A plane wave solution is also an eigenfunction of the translation operator:
Which linear combination of plane waves (having the same ) is an eigenfunction of ?
We need a solution to the Helmholtz equation of the form
then we shall have what we are looking for, namely a solution which is also an eigenfunction of the rotation operator.
Using the polar representation of , and cancelling out the factor , we have
or with ,
In other words, must satisfy Bessel's equation.
The first impulse is to solve this equation using infinite series. However, we shall take note of STOKE'S observation: ``series solutions have the advantage of being generally applicable, but are wholly devoid of elegance''. In our case ``elegance'' means ability to capture the geometric and physical properties of the Euclidean plane.
Instead of a series solution, we shall take the question on the previous page seriously and construct an appropriate superposition of plane wave solutions with a direction-dependent phase shift that varies linearly ( ) from one plane wave to the next. Such a phase shift is expressed by the phase factor
where is a constant. The superposition is therefore given by
This superposition has the desired form
provided the effect of the -dependence in the integration limits can be removed. In other words, expression (5.2), which is a solution of
is an eigenfunction of if
can be shown to be independent of . In that case , and it necessarily satisfies
which is Bessel's equation, with equal to any complex constant.
Let us, therefore, consider more closely the complex line integral
Here we assume, for the time being, that because
a product of two positive numbers. The integration contour is a curve in the complex -plane, whose points are
We shall find that the chosen integration contour will start far away from the origin at a point with large positive or negative imaginary part, , and terminate at another such point, again with or . This choice has a dual purpose. (i) It guarantees, as we shall see, that the contour integral will be independent of the real angle , which is the amount by which the two end points get shifted horizontally in the complex -plane, and (ii) it guarantees, as we shall see, that the integral converges. The value of the integral itself is independent of the integration path because the integrand is analytic in the whole complex -plane.
Where shall the starting and termination points of the contour integral be
located? This question is answered by the asymptotic behaviour of the
dominant terms in the exponent of the integrand,
To obtain an integral which converges, one must have at both endpoints. This implies that if , then the value of must satisfy
On the other hand, if , then the value of must satisfy
Thus the integration contour can start and terminate only in one of the shaded regions in the complex -plane of Figure 5.3.
There are only two basic contour integrals that one needs to consider, and they give rise to the two kinds of fundamental functions. They are , the Hankel function of the first kind , and , the Hankel function of the second kind. All other integration contours give rise to contour integrals which merely are linear combinations of these two fundamental functions.
Moving forward, we shall use in the next subsection these two functions to deduce of their mathematical wave mechanical properties and applications.
along the curve (in the complex -plane below) in terms of the two kinds of Hankel functions and
converges if the integration limits of the integration path are extended to infinity in each of a pair of such strips.
can be expressed in terms of a Hankel function. Which kind and which order?
is a solution to the wave Eq.(5.3) whenever the two constants (``frequency'') and (``wave number'') satisfy the dispersion relation
Then, using , (with ) and , and the hyperbolic angle addition formula, rewrite the phase and hence the wave function in terms of and .
where is that superposition.