On an initial value problem in the theory of two-dimensional transonic flow patterns
Stefan Bergman · Pacific Journal of Mathematics · 1970
In the case of the differential equation where N is an analytic function, the integral operator of the first kind = Γ E(λ, θ, t)f(ζ(l -Jί = ~l transforms analytic functions of a complex variable ζ = λ -f id into solutions of L(^) = 0.Here E is a fixed function which depends only on L, while f(ζ) is an arbitrary analytic function of the complex variable ζ; f is assumed to be regular at ζ = 0. Using this operator, one shows that many theorems valid for analytic functions of the complex variable can be generalized for the solutions ψ of h(ψ) = 0. Continuing ψ(2., β) to complex values U = λ + %Λ and setting λ = 0, one shows that many theorems in the theorems in the theory of functions of a real variable can be generalized to the case of solutions of By change of the variables, l(x) > 0 for x 0,1(0) = 0, when considered for x < 0 can be reduced to the equation L(^) = 0.The variables can be chosen so that U = 0 corresponds to x -0.However, in this case the function N(λ) becomes singular at λ = 0. Nevertheless, one can apply the theory of the so-called integral operators of the second kind.If ^(0, θ) = X x (d) and lim ψ M (M, θ) -X 2 (0) are given, one can determine the function /.Here M is the Mach number.In this way one can determine from Xι and % 2 the location and character of singularities of φ in the subsonic region.When considering φ in the supersonic region, one can show that some theorems on functions of one real variable can be generalized to the case of certain sets of particular solutions φXΛ, θ), v = 1, 2, , of H(^) = 0.