Pi(n,m) + \Pi(n,\phi,m), k \in \mathbb{Z}.

    **Plots**

    .. literalinclude :: /modules/mpmath/plots/ellippi.py
    .. image :: /modules/mpmath/plots/ellippi.png

    **Examples for the complete integral**

    Some basic values and limits::

        >>> from sympy.mpmath import *
        >>> mp.dps = 25; mp.pretty = True
        >>> ellippi(0,-5); ellipk(-5)
        0.9555039270640439337379334
        0.9555039270640439337379334
        >>> ellippi(inf,2)
        0.0
        >>> ellippi(2,inf)
        0.0
        >>> abs(ellippi(1,5))
        +inf
        >>> abs(ellippi(0.25,1))
        +inf

    Evaluation in terms of simpler functions::

        >>> ellippi(0.25,0.25); ellipe(0.25)/(1-0.25)
        1.956616279119236207279727
        1.956616279119236207279727
        >>> ellippi(3,0); pi/(2*sqrt(-2))
        (0.0 - 1.11072073453959156175397j)
        (0.0 - 1.11072073453959156175397j)
        >>> ellippi(-3,0); pi/(2*sqrt(4))
        0.7853981633974483096156609
        0.7853981633974483096156609

    **Examples for the incomplete integral**

    Basic values and limits::

        >>> ellippi(0.25,-0.5); ellippi(0.25,pi/2,-0.5)
        1.622944760954741603710555
        1.622944760954741603710555
        >>> ellippi(1,0,1)
        0.0
        >>> ellippi(inf,0,1)
        0.0
        >>> ellippi(0,0.25,0.5); ellipf(0.25,0.5)
        0.2513040086544925794134591
        0.2513040086544925794134591
        >>> ellippi(1,1,1); (log(sec(1)+tan(1))+sec(1)*tan(1))/2
        2.054332933256248668692452
        2.054332933256248668692452
        >>> ellippi(0.25, 53*pi/2, 0.75); 53*ellippi(0.25,0.75)
        135.240868757890840755058
        135.240868757890840755058
        >>> ellippi(0.5,pi/4,0.5); 2*ellipe(pi/4,0.5)-1/sqrt(3)
        0.9190227391656969903987269
        0.9190227391656969903987269

    Complex arguments are supported::

        >>> ellippi(0.5, 5+6j-2*pi, -7-8j)
        (-0.3612856620076747660410167 + 0.5217735339984807829755815j)

    Some degenerate cases::

        >>> ellippi(1,1)
        +inf
        >>> ellippi(1,0)
        +inf
        >>> ellippi(1,2,0)
        +inf
        >>> ellippi(1,2,1)
        +inf
        >>> ellippi(1,0,1)
        0.0

    """
    if len(args) == 2:
        n, m = args
        complete = True
        z = phi = ctx.pi/2
    else:
        n, phi, m = args
        complete = False
        z = phi
    if not (ctx.isnormal(n) and ctx.isnormal(z) and ctx.isnormal(m)):
        if ctx.isnan(n) or ctx.isnan(z) or ctx.isnan(m):
            raise ValueError
        if complete:
            if m == 0:
                if n == 1:
                    return ctx.inf
                return ctx.pi/(2*ctx.sqrt(1-n))
            if n == 0: return ctx.ellipk(m)
            if ctx.isinf(n) or ctx.isinf(m): return ctx.zero
        else:
            if z == 0: return z
            if ctx.isinf(n): return ctx.zero
            if ctx.isinf(m): return ctx.zero
        if ctx.isinf(n) or ctx.isinf(z) or ctx.isinf(m):
            raise ValueError
    if complete:
        if m == 1:
            if n == 1:
                return ctx.inf
            return -ctx.inf/ctx.sign(n-1)
        away = False
    else:
        x = z.real
        ctx.prec += max(0, ctx.mag(x))
        pi = +ctx.pi
        away = abs(x) > pi/2
    if away:
        d = ctx.nint(x/pi)
        z = z-pi*d
        P = 2*d*ctx.ellippi(n,m)
        if ctx.isinf(P):
            return ctx.inf
    else:
        P = 0
    def terms():
        if complete:
            c, s = ctx.zero, ctx.one
        else:
            c, s = ctx.cos_sin(z)
        x = c**2
        y = 1-m*s**2
        RF = ctx.elliprf(x, y, 1)
        RJ = ctx.elliprj(x, y, 1, 1-n*s**2)
        return s*RF, n*s**3*RJ/3
    return ctx.sum_accurately(terms) + P
def ellippi ( ctx , * args ) : if len ( args ) == 2 : n , m = args complete = True z = phi = ctx . pi / 2 else : n , phi , m = args complete = False z = phi if not ( ctx . isnormal ( n ) and ctx . isnormal ( z ) and ctx . isnormal ( m ) ) : if ctx . isnan ( n ) or ctx . isnan ( z ) or ctx . isnan ( m ) : raise ValueError if complete : if m == 0 : if n == 1 : return ctx . inf return ctx . pi / ( 2 * ctx . sqrt ( 1 - n ) ) if n == 0 : return ctx . ellipk ( m ) if ctx . isinf ( n ) or ctx . isinf ( m ) : return ctx . zero else : if z == 0 : return z if ctx . isinf ( n ) : return ctx . zero if ctx . isinf ( m ) : return ctx . zero if ctx . isinf ( n ) or ctx . isinf ( z ) or ctx . isinf ( m ) : raise ValueError if complete : if m == 1 : if n == 1 : return ctx . inf return - ctx . inf / ctx . sign ( n - 1 ) away = False else : x = z . real ctx . prec += max ( 0 , ctx . mag ( x ) ) pi = + ctx . pi away = abs ( x ) > pi / 2 if away : d = ctx . nint ( x / pi ) z = z - pi * d P = 2 * d * ctx . ellippi ( n , m ) if ctx . isinf ( P ) : return ctx . inf else : P = 0 def terms ( ) : if complete : c , s = ctx . zero , ctx . one else : c , s = ctx . cos_sin ( z ) x = c ** 2 y = 1 - m * s ** 2 RF = ctx . elliprf ( x , y , 1 ) RJ = ctx . elliprj ( x , y , 1 , 1 - n * s ** 2 ) return s * RF , n * s ** 3 * RJ / 3 return ctx . sum_accurately ( terms ) + P