APPLICATIONS OF COMPLEX VARIABLES
from which it follows thatC(∞)=S(∞)=^12. Clearly,C(−∞)=S(−∞)=−^12.
We are now in a position to examine these two equivalent ways of evaluatingI 0in terms of sums of infinitesmal vectors in the complex plane. When the integral∫
∞
−∞exp(−z
(^2) )dzis evaluated as a real integral, or a complex one along the real
z-axis, each elementdzgenerates a vector of length exp(−z^2 )dzin an Argand
diagram, usually called theamplitude–phase diagramfor the integral. For this
integration, whilst all vector contributions lie along the real axis, they do differ in
magnitude, starting vanishingly small, growing to a maximum length of 1×dz,
and then reducing until they are again vanishingly small. At any stage, their
vector sum (in this case, the same as their algebraic sum) is a measure of the
indefinite integral
I(x)=
∫x
−∞
exp(−z^2 )dz. (25.70)
The total length of the vector sum whenx→∞is, of course,
√
π, and it should
not 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 the
integral is real. This ‘ordinary’ view of evaluating the integral generates the same
amplitude–phase diagram as does the method of steepest descents. This is because
for 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 fromutou+du,isof
the form
g(u)du=cos(^12 πu^2 )du+isin(^12 πu^2 )du.
As infinitesimal vectors in the amplitude–phase diagram,allg(u)duhave thesame
magnitudedu, but their directions change continuously. Nearu=0,whereu^2
is small, the change is slow and each vector element is approximately equal to√
2 πexp(−iπ/4)du; these contributions are all in phase and add up to a significant
vector contribution in the directionθ=−π/4. This is illustrated by the central
part of the curve in part (b) of figure 25.13, in which the amplitude–phase diagram
for the ‘ordinary’ integration, discussed above, is drawn as part (a).
Part (b) of the figure also shows that the vector representing the indefinite
integral (25.70) initially (slarge and negative) spirals out, in a clockwise sense,
from around the point 0 +i0 in the amplitude–phase diagram and ultimately (s
large 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 the
diffraction of light at a straight edge, the relevant limits of integration are
typically−∞and some finite valuex. Then, as can be seen, the resulting vector
sum is complex in general, with its magnitude (the distance from 0 +i0tothe
point on the spiral corresponding toz=x) growing steadily forx<0 but
showing oscillations whenx>0.