General analytical formulas describing the wave pattern in the linear regime of an electromagnetic wave impinging vertically upon a horizontally stratified ionosphere, where the effective refractive index squared has an infinity, a first-order pole, are derived within the uniform approximation employed in wave propagation problems in previous papers by the author and B. Thidé. A great advantage of this approximation is that the formulas are valid also at the reflection and transition points of the problem. Particular attention is paid to the case of monotonically increasing electron density with height in the relevant region of the ionosphere. Important applications of the formulas derived are the X mode wave when the wave frequency is higher than the electron gyrofrequency and the 0 mode wave when the wave frequency is lower than the electron gyrofrequency. The type of wave equation studied here may occur also in other wave propagation problems.