コード例 #1
0
def constantRangeIntegral(s, sLimit):
    logArg = 1. - s / sLimit
    retVal = clog(logArg)
    retVal *= 2 * sLimit**2
    retVal += s * (2 * sLimit + s)
    retVal /= 2 * sLimit**2 * s**3
    return retVal
コード例 #2
0
def A(s, mPart, renScale2, allowSubThr=False):
    sig = sigma(s, mPart, allowSubThr=allowSubThr)
    retVal = log(
        mPart**2 / renScale2) + 8 * mPart**2 / s - 5. / 3. + sig**3 * clog(
            (sig + 1) /
            (sig -
             1.))  # +1.j pi to obtain the real part (at least above threshold)
    return retVal
コード例 #3
0
def DIPPR_EQ114_integral_by_T(T, Tc, a, b, c, d):
    """T-Integral of DPPR Equation #114."""
    return (-a**2 * Tc * clog(T - Tc).real + d**2 * T**6 / (30 * Tc**5) -
            T**5 * (c * d + 2 * d**2) / (10 * Tc**4) + T**4 *
            (c**2 + 6 * c * d + 6 * d**2) / (12 * Tc**3) - T**3 *
            (a * d + c**2 + 3 * c * d + 2 * d**2) / (3 * Tc**2) + T**2 *
            (2 * a * c + 2 * a * d + c**2 + 2 * c * d + d**2) / (2 * Tc) + T *
            (-2 * a * c - a * d + b - c**2 / 3 - c * d / 2 - d**2 / 5))
コード例 #4
0
def constantRangeIntegral(s, sLimit):
    logArg = 1. - s / sLimit
    #	retVal  = log(abs(logArg))
    #	if logArg < 0.:
    #		retVal += 1.j*pi
    retVal = clog(logArg)
    retVal *= 2 * sLimit**2
    retVal += s * (2 * sLimit + s)
    retVal /= 2 * sLimit**2 * s**3
    return retVal
コード例 #5
0
 def _set_coordonnees(self, x, y):
     zA = self.__point1.z
     zB = self.__point2.z
     z = x + 1j * y
     try:
         if abs(zB - zA) > param.tolerance:
             # TODO: cas où l'angle n'est pas en radian
             self.__angle.val = clog((z - zA) / (zB - zA)).imag
     except (OverflowError, ZeroDivisionError):
         if param.debug:
             print_error()
コード例 #6
0
ファイル: polyedres.py プロジェクト: wxgeo/geophar
 def _set_coordonnees(self, x, y):
     zA = self.__point1.z
     zB = self.__point2.z
     z = x + 1j*y
     try:
         if abs(zB - zA) > param.tolerance:
        # TODO: cas où l'angle n'est pas en radian
             self.__angle.val = clog((z - zA)/(zB - zA)).imag
     except (OverflowError, ZeroDivisionError):
         if param.debug:
             print_error()
コード例 #7
0
ファイル: polygones.py プロジェクト: Grahack/geophar
 def _set_coordonnees(self, x, y):
     zA = self.__sommet_principal.z
     zB = self.__point2.z
     z = x + 1j*y
     try:
         if issympy(zA, zB):
             self.__angle = sarg((z - zA)/(zB - zA))
         else:
             self.__angle = clog((z - zA)/(zB - zA)).imag
     except (OverflowError, ZeroDivisionError):
         if param.debug:
             print_error()
コード例 #8
0
ファイル: polygones.py プロジェクト: jpg32/wxgeometrie
 def _set_coordonnees(self, x, y):
     zA = self.__sommet_principal.z
     zB = self.__point2.z
     z = x + 1j * y
     try:
         if issympy(zA, zB):
             self.__angle = sarg((z - zA) / (zB - zA))
         else:
             self.__angle = clog((z - zA) / (zB - zA)).imag
     except (OverflowError, ZeroDivisionError):
         if param.debug:
             print_error()
コード例 #9
0
ファイル: polygones.py プロジェクト: Grahack/geophar
 def _set_coordonnees(self, x, y):
     zA = self.__point1.z
     zB = self.__point2.z
     zI = (zA + zB)/2
     z = x + 1j*y
     try:
         if issympy(zI):
             self.__angle = sarg((z - zI)/(zB - zI))/2
         else:
             self.__angle = clog((z - zI)/(zB - zI)).imag/2
     except (OverflowError, ZeroDivisionError):
         if param.debug:
             print_error()
コード例 #10
0
ファイル: polygones.py プロジェクト: jpg32/wxgeometrie
 def _set_coordonnees(self, x, y):
     zA = self.__point1.z
     zB = self.__point2.z
     zI = (zA + zB) / 2
     z = x + 1j * y
     try:
         if issympy(zI):
             self.__angle = sarg((z - zI) / (zB - zI)) / 2
         else:
             self.__angle = clog((z - zI) / (zB - zI)).imag / 2
     except (OverflowError, ZeroDivisionError):
         if param.debug:
             print_error()
コード例 #11
0
def DIPPR_EQ114_integral_by_T_over_T(T, Tc, a, b, c, d):
    """T-Integral of DPPR Equation #114 divided by T."""
    return (
        -a**2 *
        clog(T +
             (-60 * a**2 * Tc + 60 * a * c * Tc + 30 * a * d * Tc -
              30 * b * Tc + 10 * c**2 * Tc + 15 * c * d * Tc + 6 * d**2 * Tc) /
             (60 * a**2 - 60 * a * c - 30 * a * d + 30 * b - 10 * c**2 -
              15 * c * d - 6 * d**2)).real + d**2 * T**5 / (25 * Tc**5) -
        T**4 * (c * d + 2 * d**2) / (8 * Tc**4) + T**3 *
        (c**2 + 6 * c * d + 6 * d**2) / (9 * Tc**3) - T**2 *
        (a * d + c**2 + 3 * c * d + 2 * d**2) / (2 * Tc**2) + T *
        (2 * a * c + 2 * a * d + c**2 + 2 * c * d + d**2) / Tc +
        (30 * a**2 - 60 * a * c - 30 * a * d + 30 * b - 10 * c**2 -
         15 * c * d - 6 * d**2) *
        clog(T +
             (-30 * a**2 * Tc + 60 * a * c * Tc + 30 * a * d * Tc - 30 * b *
              Tc + 10 * c**2 * Tc + 15 * c * d * Tc + 6 * d**2 * Tc + Tc *
              (30 * a**2 - 60 * a * c - 30 * a * d + 30 * b - 10 * c**2 -
               15 * c * d - 6 * d**2)) /
             (60 * a**2 - 60 * a * c - 30 * a * d + 30 * b - 10 * c**2 -
              15 * c * d - 6 * d**2)).real / 30)
コード例 #12
0
def Lastovka_Shaw_Indefinite_Integral_Over_T(T, MW, similarity_variable, term):
    a = similarity_variable
    a2 = a * a
    B11 = 0.73917383
    B12 = 8.88308889
    C11 = 1188.28051
    C12 = 1813.04613
    B21 = 0.0483019
    B22 = 4.35656721
    C21 = 2897.01927
    C22 = 5987.80407
    S = (term * clog(T) +
         (-B11 - B12 * a) * clog(cexp((-C11 - C12 * a) / T) - 1.) +
         (-B11 * C11 - B11 * C12 * a - B12 * C11 * a - B12 * C12 * a2) /
         (T * cexp((-C11 - C12 * a) / T) - T) -
         (B11 * C11 + B11 * C12 * a + B12 * C11 * a + B12 * C12 * a2) / T)
    S += ((-B21 - B22 * a) * clog(cexp((-C21 - C22 * a) / T) - 1.) +
          (-B21 * C21 - B21 * C22 * a - B22 * C21 * a - B22 * C22 * a2) /
          (T * cexp((-C21 - C22 * a) / T) - T) -
          (B21 * C21 + B21 * C22 * a + B22 * C21 * a + B22 * C22 * a**2) / T)
    # There is a non-real component, but it is only a function of similariy
    # variable and so will always cancel out.
    return MW * S.real
コード例 #13
0
def ghiggs(t):
    if t<=1:
        return csqrt(1-1/t)/2. * ( clog((1 + csqrt(1-1/t))/(1 - csqrt(1-1/t)))-pi*1j)
    else:
        return csqrt(1/t-1)*casin(csqrt(t))
コード例 #14
0
ファイル: dippr.py プロジェクト: lilanyu/biosteam
def EQ114(T, Tc, A, B, C, D, order=0):
    r'''DIPPR Equation #114. Rarely used, normally as an alternate liquid
    heat capacity expression. All 4 parameters are required, as well as
    critical temperature.

    .. math::
        Y = \frac{A^2}{\tau} + B - 2AC\tau - AD\tau^2 - \frac{1}{3}C^2\tau^3
        - \frac{1}{2}CD\tau^4 - \frac{1}{5}D^2\tau^5

        \tau = 1 - \frac{T}{Tc}

    Parameters
    ----------
    T : float
        Temperature, [K]
    Tc : float
        Critical temperature, [K]
    A-D : float
        Parameter for the equation; chemical and property specific [-]
    order : int, optional
        Order of the calculation. 0 for the calculation of the result itself;
        for 1, the first derivative of the property is returned, for
        -1, the indefinite integral of the property with respect to temperature
        is returned; and for -1j, the indefinite integral of the property
        divided by temperature with respect to temperature is returned. No 
        other integrals or derivatives are implemented, and an exception will 
        be raised if any other order is given.

    Returns
    -------
    Y : float
        Property [constant-specific; if order == 1, property/K; if order == -1,
                  property*K; if order == -1j, unchanged from default]

    Notes
    -----
    The derivative with respect to T, integral with respect to T, and integral
    over T with respect to T are computed as follows. All expressions can be
    obtained with SymPy readily.
    
    .. math::
        \frac{d Y}{dT} = \frac{A^{2}}{T_{c} \left(- \frac{T}{T_{c}} 
        + 1\right)^{2}} + \frac{2 A}{T_{c}} C + \frac{2 A}{T_{c}} D \left(
        - \frac{T}{T_{c}} + 1\right) + \frac{C^{2}}{T_{c}} \left(
        - \frac{T}{T_{c}} + 1\right)^{2} + \frac{2 C}{T_{c}} D \left(
        - \frac{T}{T_{c}} + 1\right)^{3} + \frac{D^{2}}{T_{c}} \left(
        - \frac{T}{T_{c}} + 1\right)^{4}
        
    .. math::
        \int Y dT = - A^{2} T_{c} \log{\left (T - T_{c} \right )} + \frac{D^{2}
        T^{6}}{30 T_{c}^{5}} - \frac{T^{5}}{10 T_{c}^{4}} \left(C D + 2 D^{2}
        \right) + \frac{T^{4}}{12 T_{c}^{3}} \left(C^{2} + 6 C D + 6 D^{2}
        \right) - \frac{T^{3}}{3 T_{c}^{2}} \left(A D + C^{2} + 3 C D 
        + 2 D^{2}\right) + \frac{T^{2}}{2 T_{c}} \left(2 A C + 2 A D + C^{2} 
        + 2 C D + D^{2}\right) + T \left(- 2 A C - A D + B - \frac{C^{2}}{3} 
        - \frac{C D}{2} - \frac{D^{2}}{5}\right)
        
    .. math::
        \int \frac{Y}{T} dT = - A^{2} \log{\left (T + \frac{- 60 A^{2} T_{c}
        + 60 A C T_{c} + 30 A D T_{c} - 30 B T_{c} + 10 C^{2} T_{c}
        + 15 C D T_{c} + 6 D^{2} T_{c}}{60 A^{2} - 60 A C - 30 A D + 30 B 
        - 10 C^{2} - 15 C D - 6 D^{2}} \right )} + \frac{D^{2} T^{5}}
        {25 T_{c}^{5}} - \frac{T^{4}}{8 T_{c}^{4}} \left(C D + 2 D^{2}
        \right) + \frac{T^{3}}{9 T_{c}^{3}} \left(C^{2} + 6 C D + 6 D^{2}
        \right) - \frac{T^{2}}{2 T_{c}^{2}} \left(A D + C^{2} + 3 C D
        + 2 D^{2}\right) + \frac{T}{T_{c}} \left(2 A C + 2 A D + C^{2} 
        + 2 C D + D^{2}\right) + \frac{1}{30} \left(30 A^{2} - 60 A C 
        - 30 A D + 30 B - 10 C^{2} - 15 C D - 6 D^{2}\right) \log{\left 
        (T + \frac{1}{60 A^{2} - 60 A C - 30 A D + 30 B - 10 C^{2} - 15 C D
        - 6 D^{2}} \left(- 30 A^{2} T_{c} + 60 A C T_{c} + 30 A D T_{c} 
        - 30 B T_{c} + 10 C^{2} T_{c} + 15 C D T_{c} + 6 D^{2} T_{c}
        + T_{c} \left(30 A^{2} - 60 A C - 30 A D + 30 B - 10 C^{2} - 15 C D
        - 6 D^{2}\right)\right) \right )}

    Strictly speaking, the integral over T has an imaginary component, but
    only the real component is relevant and the complex part discarded.

    Examples
    --------
    Hydrogen liquid heat capacity; DIPPR coefficients normally in J/kmol/K.

    >>> EQ114(20, 33.19, 66.653, 6765.9, -123.63, 478.27)
    19423.948911676463

    References
    ----------
    .. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
       DIPPR/AIChE
    '''
    if order == 0:
        t = 1. - T / Tc
        return (A**2. / t + B - 2. * A * C * t - A * D * t**2. -
                C**2. * t**3. / 3. - C * D * t**4. / 2. - D**2 * t**5. / 5.)
    elif order == 1:
        return (A**2 / (Tc * (-T / Tc + 1)**2) + 2 * A * C / Tc + 2 * A * D *
                (-T / Tc + 1) / Tc + C**2 * (-T / Tc + 1)**2 / Tc + 2 * C * D *
                (-T / Tc + 1)**3 / Tc + D**2 * (-T / Tc + 1)**4 / Tc)
    elif order == -1:
        return (-A**2 * Tc * clog(T - Tc).real + D**2 * T**6 / (30 * Tc**5) -
                T**5 * (C * D + 2 * D**2) / (10 * Tc**4) + T**4 *
                (C**2 + 6 * C * D + 6 * D**2) / (12 * Tc**3) - T**3 *
                (A * D + C**2 + 3 * C * D + 2 * D**2) / (3 * Tc**2) + T**2 *
                (2 * A * C + 2 * A * D + C**2 + 2 * C * D + D**2) / (2 * Tc) +
                T * (-2 * A * C - A * D + B - C**2 / 3 - C * D / 2 - D**2 / 5))
    elif order == -1j:
        return (
            -A**2 *
            clog(T +
                 (-60 * A**2 * Tc + 60 * A * C * Tc + 30 * A * D * Tc - 30 *
                  B * Tc + 10 * C**2 * Tc + 15 * C * D * Tc + 6 * D**2 * Tc) /
                 (60 * A**2 - 60 * A * C - 30 * A * D + 30 * B - 10 * C**2 -
                  15 * C * D - 6 * D**2)).real + D**2 * T**5 / (25 * Tc**5) -
            T**4 * (C * D + 2 * D**2) / (8 * Tc**4) + T**3 *
            (C**2 + 6 * C * D + 6 * D**2) / (9 * Tc**3) - T**2 *
            (A * D + C**2 + 3 * C * D + 2 * D**2) / (2 * Tc**2) + T *
            (2 * A * C + 2 * A * D + C**2 + 2 * C * D + D**2) / Tc +
            (30 * A**2 - 60 * A * C - 30 * A * D + 30 * B - 10 * C**2 -
             15 * C * D - 6 * D**2) *
            clog(T + (-30 * A**2 * Tc + 60 * A * C * Tc + 30 * A * D * Tc -
                      30 * B * Tc + 10 * C**2 * Tc + 15 * C * D * Tc +
                      6 * D**2 * Tc + Tc *
                      (30 * A**2 - 60 * A * C - 30 * A * D + 30 * B -
                       10 * C**2 - 15 * C * D - 6 * D**2)) /
                 (60 * A**2 - 60 * A * C - 30 * A * D + 30 * B - 10 * C**2 -
                  15 * C * D - 6 * D**2)).real / 30)
    else:
        raise Exception(order_not_found_msg)
コード例 #15
0
ファイル: special.py プロジェクト: kkremitzki/fluids
def py_catan(x):
    # Implemented only because micropython is missing this function
    return 0.5j * (clog(1.0 - 1.0j * x) - clog(1.0 + 1.0j * x))
コード例 #16
0
def ghiggs(t):
    if t <= 1:
        return csqrt(1 - 1 / t) / 2. * (clog(
            (1 + csqrt(1 - 1 / t)) / (1 - csqrt(1 - 1 / t))) - pi * 1j)
    else:
        return csqrt(1 / t - 1) * casin(csqrt(t))
コード例 #17
0
ファイル: dippr.py プロジェクト: CalebBell/thermo
def EQ114(T, Tc, A, B, C, D, order=0):
    r'''DIPPR Equation #114. Rarely used, normally as an alternate liquid
    heat capacity expression. All 4 parameters are required, as well as
    critical temperature.

    .. math::
        Y = \frac{A^2}{\tau} + B - 2AC\tau - AD\tau^2 - \frac{1}{3}C^2\tau^3
        - \frac{1}{2}CD\tau^4 - \frac{1}{5}D^2\tau^5

        \tau = 1 - \frac{T}{Tc}

    Parameters
    ----------
    T : float
        Temperature, [K]
    Tc : float
        Critical temperature, [K]
    A-D : float
        Parameter for the equation; chemical and property specific [-]
    order : int, optional
        Order of the calculation. 0 for the calculation of the result itself;
        for 1, the first derivative of the property is returned, for
        -1, the indefinite integral of the property with respect to temperature
        is returned; and for -1j, the indefinite integral of the property
        divided by temperature with respect to temperature is returned. No 
        other integrals or derivatives are implemented, and an exception will 
        be raised if any other order is given.

    Returns
    -------
    Y : float
        Property [constant-specific; if order == 1, property/K; if order == -1,
                  property*K; if order == -1j, unchanged from default]

    Notes
    -----
    The derivative with respect to T, integral with respect to T, and integral
    over T with respect to T are computed as follows. All expressions can be
    obtained with SymPy readily.
    
    .. math::
        \frac{d Y}{dT} = \frac{A^{2}}{T_{c} \left(- \frac{T}{T_{c}} 
        + 1\right)^{2}} + \frac{2 A}{T_{c}} C + \frac{2 A}{T_{c}} D \left(
        - \frac{T}{T_{c}} + 1\right) + \frac{C^{2}}{T_{c}} \left(
        - \frac{T}{T_{c}} + 1\right)^{2} + \frac{2 C}{T_{c}} D \left(
        - \frac{T}{T_{c}} + 1\right)^{3} + \frac{D^{2}}{T_{c}} \left(
        - \frac{T}{T_{c}} + 1\right)^{4}
        
    .. math::
        \int Y dT = - A^{2} T_{c} \log{\left (T - T_{c} \right )} + \frac{D^{2}
        T^{6}}{30 T_{c}^{5}} - \frac{T^{5}}{10 T_{c}^{4}} \left(C D + 2 D^{2}
        \right) + \frac{T^{4}}{12 T_{c}^{3}} \left(C^{2} + 6 C D + 6 D^{2}
        \right) - \frac{T^{3}}{3 T_{c}^{2}} \left(A D + C^{2} + 3 C D 
        + 2 D^{2}\right) + \frac{T^{2}}{2 T_{c}} \left(2 A C + 2 A D + C^{2} 
        + 2 C D + D^{2}\right) + T \left(- 2 A C - A D + B - \frac{C^{2}}{3} 
        - \frac{C D}{2} - \frac{D^{2}}{5}\right)
        
    .. math::
        \int \frac{Y}{T} dT = - A^{2} \log{\left (T + \frac{- 60 A^{2} T_{c}
        + 60 A C T_{c} + 30 A D T_{c} - 30 B T_{c} + 10 C^{2} T_{c}
        + 15 C D T_{c} + 6 D^{2} T_{c}}{60 A^{2} - 60 A C - 30 A D + 30 B 
        - 10 C^{2} - 15 C D - 6 D^{2}} \right )} + \frac{D^{2} T^{5}}
        {25 T_{c}^{5}} - \frac{T^{4}}{8 T_{c}^{4}} \left(C D + 2 D^{2}
        \right) + \frac{T^{3}}{9 T_{c}^{3}} \left(C^{2} + 6 C D + 6 D^{2}
        \right) - \frac{T^{2}}{2 T_{c}^{2}} \left(A D + C^{2} + 3 C D
        + 2 D^{2}\right) + \frac{T}{T_{c}} \left(2 A C + 2 A D + C^{2} 
        + 2 C D + D^{2}\right) + \frac{1}{30} \left(30 A^{2} - 60 A C 
        - 30 A D + 30 B - 10 C^{2} - 15 C D - 6 D^{2}\right) \log{\left 
        (T + \frac{1}{60 A^{2} - 60 A C - 30 A D + 30 B - 10 C^{2} - 15 C D
        - 6 D^{2}} \left(- 30 A^{2} T_{c} + 60 A C T_{c} + 30 A D T_{c} 
        - 30 B T_{c} + 10 C^{2} T_{c} + 15 C D T_{c} + 6 D^{2} T_{c}
        + T_{c} \left(30 A^{2} - 60 A C - 30 A D + 30 B - 10 C^{2} - 15 C D
        - 6 D^{2}\right)\right) \right )}

    Strictly speaking, the integral over T has an imaginary component, but
    only the real component is relevant and the complex part discarded.

    Examples
    --------
    Hydrogen liquid heat capacity; DIPPR coefficients normally in J/kmol/K.

    >>> EQ114(20, 33.19, 66.653, 6765.9, -123.63, 478.27)
    19423.948911676463

    References
    ----------
    .. [1] Design Institute for Physical Properties, 1996. DIPPR Project 801
       DIPPR/AIChE
    '''
    if order == 0:
        t = 1.-T/Tc
        return (A**2./t + B - 2.*A*C*t - A*D*t**2. - C**2.*t**3./3. 
                - C*D*t**4./2. - D**2*t**5./5.)
    elif order == 1:
        return (A**2/(Tc*(-T/Tc + 1)**2) + 2*A*C/Tc + 2*A*D*(-T/Tc + 1)/Tc 
                + C**2*(-T/Tc + 1)**2/Tc + 2*C*D*(-T/Tc + 1)**3/Tc 
                + D**2*(-T/Tc + 1)**4/Tc)
    elif order == -1:
        return (-A**2*Tc*clog(T - Tc).real + D**2*T**6/(30*Tc**5) 
                - T**5*(C*D + 2*D**2)/(10*Tc**4) 
                + T**4*(C**2 + 6*C*D + 6*D**2)/(12*Tc**3) - T**3*(A*D + C**2 
                + 3*C*D + 2*D**2)/(3*Tc**2) + T**2*(2*A*C + 2*A*D + C**2 + 2*C*D 
                + D**2)/(2*Tc) + T*(-2*A*C - A*D + B - C**2/3 - C*D/2 - D**2/5))
    elif order == -1j:
        return (-A**2*clog(T + (-60*A**2*Tc + 60*A*C*Tc + 30*A*D*Tc - 30*B*Tc 
                + 10*C**2*Tc + 15*C*D*Tc + 6*D**2*Tc)/(60*A**2 - 60*A*C 
                - 30*A*D + 30*B - 10*C**2 - 15*C*D - 6*D**2)).real 
                + D**2*T**5/(25*Tc**5) - T**4*(C*D + 2*D**2)/(8*Tc**4) 
                + T**3*(C**2 + 6*C*D + 6*D**2)/(9*Tc**3) - T**2*(A*D + C**2
                + 3*C*D + 2*D**2)/(2*Tc**2) + T*(2*A*C + 2*A*D + C**2 + 2*C*D
                + D**2)/Tc + (30*A**2 - 60*A*C - 30*A*D + 30*B - 10*C**2
                - 15*C*D - 6*D**2)*clog(T + (-30*A**2*Tc + 60*A*C*Tc 
                + 30*A*D*Tc - 30*B*Tc + 10*C**2*Tc + 15*C*D*Tc + 6*D**2*Tc 
                + Tc*(30*A**2 - 60*A*C - 30*A*D + 30*B - 10*C**2 - 15*C*D 
                - 6*D**2))/(60*A**2 - 60*A*C - 30*A*D + 30*B - 10*C**2 
                - 15*C*D - 6*D**2)).real/30)
    else:
        raise Exception(order_not_found_msg)
コード例 #18
0
	def A(self, s, mPart, sheet):
		sig    = self.sigma(s, mPart)
		retVal = log(mPart**2/self.mu**2) + 8*mPart**2/s - 5./3. + sig**3*clog((sig+1)/(sig-1.)) # +1.j pi to obtain the real part (at least above threshold)
		if sheet == 2:
			retVal += 2*pi* (4*mPart**2/s-1.+0.j)**1.5
		return retVal