expintegrals.py

from .functions import defun, defun_wrapped

@defun_wrapped
def _erf_complex(ctx, z):
    z2 = ctx.square_exp_arg(z, -1)
    #z2 = -z**2
    v = (2/ctx.sqrt(ctx.pi))*z * ctx.hyp1f1((1,2),(3,2), z2)
    if not ctx._re(z):
        v = ctx._im(v)*ctx.j
    return v

@defun_wrapped
def _erfc_complex(ctx, z):
    if ctx.re(z) > 2:
        z2 = ctx.square_exp_arg(z)
        nz2 = ctx.fneg(z2, exact=True)
        v = ctx.exp(nz2)/ctx.sqrt(ctx.pi) * ctx.hyperu((1,2),(1,2), z2)
    else:
        v = 1 - ctx._erf_complex(z)
    if not ctx._re(z):
        v = 1+ctx._im(v)*ctx.j
    return v

@defun
def erf(ctx, z):
    z = ctx.convert(z)
    if ctx._is_real_type(z):
        try:
            return ctx._erf(z)
        except NotImplementedError:
            pass
    if ctx._is_complex_type(z) and not z.imag:
        try:
            return ctx.mpc(ctx._erf(z.real))
        except NotImplementedError:
            pass
    return ctx._erf_complex(z)

@defun
def erfc(ctx, z):
    z = ctx.convert(z)
    if ctx._is_real_type(z):
        try:
            return ctx._erfc(z)
        except NotImplementedError:
            pass
    if ctx._is_complex_type(z) and not z.imag:
        try:
            return ctx.mpc(ctx._erfc(z.real))
        except NotImplementedError:
            pass
    return ctx._erfc_complex(z)

@defun
def square_exp_arg(ctx, z, mult=1, reciprocal=False):
    prec = ctx.prec*4+20
    if reciprocal:
        z2 = ctx.fmul(z, z, prec=prec)
        z2 = ctx.fdiv(ctx.one, z2, prec=prec)
    else:
        z2 = ctx.fmul(z, z, prec=prec)
    if mult != 1:
        z2 = ctx.fmul(z2, mult, exact=True)
    return z2

@defun_wrapped
def erfi(ctx, z):
    if not z:
        return z
    z2 = ctx.square_exp_arg(z)
    v = (2/ctx.sqrt(ctx.pi)*z) * ctx.hyp1f1((1,2), (3,2), z2)
    if not ctx._re(z):
        v = ctx._im(v)*ctx.j
    return v

@defun_wrapped
def erfinv(ctx, x):
    xre = ctx._re(x)
    if (xre != x) or (xre < -1) or (xre > 1):
        return ctx.bad_domain("erfinv(x) is defined only for -1 <= x <= 1")
    x = xre
    #if ctx.isnan(x): return x
    if not x: return x
    if x == 1: return ctx.inf
    if x == -1: return ctx.ninf
    if abs(x) < 0.9:
        a = 0.53728*x**3 + 0.813198*x
    else:
        # An asymptotic formula
        u = ctx.ln(2/ctx.pi/(abs(x)-1)**2)
        a = ctx.sign(x) * ctx.sqrt(u - ctx.ln(u))/ctx.sqrt(2)
    ctx.prec += 10
    return ctx.findroot(lambda t: ctx.erf(t)-x, a)

@defun_wrapped
def npdf(ctx, x, mu=0, sigma=1):
    sigma = ctx.convert(sigma)
    return ctx.exp(-(x-mu)**2/(2*sigma**2)) / (sigma*ctx.sqrt(2*ctx.pi))

@defun_wrapped
def ncdf(ctx, x, mu=0, sigma=1):
    a = (x-mu)/(sigma*ctx.sqrt(2))
    if a < 0:
        return ctx.erfc(-a)/2
    else:
        return (1+ctx.erf(a))/2

@defun_wrapped
def betainc(ctx, a, b, x1=0, x2=1, regularized=False):
    if x1 == x2:
        v = ctx.zero
    elif not x1:
        if x1 == 0 and x2 == 1:
            v = ctx.beta(a, b)
        else:
            v = x2**a * ctx.hyp2f1(a, 1-b, a+1, x2) / a
    else:
        m, d = ctx.nint_distance(a)
        if m <= 0:
            if d < -ctx.prec:
                h = +ctx.eps
                ctx.prec *= 2
                a += h
            elif d < -4:
                ctx.prec -= d
        s1 = x2**a * ctx.hyp2f1(a,1-b,a+1,x2)
        s2 = x1**a * ctx.hyp2f1(a,1-b,a+1,x1)
        v = (s1 - s2) / a
    if regularized:
        v /= ctx.beta(a,b)
    return v

@defun
def gammainc(ctx, z, a=0, b=None, regularized=False):
    regularized = bool(regularized)
    z = ctx.convert(z)
    if a is None:
        a = ctx.zero
        lower_modified = False
    else:
        a = ctx.convert(a)
        lower_modified = a != ctx.zero
    if b is None:
        b = ctx.inf
        upper_modified = False
    else:
        b = ctx.convert(b)
        upper_modified = b != ctx.inf
    # Complete gamma function
    if not (upper_modified or lower_modified):
        if regularized:
            if ctx.re(z) < 0:
                return ctx.inf
            elif ctx.re(z) > 0:
                return ctx.one
            else:
                return ctx.nan
        return ctx.gamma(z)
    if a == b:
        return ctx.zero
    # Standardize
    if ctx.re(a) > ctx.re(b):
        return -ctx.gammainc(z, b, a, regularized)
    # Generalized gamma
    if upper_modified and lower_modified:
        return +ctx._gamma3(z, a, b, regularized)
    # Upper gamma
    elif lower_modified:
        return ctx.upper_gamma(z, a, regularized)
    # Lower gamma
    elif upper_modified:
        return ctx.lower_gamma(z, b, regularized)

@defun
def lower_gamma(ctx, z, b, regularized=False):
    z = ctx.convert(z)
    b = ctx.convert(b)
    # Pole
    if ctx.isnpint(z):
        return ctx.inf
    G = [z] * regularized
    negb = ctx.fneg(b, exact=True)
    def h(z):
        T1 = [ctx.exp(negb), b, z], [1, z, -1], [], G, [1], [1+z], b
        return (T1,)
    return ctx.hypercomb(h, [z])

@defun
def upper_gamma(ctx, z, a, regularized=False):
    z = ctx.convert(z)
    a = ctx.convert(a)
    # Fast integer case, when available
    if ctx.isint(z):
        try:
            if regularized:
                # Gamma pole
                if ctx.isnpint(z):
                    return ctx.zero
                orig = ctx.prec
                try:
                    ctx.prec += 10
                    return ctx._gamma_upper_int(z, a) / ctx.gamma(z)
                finally:
                    ctx.prec = orig
            else:
                return ctx._gamma_upper_int(z, a)
        except NotImplementedError:
            pass
    # hypercomb is unable to detect the exact zeros, so handle them here
    if z == 2 and a == -1:
        return (z+a)*0
    if z == 3 and (a == -1-1j or a == -1+1j):
        return (z+a)*0
    nega = ctx.fneg(a, exact=True)
    G = [z] * regularized
    # Use 2F0 series when possible; fall back to lower gamma representation
    try:
        def h(z):
            r = z-1
            return [([ctx.exp(nega), a], [1, r], [], G, [1, -r], [], 1/nega)]
        return ctx.hypercomb(h, [z], force_series=True)
    except ctx.NoConvergence:
        def h(z):
            T1 = [], [1, z-1], [z], G, [], [], 0
            T2 = [-ctx.exp(nega), a, z], [1, z, -1], [], G, [1], [1+z], a
            return T1, T2
        return ctx.hypercomb(h, [z])

@defun
def _gamma3(ctx, z, a, b, regularized=False):
    pole = ctx.isnpint(z)
    if regularized and pole:
        return ctx.zero
    try:
        ctx.prec += 15
        # We don't know in advance whether it's better to write as a difference
        # of lower or upper gamma functions, so try both
        T1 = ctx.gammainc(z, a, regularized=regularized)
        T2 = ctx.gammainc(z, b, regularized=regularized)
        R = T1 - T2
        if ctx.mag(R) - max(ctx.mag(T1), ctx.mag(T2)) > -10:
            return R
        if not pole:
            T1 = ctx.gammainc(z, 0, b, regularized=regularized)
            T2 = ctx.gammainc(z, 0, a, regularized=regularized)
            R = T1 - T2
            # May be ok, but should probably at least print a warning
            # about possible cancellation
            if 1: #ctx.mag(R) - max(ctx.mag(T1), ctx.mag(T2)) > -10:
                return R
    finally:
        ctx.prec -= 15
    raise NotImplementedError

@defun_wrapped
def expint(ctx, n, z):
    if ctx.isint(n) and ctx._is_real_type(z):
        try:
            return ctx._expint_int(n, z)
        except NotImplementedError:
            pass
    if ctx.isnan(n) or ctx.isnan(z):
        return z*n
    if z == ctx.inf:
        return 1/z
    if z == 0:
        # integral from 1 to infinity of t^n
        if ctx.re(n) <= 1:
            # TODO: reasonable sign of infinity
            return ctx.inf
        else:
            return ctx.one/(n-1)
    if n == 0:
        return ctx.exp(-z)/z
    if n == -1:
        return ctx.exp(-z)*(z+1)/z**2
    return z**(n-1) * ctx.upper_gamma(1-n, z)

@defun_wrapped
def li(ctx, z, offset=False):
    if offset:
        if z == 2:
            return ctx.zero
        return ctx.ei(ctx.ln(z)) - ctx.ei(ctx.ln2)
    if not z:
        return z
    if z == 1:
        return ctx.ninf
    return ctx.ei(ctx.ln(z))

@defun
def ei(ctx, z):
    try:
        return ctx._ei(z)
    except NotImplementedError:
        return ctx._ei_generic(z)

@defun_wrapped
def _ei_generic(ctx, z):
    # Note: the following is currently untested because mp and fp
    # both use special-case ei code
    if z == ctx.inf:
        return z
    if z == ctx.ninf:
        return ctx.zero
    if ctx.mag(z) > 1:
        try:
            r = ctx.one/z
            v = ctx.exp(z)*ctx.hyper([1,1],[],r,
                maxterms=ctx.prec, force_series=True)/z
            im = ctx._im(z)
            if im > 0:
                v += ctx.pi*ctx.j
            if im < 0:
                v -= ctx.pi*ctx.j
            return v
        except ctx.NoConvergence:
            pass
    v = z*ctx.hyp2f2(1,1,2,2,z) + ctx.euler
    if ctx._im(z):
        v += 0.5*(ctx.log(z) - ctx.log(ctx.one/z))
    else:
        v += ctx.log(abs(z))
    return v

@defun
def e1(ctx, z):
    try:
        return ctx._e1(z)
    except NotImplementedError:
        return ctx.expint(1, z)

@defun
def ci(ctx, z):
    try:
        return ctx._ci(z)
    except NotImplementedError:
        return ctx._ci_generic(z)

@defun_wrapped
def _ci_generic(ctx, z):
    if ctx.isinf(z):
        if z == ctx.inf: return ctx.zero
        if z == ctx.ninf: return ctx.pi*1j
    jz = ctx.fmul(ctx.j,z,exact=True)
    njz = ctx.fneg(jz,exact=True)
    v = 0.5*(ctx.ei(jz) + ctx.ei(njz))
    zreal = ctx._re(z)
    zimag = ctx._im(z)
    if zreal == 0:
        if zimag > 0: v += ctx.pi*0.5j
        if zimag < 0: v -= ctx.pi*0.5j
    if zreal < 0:
        if zimag >= 0: v += ctx.pi*1j
        if zimag <  0: v -= ctx.pi*1j
    if ctx._is_real_type(z) and zreal > 0:
        v = ctx._re(v)
    return v

@defun
def si(ctx, z):
    try:
        return ctx._si(z)
    except NotImplementedError:
        return ctx._si_generic(z)

@defun_wrapped
def _si_generic(ctx, z):
    if ctx.isinf(z):
        if z == ctx.inf: return 0.5*ctx.pi
        if z == ctx.ninf: return -0.5*ctx.pi
    # Suffers from cancellation near 0
    if ctx.mag(z) >= -1:
        jz = ctx.fmul(ctx.j,z,exact=True)
        njz = ctx.fneg(jz,exact=True)
        v = (-0.5j)*(ctx.ei(jz) - ctx.ei(njz))
        zreal = ctx._re(z)
        if zreal > 0:
            v -= 0.5*ctx.pi
        if zreal < 0:
            v += 0.5*ctx.pi
        if ctx._is_real_type(z):
            v = ctx._re(v)
        return v
    else:
        return z*ctx.hyp1f2((1,2),(3,2),(3,2),-0.25*z*z)

@defun_wrapped
def chi(ctx, z):
    nz = ctx.fneg(z, exact=True)
    v = 0.5*(ctx.ei(z) + ctx.ei(nz))
    zreal = ctx._re(z)
    zimag = ctx._im(z)
    if zimag > 0:
        v += ctx.pi*0.5j
    elif zimag < 0:
        v -= ctx.pi*0.5j
    elif zreal < 0:
        v += ctx.pi*1j
    return v

@defun_wrapped
def shi(ctx, z):
    # Suffers from cancellation near 0
    if ctx.mag(z) >= -1:
        nz = ctx.fneg(z, exact=True)
        v = 0.5*(ctx.ei(z) - ctx.ei(nz))
        zimag = ctx._im(z)
        if zimag > 0: v -= 0.5j*ctx.pi
        if zimag < 0: v += 0.5j*ctx.pi
        return v
    else:
        return z * ctx.hyp1f2((1,2),(3,2),(3,2),0.25*z*z)

@defun_wrapped
def fresnels(ctx, z):
    if z == ctx.inf:
        return ctx.mpf(0.5)
    if z == ctx.ninf:
        return ctx.mpf(-0.5)
    return ctx.pi*z**3/6*ctx.hyp1f2((3,4),(3,2),(7,4),-ctx.pi**2*z**4/16)

@defun_wrapped
def fresnelc(ctx, z):
    if z == ctx.inf:
        return ctx.mpf(0.5)
    if z == ctx.ninf:
        return ctx.mpf(-0.5)
    return z*ctx.hyp1f2((1,4),(1,2),(5,4),-ctx.pi**2*z**4/16)