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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def _func(a, p, r, v):
 """ calculates f-hat for the coefficients in a, probability p, sample mean difference r, and degrees of freedom v. """

    # eq. 2.3
    f = a[0]*math.log(r-1.) + \
        a[1]*math.log(r-1.)**2 + \
        a[2]*math.log(r-1.)**3 + \
        a[3]*math.log(r-1.)**4

    # eq. 2.7 and 2.8 corrections
    if r == 3:
        f += -0.002 / (1. + 12. * _phi(p)**2)

        if v <= 4.364:
            f += 1./517. - 1./(312.*(v,1e38)[np.isinf(v)])
        else:
            f += 1./(191.*(v,1e38)[np.isinf(v)])

    return -f