JWKB representation for equations with infinite order turning point (Q1374360)

The author considers the linear differential equation \[ u''- a(t)\xi^2u+ b(t)\xi u=0\tag{1} \] depending on the large parameter \(\xi\). Here \(J=[0,T]\) is a compact interval, \(a,b\in C^\infty(J)\) and \(t=0\) is a single turning point of infinite order for (1), that is \(a^{(n)}(0)= 0\), \(n= 0,1,2,\dots\). Linear independent solutions of (1) are constructed using representation by means of complex-valued phase functions and amplitudes. Conditions on the coefficients of (1) are proposed that lead to amplitudes with exponential growth as the parameter \(\xi\) increases.