tokens: transpose. We use factorization with the matrix (see Silant’ev et al. 2017a ): Here , . The matrices and have the forms It is useful to introduce the vector K ( τ ): The matrix product of matrices and is denoted as . It is easy to verify that equation ( 7 ) gives rise to the conservation law of radiative flux. 3 THE INTEGRAL EQUATION FOR <bold><italic>K</italic></bold>(<italic>τ</italic>) The formal solution of equation ( 7 ) gives the integral dependence of I 0 ( μ, τ ) on K ( τ ). Substitution of this solution in equation ( 10 ) gives a homogeneous integral equation for K ( τ ): We are interested in the non-zero solution of this equation. The kernel has the form The matrices and have the forms Note that , and . The general theory to calculate the vector K ( τ ) was presented in Silant’ev, Alekseeva Novikov ( 2015 ). Recall that, according to this theory, the vector K ( τ ) has a solution through the resolvent matrix : Here and in what follows we omit the subscription ‘hom’ . The resolvent matrix obeys the equation The kernel of the equation for is symmetric: . This gives rise to the relation . The double Laplace transform of R ( τ, τ ′) over parameters a and b is equal to This property demonstrates that the matrix can be calculated, if we know the matrices and (see Silant’ev et al. 2015 ). For this reason, it is useful to study the equation for , which follows from equation ( 15 ): Using the particular form of and equation ( 16 ), we obtain the following relation: Here is the unit matrix. Hereafter in our theory, we use a = μ and a = μ /(1 + h ). Introducing the new values we obtain the following equations: Note that the H-functions E ( μ ) and K ( μ ) can be obtained from the algebraic system of equations, if we know the other functions – A ( μ ), C ( μ ), D ( μ ), G ( μ ), M ( μ ) and N ( μ ). These functions do not occur in the Milne problem. 4 FORMULAE FOR THE MILNE PROBLEM The specific feature of the Milne problem is that we need to solve an integral equation (see equation 11 ) for K ( τ ), which does not have a free term. Using the vector K ( τ ), one can obtain the solution of transfer equation ( 7 ). The vector I 0 (0, μ ) describes the emerging radiation. This vector has the form i.e. this expression depends on the Laplace transforms of K ( τ ) over variable τ . From equation ( 22 ), we obtain the particular formulae Further, we follow the simple approach of Sobolev ( 1969 ), generalizing his method for the vector case and magnetized atmosphere. The final formulae depend on K (0). Taking τ = 0 in equation ( 11 ), we obtain Now we find the relations and with the matrices and . Let us obtain the equation for derivative d K ( τ )/d τ , taking into account the fact that the kernel depends on the difference ( τ − τ ′): Equation ( 26 ) is a non-homogeneous integral equation. According to the general theory of integral equations (see Tricomi 1957 ; Smirnov 1964 ), the general solution of equation ( 26 ) consists of the sum of two terms –
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