4.21.3 Half-integer and third-integer subscripts: With

the series (19.34) becomes

similarly, one finds for n = -1/2

Hence, in general, we set

especially

Bessel's differential equation then yields easily the differential equation for yn

We will meet this equation again in the theory of spherical functions. We will show here that its finite solution for r = 0 can be obtained by the following process from y0:

It is best to start the proof from (19.34). If p is an arbitrary subscript (in the present case p = 1/2), then

Differentiate this equation n-times with respect to r ²/2 and obtain on the right hand side

Just renaming of the summation index (m = m - n, m = n + m) yields

On the other hand, during the same differentiation (due to d(r ²/2) = r dr), the left hand side becomes

Comparing (13a,b), one obtains the general relation

for p = 1/2, it is, due to (10), identical to (12).

If you change (13) so that one does not differentiate the quotient, but the product of In(r) and (r²/2)p n times with respect to (r ²/2), one finds, instead of (13c),

If one now sets again p = 1/2 and uses the definition (11), one obtains for (12) the extension

Equations (12) and (13) yield readily the recursion formulae for

Corresponding recursion formulae for have been discussed in 4.19.4.

Using (14), one readily computes successively from y sin r/r

and concludes that one can express all for integer n in an elementary form in terms of sin and cos. This representation confirms at the same time the earlier note about the non-logarithmic character of the half-integer Bessel functions.

We define the Hankel function zn, corresponding to yn, by the equation, analogous to (11),

(you must insert here in your mind the superscripts 1 and 2). We are especially interested in z0, which follow from H½ which itself is computed from (19.31) and (19.30) as well as (11):

Hence, by (15),

The following remarks relate to the notation: Our notation agrees with the original definition of yn in Heine's Handbook of Spherical Functions and the symbols used in Frank-Mises. They differ by a factor r from those used by other authors who write instead of (11)

which at times is convenient.

P.Debye, Ann. Physik, 30 (1909), B. van der Pol and H. Bremmer, Phil.Mag. 24 (1937). Further proofs: G.N.Watson Theory of Bessel Functions, p. 56. The function yn defined in (16) is denoted in technical acoustics by Sn; the function Cn also used there would in the present terminology be called a Neumann Function; it is proportional to zm1- zm2 .

Corresponding to Equations (14a) for yn, one can express the zn in elementary form by eir). This tells the same thing as the fact, derived in 4.21.4, that the Hankel asymptotic expansions stop in the case of half-integer subscripts.

A strange, elegant form assumes the differential equation for Z±½(kr) if one replaces there the dependent and independent variables by

The direct calculation would lead to extensive transformations. You avoid them and see simultaneously the capability of the relations (17) - (20) of being transformed, if one starts from the conform transformation

by which

with becomes

cf. (23.17). In particular, if one sets in (2) f proportional to a power of x + ih, for example, if one sets

then

The solution of (1)

then becomes that of (2)

whence

If one still sets then one sees after simple manipulations that

with the two readily verified series

They can be interrelated, due to the connection between F and v to the two in (4) combined solutions of v; interpreting as constant factors, one must have:

If one substitutes here the power series for I from (19.34), the comparison with (6) yields

For m = 3, Equations (5), (6), (7), (7a) become Equations (18), (19), (20) of the main text.

For m = 2, (5) specializes itself to the differential equation of the trigonometric functions and F0,F1 to cos r, sinr. For m=1, this series representation fails, because r = 0 is a singular point of the differential equation (5).

(Continued from above footnote) The differential equation then becomes with the substitution (17)

If one sets its integrals equal to series with undetermined coefficients, starting with r 0 and r 1, one finds directly

Equation (19.34) confirms that the first is proportional to I, the second to I, i.e., that

We will again encounter the functions I±½ at the end of 4.21.4.

4.21.4 Generalization of the saddle point method according to Debye: Although we will only require in the later applications the asymptotic limit of the Bessel functions, stated in 4.19, we will still refer to certain extensions, first derived by Hankel, which advance with negative powers of r and the first term of which is that asymptotic limiting value. These series are actually divergent (as expansions at an essential, singular point), but are often called semi-convergent. At first, their terms decrease strongly, and then increase from a certain place rapidly to infinity; obviously, one must truncate at this location, in order to be able to employ them as approximation formulae.

They are obtained most readily from the differential equation for the Hankel functions, if we enter them in the relevant series form and compute the available coefficients as in the case of a convergent series, i.e., by setting to zero the arising factors of every individual power, which obviously is not quite strict. In connection with (19.55) and (19.56), we write

and find in (19.11), disregarding the common factors, the terms multiplied by r-m-3/2; we thus obtain readily

Here, successive rows correspond to the successive terms

of (19.11). The sum of all three lines, set equal to zero, yields the simple, indeed only two terms recursion formula

Hence, with a0 = 1,

If we employ Hankel's symbol

where, however, (n, 0) is to equal 1, we arrive, in general, at

Hence, the series for H1 and H2, combined in (22), assume the final form

Moreover, half their sum yields still

In Exercise 4.5, we will employ a corresponding method to derive the leading term (here taken over from the saddle point method) of the series (27), (28) from Bessel's differential equation, truncated for large r, however, without the known normalization factor, which naturally is not determined by the differential equation.

Regarding the range of validity of such asymptotic series in the complex regime, there exist since the enormous effort of Henri Poincaré (1854 - 1012, acta Math. Vol. 8 1886) extensive mathematical investigations, which naturally we cannot discuss here. An exception from the divergence occurs with the series for half-integer subscripts n = n + 1/2, which, according to the definition of the symbol (m, n), truncate at the n-th term and represent exactly the corresponding Bessel function. One obtains in this manner the elementary expressions for zn, yn.

However, our work so far is essentially limited by the demand it fails when n goes simultaneously with r to infinity. This is the case in all optical problems, which are on the border between geometrical optics (optics of very small wave lengths) and actual wave optics. It was during the study of a problem of this kind, namely, that of the rainbow (radius of rain drops approximately equal to the wave length of the light), that Peter Joseph Wilhelm Debye (1884 - 1966), Math. Ann., Vol. 67, 1909, Bayr. Akad. 1910) discovered his fundamental extensions of the Hankel asymptotic series. In order to understand their origin, we must generalize the saddle point method.

The exponent in the representation (19.22) of the Hankel functions

now depends on the two large numbers r and n. For the sake of brevity, we assume here that r and at first also n is real positive and set, depending on whether n is larger or smaller than r,

moreover, as in Fig. 18, we employ as integration variable

a) Then, for n < r,

The saddle point F'(b) = 0 is given by

it lies for H1 and H2, respectively, at corresponding to the former values w0 = 0 and p, respectively, which, by (30c), become . Equation (31) now yields

and for F(b) the expansion which is truncated with the quadratic term

Introduce instead of b the arc length s, measured from the saddle point , and set

where, due to the sign in the last equation, we direct the reader's attention to the corresponding discussion of signs in (19.54b). The integration over the neighbourhood of the corresponding saddle point then yields

It can again be reduced to Laplace's integral and then becomes

In the limit , Equation (32) becomes the earlier representation (19.55), (19.56).

b) The same manipulation is also successful in the case n > r, where one replaces, following (30b), cos a by cosh a and therefore (31) by

Among the two saddle points that one is decisive, which has the higher placed passage, i.e., b=-ia. For this one, F"(b0) = r sinh a. Thus, whence one find instead of (32)

Debye has added to the limits (32), (33) series expansions of the Hankel kind, but we will here pass them by.

c) There remains only the transition case n ~ r in which, by (30a,b), a ~ 0 and the representations (32), (33) fail due to the denominator respectively. This fact suggests that also F"(b0), from which arose the denominator, is approaching zero and that only the third term of the Taylor expansion for F(b) has a value appreciably other than 0. Hence, there arises the need for a more exact approximation of the neighbourhood of the saddle point, which was achieved best by Watson (Chapter VIII, p. 252, Theory of Bessel Functions, Cambridge, 1922), who employed for the preceding calculations instead of (31a) an expansion which is performed up to the third order term in (b - b0). The integrals of the Airy type (cf. below) occurring instead of Laplace's integrals can likewise be executed strictly. Thus, one finds: In the case

on the other hand, for n > r (n = r cosh a as in (30b):

Taken together, the Watson's formulae (34) and (35) cover the entire asymptotic range of the values of the Bessel-Hankel functions, including the limiting case n ~ r, with which we are dealing. In this case, we are, according to (30a) as well as to (30b), in the neighbourhood of a = 0. Hence, we can replace H1/3 by its limiting value for small argument, which, according to (19.31) and (19.30), is (I1/3 can here be neglected compared with I-1/3)

Due to z = (n/3)tan3a in (34), I-1/3 becomes here proportional to 1/tan a, which factor cancels the factor tan a on the right hand side; after required cancellations, Equation (34) yields

As it should be, one obtains the same expression as from (35). The corresponding limiting value of I is

It agrees with the initial results of Debye.

It is also easy to see that (34), (35), when n is not too close to r, agrees with Debye's formulae (32), (33). In that case, one can substitute for H½ (large argument and small subscript) Hankel's limit (19.55, 56):

whereby (34) simplifies to

Since n = r cos a, this result does agree with (32). In the same way, one shows that (35) and (33) agree.

Finally, we must still deal briefly with the problem of the roots of Hn1,2(r) = 0. However, since we so far have assumed that n and r are real, we must now admit arbitrary complex values of n; due to its physical meaning, we can also now assume that r is real. As regards the parameter a, the sign of which after its definition in (30a) is not defined, we will agree that its real part is to be positive.

Actually, it looks as if, according to the preceding equations (32), (33), there might not be roots for complex n, because the exponential function does not vanish for any finite value of the exponent. However, those representations arose, for example, in b) in that, of the two saddle points, only the one with higher pass height was selected. If both are of equal height and, moreover, the required integration passage (from lowland to lowland) can be led over both passes, then there arises as the sum of the two then available exponential expressions a trigonometric function which creates the possibility of zeroes.

Relating to the notations of Equations (30) and (32), we denote the two saddle points by and their corresponding exponential functions in H by

The heights of the respective passes is determined by the real part of the exponent. Thus, equal height means equal real parts of the two exponents or, since one can divide by r, assumed to be real, equality of the imaginary parts of ; this is equivalent to

Hence, especially for small a,

i.e., there are three branches through the origin, inclined to each other by the angle p/3, one of which coincides with the real a-axis. Also, when a is finite, the real axis remains a solution of (38), while the two other branches become two curves, reflected in the imaginary a-axis.

Corresponding to the basic lay-out of the integration path for the two Hankel-functions, shown in Fig. 19, this path can for H1 only then be placed consecutively with meaning over the two saddle points, if these lie on the branch of (38), which passes from the second to the fourth quadrant, i.e., if a has for a positive real part a negative imaginary part, whence n = r cos a (for real r) has a positive imaginary part. In contrast to this, the integration path for H2 must be placed from the third to the first quadrant so that a has a positive and n a negative imaginary part.

On the basis of these considerations, we obtain as superposition of the contributions of two saddle points, according to Equation (32), the representation

We determine at once from this equation the roots of H1n (r)= 0: They are the roots of the transcendental equation in a

where the negative sign on the right hand side must be chosen, so that a, as we had to demand, comes to lie in the fourth quadrant.

For small a, one obtains from (40) by expansion in a series

and, after correct choice of the third unit root,

Now, a is related to r by (30a), which, also for small a, with cos a = 1 - a²/2, yields

These roots of H1n(r) = 0 lie in the positive imaginary n-half-plane, which we will employ below, and there are an infinity of them. (Solved for r, Equation (41) yields values of the roots of r in the negative-imaginary r-half-plane). By (41), n and r are of equal order of magnitude, as we have assumed at the start, whence Equation (41) is the solution of the root problem under consideration.

Obviously, the saddle-point method is very general. It transfers from the representation of the Hankel functions to the evaluation of arbitrary integrals of the form

where the function F depends, apart from the integration variables w, on one or several large numbers r, n, ··· and the path W in the complex domain starts from a region Lim eF(w, ···) = 0 and leads to just such a region.The same peculiarities as in the case of the Hankel functions in the limiting situation n ~ r generally reappear in integrals of the type (42), if the saddle point F' = 0 moves towards a location F" = 0. This happens in Airy's refraction theory of the rainbow. Indeed, the phenomenon of the rainbow is linked to the appearance of a turning point (F" = 0) in the wave front, which during an asymptotic approach fuses with the saddle point F' = 0. The computation of the corresponding Airy integral leads then, as in (34) and (35), to the functions H1/3 or , what is the same, the functions I±1/3.

4.22 Spherical Functions and Potential Theory:

4.22.1 The Generating Function: The potential theory provides the simplest access to the theory of spherical functions. We begin with the so-called Newton potential 1/r and obtain by the displacement of the origin x = y = z = 0 to x0 , y0 , z0

The spherical polar co-ordinates r, , j used here have already been placed in such a manner that their polar axis = 0 passes through the point x0 , y0 , z0. Then x0 = y0 = 0, z0 = r0, according to the general relation

One can expand (1) in ascending or descending powers of r, depending on whether r<r0 or r>r0. We call Pn the coefficient belonging to the ascending or descending powers and have

The Pn must be the same in both expansions, since both must become each other for r = r0 and . The point r = r0, = 0 is singular; the sphere has here the same role as in two dimensions the convergence circle through the next pole of the Taylor series.

The Pn, defined by (3), are polynomials of degree n in cos . We call them spherical functions and will show below that they coincide with the polynomials Pn introduced in 1.5. The function 1/R is called the generating function of the spherical functions.

4.22.2 Differential and difference equations: We will find first the differential equation of the spherical functions. The basic equation Du = 0 of potential theory, which is satisfied by the function 1/R, we write in the form

Since it must apply for every term of the expansion (3), we obtain from the n-th term of the first line in (3) after extraction of the factor rn-2/r0n-1

the second line yields the same, but with the factor rn0/rn-3. We now replace the argument of Pn by z, writing

and noting that

Then (5) yields

or also

Imagine (6) written down for a second subscript l and form, according to the scheme of Green's theorem, by cross multiplication with P1 and P2

The physical range of values of the variable z ranges from z = -1 ( = p) to z = +1 ( = 0). If we integrate over this range, we find for the orthogonality condition

since the left hand side of the second line of (7), integrated with respect to z , vanishes, provided the P are not singular there, which according to their definition in (3) is not the case.

We will now show that our present Pn also satisfies our normalization condition (5.7) in that

Indeed, (1) yields for cos = ±1

Comparing this with (3), we arrive at the, somewhat more general than (9), equation

However, we have seen in 1.5 that the Pn there are completely determined by their orthogonality (8) and normalization (9). Thus, our present definition by the generating function leads to the same functions Pn as the method of least squares in 1.5. In particular, the former representation (1.5.8) also applies:

and consequently also, by (5.12),

The Pn are even or odd functions of z, depending of whether n is even or odd:

Our generating function yields therefore, apart from the differential equation for the variable z, also a difference equation for the index n. For example, we rewrite the first line of Equation (3) with the abbreviation a = r/r0:

whence by logarithmic differentiation with respect to a

and hence

If one now compares the factors of equal powers of a on both sides, for example, the factors of an , one finds

or, rearranged,

Naturally, the same recursion formula follows from the second row of (3).

On the other hand, Equation (11) yields by logarithmic differentiation with respect to z a mixed differential-difference equation. Instead of (11a), one obtains at first

and after reorganization and setting the factor of an+1 to zero

If one multiplies this equation by 2n + 1 and adds to it twice Equation (11b), after it has been differentiated with respect to z , one finds

Reorganized, this is the differential recursion relation

4.22.3 The Associate Spherical Functions: However, moreover, the potential equation invites to take into account, beside the particular solution depending on r and

the particular solution depending on r, and j

by introducing certain spherical functions Pmn, associated with Pn, where m is an integer which for the time being will be assumed to be positive. For their definition serves the differential equation, following from (4),

written analogously to (6) and (6a)

William Thomson (Lord Kelvin, 1824 - 1907) and Peter Guthrie Tait (1831 - 1901) called our initial Pn zonal and our associate ones tesseral spherical functions. The former subdivide by their zero lines the surface of the sphere into zones of latitude of different signs, the latter into quadrangles (tesseral) of different signs, bounded by circles of latitude and longitude. Also, these associate or tesseral spherical functions are in the case of different subscripts, but equal superscripts, orthogonal to each other; in fact, one concludes, as in the case of (7), from the presently valid differential equation (13a) that

In order to derive an analytic expression for the Pmn, we expand them at z = ±1 (North and South pole of the unit sphere) in powers of , corresponding to an approach analogous to (19.36)

Following (19.37), we obtain as equation for l from (13b)

(The other root l = -m/2 must be excluded for reasons of continuity.) We combine the corresponding root factor, occurring at z = ±1, in

and write

For v, introduced here, one obtains, by conversion from (13b), the differential equation

which now must admit solution merely by power series in . However, we need not examine their recursion relations, since we can obtain the required integral of (17) in closed form from (6a). In fact, differentiating (6a) m-times with respect to z and applying the known relation for repeated differentiation of a product, for example,

we arrive exactly at the differential expression {} in (17), but applied to the m-th derivative of Pn. Hence, we conclude that we obtain a possible solution of (17) by setting

Using (16) as well as (10), we thus obtain for our associate spherical function the simple representation, which simultaneously shall serve for fixing the hitherto open normalization factor:

Hence, Pmn, just as Pn, is a polynomial of degree n (at least for even m; for odd m, it has the form times a polynomial of degree n - 1. Moreover, we see from (18) that

The last statement follows obviously from the fact that for m > n the order of the differentiation in (18) is larger than the degree of the function, to be differentiated.

4.22.4 About the Associate Functions with Negative superscript m: Hitherto, we have assumed that the superscript m is positive, for example in (17b), where we performed an m-fold differentiation with respect to z . However, our final representation (18) can be extended without problems to negative m provided only that Hence, we can and will extend (18) to all 2n+1 values Also for negative m, Pmn is a polynomial of degree n (in the same sense as in the case of a positive m); in fact, its becoming infinite, which is indicated for negative m by the first factor (1-z²)m/2 at z = ±1, is removed by the second factor in (18), which vanishes, due to the by |m| reduced order of differentiation at = ±1 of corresponding higher order. Moreover, Equation (18) also satisfies for negative m the differential equation (which only depends on m²). Consequently, Pmn can differ with negative m from P|m|n only by a constant factor C, which most conveniently is determined by a comparison of the highest power of z in the expressions, obtained according to (18) from P-mn and P+mn . Thus, one finds

i.e.,

This equation is seen to be generally valid, i.e., for positive as well as for negative values of m.

Our general definition of Pmn, which differs from that in the older literature, has proved itself in wave mechanics and also serves in the sequel to simplify writing and explanations.

In the older literature, the superscript of Pmn is always viewed to be positive and the dependence on j is given by cos mj or sin mj. It is much simpler to start with exponentials, as we have done in (12a), where, however, the restriction to positive m vanishes.

More strongly than the present definition deviates from the established one that was recommended by C.G. Darwin (Proc. Roy. Soc. London 115, 1927), in which the factor (n-m)! has been added in the numerator of (18) . Equation (18b) then simplifies to

However, this definition has the consequence that then also the classical expression for the Legendre functions Pn = P0n is changed, which we want to avoid.

Note also that some authors, especially E.W.Hobson in his Theory of Spherical and Ellipsoidal Harmonics, Cambridge 1931, adds to the definition of Pmn in (18) the factor (-1)m, which, however, is of no consequence for our present work.

Hence, we have exactly 2n + 1 possible associated Pmn, one coinciding with Pn, the others in pairs apart from the constant factor C are equal to each other and, moreover, differ by the factor eimj, which is added in (12a).

4.22.5 Surface Spherical Functions and Representation of Arbitrary Functions: We call most general surface spherical function (Maxwell's term) the expression with 2n + 1arbitrary constants

Multiplied by rn (or r-n-1), Yn is the most general, in the co-ordinates x, y, z homogeneous, potential of order n and -n -1, respectively (Maxwell solid harmonics). It is composed of the special, also homogeneous potentials unm in (12a),

as, on the other hand, the general inhomogeneous solution of the potential equation (4) involves the sum of its homogeneous components

By specializing the representation on the sphere with radius 1 and letting u there be the arbitrary function f(x, y), one obtains

By using the notation An,m in place of Am in (19), we have indicated that the constants available in every Ym are, of course, independent and differ from the constants of every other Yn. The series (19b) and (20) express that the boundary value problem of potential theory for a sphere (for example, a unit sphere) is always soluble for the interior [factor rn in (19b)] and exterior (factor r-n-1) for arbitrarily prescribed values f(q, j) of the potential on the sphere. We will deal with the task of direct construction of the Green function and from there draw conclusions regarding these series. The first strict proof of (20), under very general assumptions regarding f(q, j), was given in 1837 by Peter Gustav Le-jeune Dirichlet (1805 - 1877).

4.22.6 Integral Representation of the Spherical Functions: Consider the special, in x, y, z homogeneous, function of n-th degree

which like every function f(z + ix) or f(z + iy) , etc., satisfies obviously the equation D u = 0. The factor of rn in (21) is therefore a surface spherical harmonic Yn. If we average it with respect to j, whereby it becomes a function of z only, we obtain the zonal spherical function

On the other hand, if we form, following (1.12), the m-th Fourier coefficient of Yn, we obtain the associate (tesseral) spherical function

The integral representation (22) occurred already in the Mécanique céleste of Pierre-Simon, Marquis de Laplace (1749 - 1827). One sees that it is correctly normalized by the denominator 2p by setting z=1; in fact, then the integrand equals 1 and therefore indeed Pn(1) = 1. In contrast, in (23), the normalization factor has to be determined. One finds* by a comparison with the normalization, obtained in (18), that

A match to (22) results, if we replace in (21) n by -n - 1, which, as we know, certainly is possible. Then we obtain as a match to (22)

which is also correctly normalized.

* For example, by letting in (23) and (18) , where the integration in (23), apart from the factors z n and e-imj , reduces to

During the binomial expansion of this expression, one need only take account of the term with eimj, since all other terms vanish during the integration with respect to j. The factor e-imp/2 in (23a) arises from the factor in (18). Concerning the limit in (18), cf. the similar computation of (18b).

4.22.7 A Recursion Formula for the Associate Functions: Referring to the recursion formulae (11b) and (11c) for the zonal spherical functions, we differentiate (11b) m times with respect toz, use in the central term Rule (17a) and multiply every term by . By (18), we find

On the other hand, one finds from (11e) by (m - 1)-fold differentiation with respect to z and repeated multiplication by

Equations (24) and (25) yield then, by elimination of the term with , the (11b) generalizing recurrence formula

Due to its derivation by m-fold differentiation, this equation is valid only for positive m. However, it transfers immediately also to negative m, if we take into account our general definition of the Pmn in Equation (18) and recompute, using (18b), Pmn into P-mn.

4.22.8 Normalization of the Associate Functions: We know from Equation (10a) the value of the normalizing integral for m = 0. We denote it by Nn or also N0n. Its evaluation in 1.5 was based on the symbol Dk,l in (1.5.9). First of all, consider as generalized form of the normalizing integral

Written with the symbol Dk,l, (27) becomes as a consequence of our generally valid definition (18)

Step by step m-fold lowering of the higher and raising of the lower differentiation symbol D yields, since the terms free from the integration vanish every time for z = ±1,

We have substituted here at the end the value of N0n from (10a). As a rule, one considers as normalizing integral

However, this can be obtained directly from (29) by using (18b). It yields

Its direct derivation, like that of (28), would be somewhat more involved.

We yet note that we will perform in the next chapter a basic normalization of eigen-functions to 1. If we denote the thus normalized associate function by Pmn, then we have for them

from which a comparison with (30) yields

4.22.9 The Addition Theorem of the Spherical Functions:

Our proof will rest on an auxiliary theorem which we will only prove with the methods of the next chapter, so that it must here be viewed as an axiom: The spherical harmonic

depends only on the mutual position of the points on the sphere's surface, i.e, it has an invariant which is independent of the co-ordinate system. If we now shift the co-ordinate system of by placing the polar axis of a new Q,F-system through the point , this point receives the co-ordinate Q0 = 0 (its F0 becomes indefinite), while the Q-co-ordinate of the former point is now given by

However, with Q0 = 0, all terms in (32) vanish except for m = 0. The right hand side of (32) then becomes the product of the two zonal spherical functions Pn(cos Q) and Pn(1). Thanks to the assumed invariance of Yn, one has then

This is the symmetric form of the addition theorem under consideration, in which the structure is expressed in a convincingly simple manner. We obtain the formulation, used as a rule in the literature, by expressing Pmn in terms of Pmn. Then (34) yields

or, in real terms,

However, it is clear that the true and natural structure of the addition theorem becomes stepwise lost during the transitions .

Another, also rather clear form of the addition theorem is obtained from (35) by interchanging one of the superscripts m with -m and using (18b), i.e.,

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