Mathematical Methods for Physics and Engineering - Matematica.NET

APPLICATIONS OF COMPLEX VARIABLESIm sLRe sλFigure 25.6 The integration path of the inverse Laplace transform is alongthe infinite line L. Thequantityλ must be positive and large enough for allpoles of the integrand to lie to the left of L.25.5 Inverse Laplace transformAs a further example of the use of contour integration we now discuss a methodwhereby the process of Laplace transformation, discussed in chapter 13, can beinverted.It will be recalled that the Laplace transform ¯f(s) ofafunctionf(x), x ≥ 0, isgiven by¯f(s) =∫ ∞0e −sx f(x) dx, Re s>s 0 . (25.24)In chapter 13, functions f(x) were deduced from the transforms by means of aprepared dictionary. However, an explicit formula for an unknown inverse maybe written in the form of an integral. It is known as the Bromwich integral and isgiven byf(x) = 12πi∫ λ+i∞λ−i∞e sx¯f(s) ds, λ > 0, (25.25)where s is treated as a complex variable and the integration is along the line Lindicated in figure 25.6. The position of the line is dictated by the requirementsthat λ is positive and that all singularities of ¯f(s) lie to the left of the line.That (25.25) really is the unique inverse of (25.24) is difficult to show for generalfunctions and transforms, but the following verification should at least make it884