Mathematical Methods for Physics and Engineering - Matematica.NET

APPLICATIONS OF COMPLEX VARIABLESfrom which it follows that C(∞) =S(∞) = 1 2 . Clearly, C(−∞) =S(−∞) =− 1 2 .We are now in a position to examine these two equivalent ways of evaluating I 0in∫terms of sums of infinitesmal vectors in the complex plane. When the integral∞−∞ exp(−z2 ) dz is evaluated as a real integral, or a complex one along the realz-axis, each element dz generates a vector of length exp(−z 2 ) dz in an Arganddiagram, usually called the amplitude–phase diagram for the integral. For thisintegration, whilst all vector contributions lie along the real axis, they do differ inmagnitude, starting vanishingly small, growing to a maximum length of 1 × dz,and then reducing until they are again vanishingly small. At any stage, theirvector sum (in this case, the same as their algebraic sum) is a measure of theindefinite integral∫ xI(x) = exp(−z 2 ) dz. (25.70)−∞The total length of the vector sum when x →∞is, of course, √ π, and it shouldnot be overlooked that the sum is a vector parallel to (actually coinciding with)the real axis in the amplitude–phase diagram. Formally this indicates that theintegral is real. This ‘ordinary’ view of evaluating the integral generates the sameamplitude–phase diagram as does the method of steepest descents. This is becausefor this particular integrand the l.s.d. never leaves the real axis.Now consider the same integral evaluated using the form of equation (25.69).Here, each contribution, as the integration variable goes from u to u + du, isofthe formg(u) du =cos( 1 2 πu2 ) du + i sin( 1 2 πu2 ) du.As infinitesimal vectors in the amplitude–phase diagram, all g(u) du have the samemagnitude du, but their directions change continuously. Near u =0,whereu 2is small, the change is slow and each vector element is approximately equal to√2π exp(−iπ/4) du; these contributions are all in phase and add up to a significantvector contribution in the direction θ = −π/4. This is illustrated by the centralpart of the curve in part (b) of figure 25.13, in which the amplitude–phase diagramfor the ‘ordinary’ integration, discussed above, is drawn as part (a).Part (b) of the figure also shows that the vector representing the indefiniteintegral (25.70) initially (s large and negative) spirals out, in a clockwise sense,from around the point 0 + i0 in the amplitude–phase diagram and ultimately (slarge and positive) spirals in, in an anticlockwise direction, to the point √ π + i0.The total curve is called a Cornu spiral. In physical applications, such as thediffraction of light at a straight edge, the relevant limits of integration aretypically −∞ and some finite value x. Then, as can be seen, the resulting vectorsum is complex in general, with its magnitude (the distance from 0 + i0 tothepoint on the spiral corresponding to z = x) growing steadily for x0.914