Deformation of the Path of Integration of the Complex Inversion Integral

Abstract

The L-transform represents an analytic function. Thus we may, by Cauchy’s theorem, alter the straight line path of integration of the complex inversion integral in a certain manner. The last example of Chapter 24 served to demonstrate such an alteration. For practical applications, one frequently employs the following modification: The vertical line is shifted to the left until the first singular point s₀ of F(s) is met. Then, the vertical line is replaced near s₀ by an arc of a circle to the right, at the same time inclining the remaining straight sections towards the left as shown in Fig. 15. The newly created path of integration offers favourable conditions of convergence for the integral, for the factor e^ts converges, for t > 0, rapidly towards zero along the straight line sections biased towards the left; whereas this factor oscillates between finite limits along the vertical lines.