def plate_enlargement_factor_analytical(amplitude, wavelength):
        r'''Calculates the enhancement factor of the sinusoidal waves of the
        plate heat exchanger. This is the multiplier for the flat plate area
        to obtain the actual area available for heat transfer. Obtained from
        the following integral:
    
        .. math::
            \phi = \frac{\text{Effective area}}{\text{Projected area}} 
            = \frac{\int_0^\lambda\sqrt{1 + \left(\frac{\gamma\pi}{2}\right)^2
            \cos^2\left(\frac{2\pi}{\lambda}x\right)}dx}{\lambda}
            
    .. math::
            \gamma = \frac{4a}{\lambda}
            
        The solution to the integral is:
            
        .. math::
            \phi = \frac{2E\left(\frac{-4a^2\pi^2}{\lambda^2}\right)}{\pi}
            
        where E is the complete elliptic integral of the second kind, 
        calculated with SciPy.
    
        Parameters
        ----------
        amplitude : float
            Half the height of the wave of the ridges, [m]
        wavelength : float
            Distance between the bottoms of two of the ridges (sometimes called 
            pitch), [m]
    
        Returns
        -------
        plate_enlargement_factor : float
            The extra surface area multiplier as compared to a flat plate
            caused the corrugations, [-]
    
        Notes
        -----
        This is the exact analytical integral, obtained via Mathematica, Maple,
        and quite a bit of trial and error. It is confirmed via numerical 
        integration. The expression normally given is an
        approximation as follows:
            
        .. math::
            \phi = \frac{1}{6}\left(1+\sqrt{1+A^2} + 4\sqrt{1+A^2/2}\right)
            
            A = \frac{2\pi a}{\lambda}
            
        Most plate heat exchangers approximate a sinusoidal geometry only.
    
        Examples
        --------
        >>> PlateExchanger.plate_enlargement_factor_analytical(amplitude=5E-4, wavelength=3.7E-3)
        1.1611862034509677
        '''
        b = 2.*amplitude
        return 2.*float(ellipe(-b*b*pi*pi/(wavelength*wavelength)))/pi
r'''Calculates the enhancement factor of the sinusoidal waves of the
        plate heat exchanger. This is the multiplier for the flat plate area
        to obtain the actual area available for heat transfer. Obtained from
        the following integral:
    
        .. math::
            \phi = \frac{\text{Effective area}}{\text{Projected area}} 
            = \frac{\int_0^\lambda\sqrt{1 + \left(\frac{\gamma\pi}{2}\right)^2
            \cos^2\left(\frac{2\pi}{\lambda}x\right)}dx}{\lambda}
            
    .. math::
            \gamma = \frac{4a}{\lambda}
            
        The solution to the integral is:
            
        .. math::
            \phi = \frac{2E\left(\frac{-4a^2\pi^2}{\lambda^2}\right)}{\pi}
            
        where E is the complete elliptic integral of the second kind, 
        calculated with SciPy.
    
        Parameters
        ----------
        amplitude : float
            Half the height of the wave of the ridges, [m]
        wavelength : float
            Distance between the bottoms of two of the ridges (sometimes called 
            pitch), [m]
    
        Returns
        -------
        plate_enlargement_factor : float
            The extra surface area multiplier as compared to a flat plate
            caused the corrugations, [-]
    
        Notes
        -----
        This is the exact analytical integral, obtained via Mathematica, Maple,
        and quite a bit of trial and error. It is confirmed via numerical 
        integration. The expression normally given is an
        approximation as follows:
            
        .. math::
            \phi = \frac{1}{6}\left(1+\sqrt{1+A^2} + 4\sqrt{1+A^2/2}\right)
            
            A = \frac{2\pi a}{\lambda}
            
        Most plate heat exchangers approximate a sinusoidal geometry only.
    
        Examples
        --------
        >>> PlateExchanger.plate_enlargement_factor_analytical(amplitude=5E-4, wavelength=3.7E-3)
        1.1611862034509677