Approximate solutions of functional equations with small delay using the WKB method

A class of functional equations containing delay terms in which the delay is small compared with an overall scale, or difference equations with slowly varying coefficients, is analysed using the WKB method. A particular example in which a periodic solution is required is analysed in detail. Results analogous to quantization conditions are discussed as are turning points. The approximate solutions obtained by the WKB method are compared with exact solutions where available in simple cases and with numerical integration and excellent agreement is obtained.