Advanced Mathematical Methods for Scientists and Engineers

a₁ a₂ Airy function an+1 analysis analytic approximation to y(x asymptotic analysis asymptotic expansion asymptotic matching asymptotic relation asymptotic series B₁ behavior of solutions behavior of y(x Bessel boundary conditions boundary layer boundary-value problem branch points c₁ c₂ coefficients compute constant contour critical point derive determine difference equation differential equation exact solution Example exponentially finite first-order formula Frobenius series full asymptotic gives initial conditions initial-value problem inner integral representation irregular singular point Laplace's method leading behavior leading-order linear local analysis nonlinear obtain optimal asymptotic approximation outer solution Padé approximants Padé sequence perturbation series perturbation theory plane polynomial power series Prob radius of convergence region result saddle point satisfies second-order Show solution y(x solve steepest-descent Substituting Taylor series tion trajectories valid values WKB approximation WKB theory y₁ Yo(X