*Michael
Fowler*

Suppose we have a complex function *f* = *u* + *iv*
of a complex variable *z* = *x* + *iy*, defined in some region
of the complex plane, where *u*, *v*, *x*, *y* are real.
That is to say,

_{}

with *u*(*x*,*y*) and *v*(*x*,*y*)
real functions in the plane.

We now assert that in this region *f*(*z*) is
differentiable, that is to say,

_{}

is well-defined. What does this tell us about the functions *u*(*x*,*y*)
and *v*(*x*,*y*), the real and imaginary parts of *f*(*z*)?

In fact, the property of differentiability for a function of
a complex variable tells us a lot! It does *not* just mean that the
function is reasonably smooth. The
crucial difference from a function of a real variable is that *D _{
}z* can approach zero

_{}

which
we can write in terms of *u*,*v*:

_{}

Equating real and imaginary parts of this equation we find:

_{}

These are called the *Cauchy-Riemann equations*.

It immediately follows that both *u*(*x*,*y*)
and *v*(*x*,*y*) must satisfy the two-dimensional Laplacian
equation,

_{}

Notice that this implies (just as for an electrostatic
potential) that *u*(*x*,*y*) cannot have an absolute minimum or
maximum inside the region of analyticity. If *df*(*z*)/*dz* = 0,
but the second-order partial derivatives are nonzero, then they must have
opposite sign, signaling a saddlepoint. In the general case, a two-dimensional
version of Gauss’ theorem can be used to show there is no local extremum.

Furthermore,

_{}

That is to say, the *contour lines of constant u(x,y) are
everywhere orthogonal to the contour lines of constant v(x,y)*. (The gradient being orthogonal to the
contour lines everywhere.)

The important point is that *just requiring
differentiability *of a function of a complex variable imposes a *strong*
constraint on its real and imaginary parts, the functions *u*(*x*,*y*)
and *v*(*x*,*y*).

It is worthwhile building a clear picture of the real and
imaginary parts of the function *z*^{2}. The real part is *x*^{2} - *y*^{2}, and
its contour lines in the square -1 to 1 are shown below. We have chosen geographic
coloring: green for valleys, brown for hills. At the origin, there is a
saddlepoint with higher ground in both directions of the real axis, lower
ground in the pure imaginary directions. The lines *x* = *y*, *x*
= -*y*
(not shown) are contours at the same level (zero) as the origin.

What about the imaginary part? Im*z*^{2} = 2*xy*
has contours:

Putting the two sets of contour lines on the same diagram it is clear that they always cut each other orthogonally:

(Incidentally, this picture has a physical realization. It represents the field lines and equipotentials of a quadrupole magnet, used for focusing beams of charged particles.)

Suppose we have a function *f*(*z*) analytic in
some region *R* of the complex plane, and at some point *z*_{0}
inside *R* the derivative *df*(*z*)/*dz* = 0. Then in the
neighborhood of *z*_{0},

_{}

Close enough to *z*_{0} we can neglect the
higher order terms, and for the case of *f *¢¢(*z*_{0})
real, the contour lines of the real and imaginary parts of *f*(*z*)
will then be exactly those we have plotted for *z*^{2} above. For *f *¢¢(*z*_{0})
complex, the plots will be rotated by an angle equal to the phase of *f *¢¢(*z*_{0}).

That is to say, for any analytic function, near any point
where *df*(*z*)/*dz* = 0, the real and imaginary parts of the function
have saddlepoints with contour maps rotated versions of those above.

We consider now integrals of the form

_{}

where *C* is some path in a region where *f*(*z*)
is analytic. This means the value of
the integral will not be affected by distorting the path, provided it stays in
the region of analyticity. (The path of
integration is usually called the *contour* of integration—we’ll call it
path here, to avoid confusion with *our* contours, which have the standard
geographic meaning, joining points having the same value of some parameter.)

Note that with the exponential form of the integrand, the *real
*part of *f*(*z*) determines the *magnitude* of the
integrand, the *imaginary* part of *f*(*z*) determines its *phase*.

The strategy is to arrange the path of integration so that
as much as possible of it is in the valleys, where the integrand is small, then
to go over the saddlepoint by the steepest possible route, which would be
staying on the imaginary axis in the case of *z*^{2} plotted
above. It is important to note that this “steepest descent” route is also a
path along which *the imaginary part of f*(*z*)* remains constant*,
so the contributions along this path are *all in phase*, that is to say,
they add coherently.

The bottom line is that by directing the path of integration
through the saddlepoint along the steepest route for the magnitude of the
integrand, the biggest contributions to the integral are all in phase. Along this path, the integral has standard
Gaussian form. If the function *f*(*z*)
is sufficiently large, it may be that the contribution of the integral away
from the saddlepoint can be neglected.
This method is therefore often valuable in cases where some parameter
becomes large: we give a number of examples to clarify this point.

We use the identity

_{}

To picture *t ^{n}e^{-t}*, here it is
for

Note that

_{}

Therefore, in the neighborhood of the maximum value of *f*(*t*)
at *t* = *n*,

_{}

For integer *n*,
the function is analytic in any finite region of the complex plane. Taking *n*
= 10, as in the real-axis graph above, and plotting the contours of Re(*t ^{n}e^{-t}*)
in the neighborhood of

It is clear that the integral along the real axis is in fact a steepest descent path. The reason we look at this straightforward case is to gain some experience about when it is reasonable to throw away all the contribution to the integral except that near the saddlepoint. If we simply take

_{}

and take the *t* integration to be over the whole real
axis, not just positive *t*, it is a Gaussian integral and

_{}

More precise, and considerably more complicated, methods
give the leading correction to this expression. It is down by a factor of 1/12*n*, so the naïve Gaussian
saddlepoint result is accurate within 1% for *n* = 10, and improves as *n*
increases.

Recall that the delta function can be defined by the limit of a Gaussian integral

_{}.

It is easy to see how this leads to

_{}

for an integral along the real axis with a function *f*(*x*)
reasonably well-behaved near the origin.
Shankar mentions that the definition also works even if *D*^{2}
is replaced by *i**D*^{2}.
In that case, the absolute value of the function is the same everywhere
on the real axis, and increases as *D*^{-1}
on taking *D
*small. The reason it still works is
that the phase oscillations are so rapid everywhere except at the origin, where
the phase is momentarily stationary, so all the contribution comes from
there.

However, it is easier to believe

_{}

on going into the complex plane. If we change variables from *x* to *x*,
where *x*^{2} = *i**x *^{2}, the
integral again becomes a simple real Gaussian. But, regarding *x* as a
complex variable, transforming to *x* is just equivalent to
rotating the axes by *p*/4, or multiplying by the square root of *i*. The steepest descent route through the
origin is now along the line at p/4 to the real axis.
So this is a perfectly good definition of the *d*-function provided we
can distort the path of integration from the real axis to the line *x* = *y*. (Strictly speaking, the path would now
include two octants of a circle at very large *R*—their contribution
vanishes in the limit of *R* going to infinity.)

The WKB semiclassical approximate solution to Schrödinger’s equation,

_{}

is reliable in regions where the wavelength (for oscillating solutions) or the decay length (for exponential solutions) changes only slightly over a distance of one wavelength or decay length respectively. For a particle trapped in a (one-dimensional) potential well, classically the particle would bounce back and forth between the two turning points where its kinetic energy vanishes. In the quantum case, these are precisely the points where the wavelength becomes infinite, so the WKB solution fails.

However, for a reasonably smooth potential it may be an adequate approximation to treat a turning point region as one where the potential is increasing linearly with distance over a sufficient range that beyond this point the WKB approximation can be used in both directions.

The solution of Schrödinger’s equation for a linearly increasing or decreasing potential is well known, it is the Airy function, the solution of the differential equation

_{}

plotted here at the left-hand turning point :

The strategy is to evaluate this function for large *x*,
both positive and negative, so that we can join together the two WKB solutions,
valid in the far regions, in a quantitative fashion.

Following Mathews and Walker (page 116) the differential equation is most simply solved by taking its Fourier transform.

If

_{}

then

_{}

Therefore

_{}

This is an exact result, but a nontrivial integral! Fortunately, we are only interested in its
value for *large* values of |*x*|,
and this is precisely where saddlepoint methods become accurate.

To see where the saddlepoints are, and how they relate to an integral along the real axis, we plot contour maps of the term in the exponential.

**Large positive x:
**In the map below, we take

Writing the integral as

_{}

(dropping irrelevant overall constants) then

_{}

(Mathews and Walker take the “large” parameter *x* out
of *f*, we’ve left it in—this doesn’t affect the final result.)

Near the positive saddlepoint

_{}

dropping higher-order terms. Since *f* ¢¢
is pure imaginary at the saddlepoint, the appropriate path for a real exponent
in the Gaussian integral is at *p*/4 to the *x*-axis. So in the path integral

_{}

where *ds* is a real parameter measuring incremental
path length, and the sign in the exponent is positive for the saddlepoint on
the left. The contributions from the
two saddlepoints give the asymptotic (large positive *x*) solutions as:

_{}

**Large negative x:** In this case, the
saddlepoint geography is quite different, although the distant geography is the
same, being dominated by the

Just as before, the real axis integration path can be moved down from the real axis into the valleys at bottom left and bottom right. (The extra contributions from linking up the new path with the real axis at infinity are zero.) It is clear from the map above that to get from the valley on the left to the one on the right means just going over the saddlepoint on the negative imaginary axis. Note from the darker shading that the other saddlepoint is at higher elevation.

The integration through the saddlepoint is parallel to the real axis, and gives

_{}

This, then, is the decaying wavefunction solution of the
Airy equation that we are looking for, and it is clear that it goes smoothly
from the exponential to the cosine form as *x* is taken along the real
axis from large negative to large positive values.