Indefinite and Definite Integrals Involving Parabolic Functions and Some Infinite Series

Abstract

If the two functions F ₁(z) and F ₂(z) satisfy the differential equations

$$\begin{array}{*{20}c} \hfill {\frac{{d^2 F_1 }} {{dz^2 }} + Q_1 \left( z \right) \cdot F_1 \left( z \right) = 0} & \hfill {\left( {1a} \right)} & \hfill {\frac{{d^2 F_2 }} {{dz^2 }} + Q_2 \left( z \right) \cdot F_2 \left( z \right) = 0} & \hfill {\left( {1b} \right)} \\ \end{array}$$

(1b)

then multiplication of the first and second equation by F ₂(z) and F ₁(z) respectively, followed by subtraction and subsequent integration leads to

$$\int\limits_{}^z {\left\{ {Q_1 \left( z \right) - Q_2 \left( z \right)} \right\} \cdot F_1 \left( z \right) \cdot F_2 \left( z \right) \cdot dz = F_1 \left( z \right) \cdot \frac{{dF_2 \left( z \right)}} {{dz}} - F_2 \left( z \right) \cdot \frac{{dF_1 \left( z \right)}} {{dz}}} .$$

(2)