Normalization of eigen-functions in an infinitely expanded region
There arises during the transition from the bounded to the unbounded region a convergence difficulty, which can only be removed by a change of the normalization method. This changed method was introduced into the theory of integral equations by Hermann Weyl and then taken over into wave mechanics.
As an example, we employ the function In(k, r), where n is arbitrary, k is determined as root of In(k,a)=0. According to (20.19), its normalization integral would be
however, it becomes divergent in the limit 
due to the asymptotic behaviour of In' (indefinite between
the bounds 
In order to make In nevertheless normalizable for , we start with the general integral
which, according
to (21.9a),
behaves like the function d(k|k')
(the symbols s, r, s in (21.9a) correspond
now to r, k'. k), in fact,
because it vanishes for 
and becomes infinite for k = k'
in such a manner that
where D is an arbitrary interval which contains the critical point k' = k. In particular, since D can be chosen arbitrarily small, so that k can be viewed to be constant in D, one can also write instead of (2)
or, for example, as we will occasionally require,
We now change the normalizing
integral (1) in the limit 
into
i.e., replace in (1) one of the two eigen-modes I by the group of neighbouring eigen-modes
and succeed thereby in averaging the previously referred to uncertainty, by, so to say, interference within the wave group. By (1a) and (2a), we then find
Following the initial instruction of Weyl, we could also replace both the eigen-functions I by eigen-differentials of the form (3a). Instead of the expression group of eigen-vibrations or wave group, wave mechanics employs usually the not very nice term wave packet.
In the preceding
work, in order to extract clearly the core of the mathematical
process, we have acted somehow generously while interchanging the
limiting processes 
(or, what is the same thing, 
.
Hence, it will be good to justify (4) again by hints on the basis
of Green's theorem. One computes with
the integral 
in
the known fashion, on the one hand, as area integral, on the
other hand, by Green's transformation, as boundary integral and
concludes:
The integration range of Green's theorem is here not the complete circle of radius a, but the periodicity range of u and v, similarly as in (25.5), namely a circular sector with angle a = 2p/n. During the integration with respect to j, there would appear on both sides the factor a, which has already been cancelled in (5a).
If one divides here by (k˛
- k'˛) and integrates with respect to k' under the
integral from k' - k - D/2 to k' =
k + D/2, then, on letting 
, one obtains on the
left hand side the normalizing integral (3). On the right hand
side, let a be so large that the In can
be computed asymptotically; it is most convenient to employ half
the sum of (19.55)
and (19.56), where the constants 
beside the exponents ±ir partly cancel each
other, partly are of no consequence for the following and can
therefore be omitted immediately. Thus, one obtains as right hand
side
This yields after suitable manipulations
Obviously, we can replace here 
for
sufficiently small D by k and place it outside the integral. If one
then substitutes in the first term of the integral x = (k
- k')a, one obtains for this
i.e., in the limit , the value p (cf.
Exercise I.5). In the second part of the integral (6), on
substituting y = (k + k')a, the
integration bounds become
It is readily seen that this integral vanishes, whence (6) becomes 1/k and at the same time (5a) becomes
in agreement with (4), whence follows immediately that the, in the above manner to 1 normalized, Bessel function In is given by
Moreover, it follows from (21.11), that the to 1 normalized function yn, which we have called Yn before, depends on yn as follows:
In fact, one has, by (7a) and (8),
and, by substitution from (8a),
which indeed, by (3), is normalization to 1.
Since (8) does not only link In+˝ and yn, but also the associated functions H1,2c+˝ and z1,2n , Equation (8a) at the same time applies for the to 1 normalized zn , which we called before Zn.
Equation (3) transfers to a general three-dimensional eigen-function in the form
where un and u'n are the eigen-functions belonging to k and k'. The to 1 normalized eigen-function is then
The external boundary value problem under consideration involves the derivation of a solution of the wave equation Du + k˛u = 0, which outside the given, entirely in the finite plane s located plane is continuous, assumes on s arbitrarily prescribed values u = U and satisfies at infinity the radiation condition. This solution, as we know, is represented most simply by the Green function G, which vanishes on s, meets the radiation condition at infinity and has at an arbitrarily selected point Q a discontinuity with the character of a unit source.
In the case of the
sphere of radius a and a source point Q with
the co-ordinates 
, we have constructed G in the form of Equation (28.22):
The radiation condition for 
was
satisfied by the factor of the first row zn(kr),
written more exactly z
1n(kr),
and meets the boundary condition for r = a by
the factor of the second row
n was there a positive integer with the result that Pn(cos q) was for all values of 
steady.
We will now try to solve the problem more economically in that we impose on the function zn(kr) not only the radiation, but also the boundary condition
Then, n must not be an integer, because the roots of zn(r) coincide with those of H1n+˝(r) = 0, which, by (21.41), are not only not integers, but also complex (of large absolute value). In general, we call nm the successive roots n1, n2, ···, located in the first quadrant of the complex n-plane, understand by S the summation over the entire system of these roots and write
The function Pn,, introduced here, is not a Legendre polynomial, but the hypergeometric series of (24.24a)
We have used in (3) 
because G is to be regular on the half ray 
, while
the half ray 
, by
assumption, contains the point Q; to this corresponds
that, by (3a), 
has the value F(-n, n + 1, 1,
0), and, in contrast, for , by (24.32), when we interchange
there z with 
,
the value
Next, we will determine the coefficient D, where Mr. F. Sauter has co-operated most effectively. First an observation: The functions z are mutually orthogonal; similarly, one has
as a consequence of the differential equation of z, which, corresponding to (21.11a), is observed by yn:
If one writes down the same differential
equation for zm
, cross-multiplies by zm and zv, respectively, one obtains from the difference of the
two equations by integration over 
Due to (2), the right hand side of (5a) vanishes at the lower end and, due to the asymptotic behaviour of z according to (19.55), at the upper end, whereby (5) has been proven. Simultaneously, (5a) yields for the normalization integral
and from it, by differentiation of the numerator and denominator with respect to m and consequent taking account of (2),
With the notation
Equation (6) becomes
Based on this orthogonality and normalization,
Equation (3), after multiplication by 
and integration from 
is
any of the roots n), yields
However, this determination of B is
still unsatisfactory, because we do not yet know G.
Hence we take the limit: We let 
after dividing (7) by 
(the
bar above n can from now on be omitted) and set, by splitting the
source point singularity from G,
Thus, we rewrite (7) into
Since now g remains finite for all
values 
and arbitrary 
, while 
becomes
infinite for 
, the contribution by g on the right hand
side of (7b) drops out. For the same reason, one can restrict in
the contribution of eikR/R the
integration variable to the immediate neighbourhood of the source
point co-ordinate r0 by setting
while one has to approximate the denominator R better:
Thus, (7b) yields
However, the limit on the right hand side is known from (24.32), i.e., it equals p/sin np . Hence, (7c) yields, with (6), the free of e and completely determined value
The Green function (3) now is
This formula becomes considerably simpler, if one introduces instead of z the to 1 normalized eigen-functions
In fact, on account of (6b), we can then rewrite (9)
This equation will be of great use later on; however, first of all, we continue with (9).
1. In (9), the Green function is represented by the same formalism for r > r0 and for r < r0, not like in (1) by two different formalisms.
2. The reciprocity of the
Green function, to be demanded in general , expresses itself in
that (9) was built symmetrically in r and r0. In contrast,
by (1), G satisfied reciprocity only in that, if r
and r0 were interchanged, the two partial
representations for r > r0 and r
< r0 were interchanged. The fact that
G is also reciprocal with respect to the two angles 
(we
set the latter especially equal to zero) allows us to simply
express by replacing in (9) as well as in (1) replace 
by
the in 
and 0 symmetric notation cos Q.
3. The orthogonality relation (5) differs essentially from our earlier formulations: zn is multiplied in (5) by zm, not, as we had expected in the case of a complex character of the eigen-function in (25.11a), by z*m; hence znzm in (5) is multiplied by the one-dimensional interval dr, not by r˛dr, as is necessary during application of Green's theorem in Exercise 5.1b.
4. It is very strange that our
representation (9) for 
seems to fail, because then, by (4),
every term of the series becomes infinite like 
,
while the function to be represented for and 
is
regular there. Hence we must view the entire half ray (not only the point , r = r0
of it) as a singular place of the representation
(9). Hence, our representation becomes more or less unusable also
near this location, i.e., in a less or more narrow cone about it.
The question how it is to be complemented inside this cone will
be deferred to later on.
5. In
contrast, Equation (9) becomes simpler for the neighbourhood of 
say
for 
where it becomes, written in the initial form (3),
Moreover, if we let here r0 tend
to 
, i.e., proceed
from the primary spherical
wave G
to a plane wave u, entering from the direction , we
can employ for Hn+˝(kr0), without trouble
arising from the complex subscript n, the Hankel
approximation (19.55), because then the argument is large compared with the
subscript, and, by (21.15) , set
Then, combining all factors which do not depend on n in the amplitude quantity A, one finds
Substituted into (10), this value yields the refraction field of the plane wave u behind a sphere of radius a under the refraction angle d. In a provisional manner, we demanded u = 0 as boundary condition for r = a; we will discuss below a boundary condition, fitted to electro-magnetic optics.
6. The great advantage of (9) above the earlier version (1) is its good convergence for large values of ka. In order to check this, we will compute asymptotically the factors in the denominator in (8) for large ka and n. With
[cf.
(21.30a)]
here, z is the abbreviation
The roots of zn = 0 are given by
By (11a), one obtains for z = zm, disregarding the slowly changing factor sin a,
i.e., taking (11) into consideration,
On the other hand, by (11a, b) for z = zm and consideration of sin zm = 0, one has
Finally, (11d,e) yield for r = ka
Substituting this into (8), one finds
By (21.41), n equals ka in first approximation, but grows with increasing m into the positive imaginary, whence sin np increases exponentially in the sequence of the terms n1, n2, ··· and D decreases exponentially due to the denominator sin np.
However, due to the factor zn (kr0), the same happens in (13). This must be computed by Debye's formula (21.32) (the higher saddle point only is decisive) and not by (11a) (both saddle points are approximately equally high). Hence, the auxiliary angle a has now a meaning which differs from (11): cos a equals n/kr0, not like before n/ka. As a consequence, zn (kr0) likewise decreases exponentially in the consecutive terms of the n-series. The same applies eventually also to the factor zn(kr) which joins in (9).
In the case of the particular problem of wireless telegraphy (Appendix VI), where a representation of type (1) would demand about 1000 terms, we manage, as we will show there, to get along with one or two terms of the corresponding series of type (9). The same example will also serve to show how this type arises out of that one by a purely mathematical transformation (in the complex plane of the index n).
7. The
structure of our Green function and its singular behaviour for
become especially clear in our representation (9a). In order to
estimate from here quite roughly the behaviour for small 
, we employ the
approximation (24.32)
and obtain from (9a)
This sum 
has a d-similar
character. In fact, expanding a d-function
in the interval 
in
terms of the to 1 normalized and orthogonalized, but otherwise
arbitrary functions Zn(kr)
we obtain formally
where indeed so far nothing has been said about the convergence of the so developed general d-series
Only its divergence for r = r0
is obvious (all positive terms). Moreover, in general, the
for 
demanded representation of zero is achieved by infinitely fast oscillations about zero. Hence, (14a) is unsuitable for a proper calculation of
G for 
.
However, one can obtain from the definition (7a) a formula for the throughout finite function g, which for r = a assumes, by (7a), the boundary value
is the polar distance measured on the sphere r
= a. Moreover, g must satisfy everywhere
outside the sphere 
the differential equation Dg + k˛g
= 0 and, at infinity, the radiation condition, whence, for
example, g can be computed as a solution of the external
boundary value problem formulated above in the form of the by
(9) represented Green function following the scheme
Thus, one obtains g as a series of
advancing n,
which for the half ray 
of
interest here, is
Since the singular point of Pn (-cos q) occurs now only under the integral sign in Cn
and is only of logarithmic order, all the coefficients Cn are of finite magnitude; however, their explicit
evaluation does not seem to be easy.
Cf. details in Watson's text, p. 239/40, regarding Whittacker's integral, which is a limiting case of our Cn.
We have introduced in this Appendix a quite new kind of singular eigen-function, which differs essentially from the otherwise in this chapter used eigen-functions - regular eigen-functions. Our singular eigen-functions
are not oscillations, but require an excitation
along the half-ray . In contrast, everyone of the two
particular solutions
employed in (1), behaves in the
entire physical field 
source
free; its excitation, if one wants
to speak of such a process, occurs outside this
region, namely at the centre of the sphere r = 0 and
besides at un at infinity.
On page 184 of the author's paper in 1912, our regular eigen-functions were called improper, our singular ones actual eigen-oscillations. The following discussion justifies the apparently paradox terminology.
We start from the fact that there exists for all vibration problems - free and forced ones, periodic and decaying in time ones - the relation
in which we can assume that c (the velocity of sound or light) is real and the transmitting medium is free of absorption. Hitherto, we have assumed that w is real and the time dependence has the form e-iwt. Then our condition z(ka) = 0, based on the with (11) identical condition
yields a complex value for n with a positive imaginary part, so that for real k also acos a has the same character, i.e., has the form
However, retaining (15), (16.5a) and (15.b), we will set n equal to a positive integer, say = n. Then the same relations yield for k and w complex values with a negative imaginary part
The boundary condition z(ka) = 0 becomes
and the formerly pure periodic time dependence e-iwt
By addition of this time factor and with
respect to the for integer n singularity free character
of 
, our singular eigen-function (14) becomes in the
entire region r> a a regular decaying oscillation
which begins for 
with an infinite
amplitude, at constant duration of oscillation and constant
decrement swings and finishes at . (Only the surface of
the sphere r = a makes here an exception, since
there all the time zn(ka) = 0.) Obviously,
there are 
such
vibrations with a zonal character. They are counted by the
integer n and the number of
infinitely many complex roots k, which the
transcendental equation zn(ka) = 0 can yield. (In
the case of a tesseral character of the spherical function, the
number of possible vibration modes would rise even to 
.)
Obviously, these damped vibrations are the physically simplest
particular solutions of our spherical problem for the boundary
condition u = 0 and as such deserve the term actual
eigen-vibrations. Since they
are linked closely to our singular eigen-functions (16), we now
understand why the latter could also yield the simplest solution
of our boundary value problem.
In the sequel, we will first extend our eigen-vibrations, developed for scalar fields, without difficulties to the electro-magnetic optical vector field, which, as we will see in Chapter VI, raises no principal difficulties. We need only observe the following points:
1. The hitherto basic boundary condition u = 0 must be replaced electro-magnetically by
which, since r = kr, is the same as
Correspondingly, one must replace the function sign h in (6a) by
3. While we could manage in the scalar case with a single field function u, we have to consider side by side in the electro-magnetic case two such functions u and v; v satisfies the same differential equation and a similar boundary condition as u.
Our decaying electro-magnetic eigen-vibrations have been known for a long time. In the case of the sphere, they were already examined in 1884 by J.J.Thomson (London Math, Soc. Proc. 15, 197 and the Recent Researches in Electricity and Magnetism, Oxford, 1893, the so-called third volume of Maxwell) as the simplest example of the problem which at that time was at the centre of interest in the Hertz oscillator. They were generalized by M.Abraham(Ann. Phys. 66 and 67 (1899) 834; Math. Ann. 52 (1899) 81. More literature in Enzykl. d. Math. Wiss. Vol. V, Artikel Abraham, 508). to the case of the extended rotational ellipsoid (bar-formed oscillator) and to the rotational paraboloid (freely ending wire). Indeed, our entire trend of thought can be transferred by the use of elliptic co-ordinates from the cylindrical and spherical functions of the representation (16) without basic changes to the field of Lamé's wave functions. Beyond this, we see now that they can also serve to construct the Green function for the outside of the ellipsoid or paraboloid and with their help solve generally the relevant boundary value problems.
We conclude with a few problems, which could be solved by the method above.
a) Dispersion at colloidal particles
In a paper of
1908, G. Mie has reduced the impressive colour phenomena in the
ultra-microscope to the dieletric constant and the conductivity
of the individual dispersing particles. He assumed the particle
to be spherical, its diameter small compared with the wave
length, i.e., 
. In this case, the series of the type (1) converge
sufficiently fast. In the opposite case 
, one succeeds with the
geometric series; however, the intermediate case raises problems.
Obviously, series of the type (9) should serve well, specialized
to an infinitely distant source point. The fact that we have
assumed the sphere to be an infinitely good conductor and Mie
assumed that there was present an arbitrarily dispersing medium
causes no great difference. One merely need replace the hitherto boundary condition (17) for the perfect conductor by a transition condition between the inner and outer space. The
convergence of the thus derived series will be the better, the
more one approaches the limiting case of geometrical optics.
b) Return-radiation of the plane wave from the surface of a completely conducting sphere.
We have discussed
the refraction field behind a sphere schematically (i.e., for the
simplified boundary condition u = 0 and scalar field
character) and represented by (10), (10a) for . At
the front of the sphere, especially for , there is observed a strange interference which has
hitherto made the normal treatment by series of the type (1)
impossible. The present analytical difficulty is expressed in our
series of the type (9) by the singularity of the half-ray .
However, we claim that it can be overcome by a later approach, if one
takes into account the present conditions for the real problem of
return-radiation .
With this
classical problem, we return to the stage from which started Debye's asymptotic investigations and all of the with their aid
undertaken excursions into the field of short waves ().
Indeed, the rainbow problem itself has found a beautiful
conclusion with the work of B. Van der Pol and H. Bremmer (Phil.
Mag. 24 (1937) 141, 825). However, there remains
also here a certain gap between the wave-optical and ray-optical
methods.
In general, it is the task of this Appendix to bridge such gaps mathematically.
The wave mechanical eigen functions of the dispersion problem in polar co-ordinates
We will indicate here briefly the individual steps which lead to the representation (30.7). More details are given in texts on wave mechanics.
For example, Atomic structure and Spectral-lines, Volume II, Chapter V, Sect. 6 and Chapter II, Sect. 9. Here too, the asymptotic equation (30.8) is derived with the aid of a complex integral representation from L, which we cannot treat here.
The parabolic co-ordinates x = r +
x , h = r - x are known to define in a plane through
the x-axis a double system of confocal parabolae, which
have as common focus the point r = 0 . The degenerate
parabolae x = 0, h = 0 coincide with the positive, negative x-axes,
respectively; the parabolae 
bound the plane
towards the side of large positive, negative x.
If you rotate the plane about the x-axis, one has with x , h and the angle of rotation j an orthogonal spatial co-ordinate system, which is related to the Cartesian system x, y, z as follows:
Hence, one has for the line-element
With its aid, one finds
Hence, the wave equation (29.1) becomes in terms of the exchange-action-energy
and in the case of independence on j
It can be separated into y = y1(x)y2(h); with b as separation constant, one finds
For 
(large positive x),
y1 must satisfy the radiation condition (28.2). Rewritten in parabolic co-ordinates, (by (1), dsx is
equal for large x to ˝dx,
i.e., 
in (28.2) becomes 
(cf. 30.7a).
Hence, one sets y = eikx/2 and (3) finally yields
The two terms with x cancel one another due to the meaning of k and W (cf.(30.1)). Hence, Equation (6) can be satisfied by a suitable choice of b :
It is easily shown that (3) is met by the above formulation of y1 not only asymptotically for large x, but is exact for all x.
Due to (7), Equation (4) becomes
y2
must satisfy for 
(large negative x) the irradation condition (28.2a); in correspondence with (5), it becomes
Hence one would have to set, in first approximation for large h,
However, this would not be an exact solution of (8), whence we write more generally
After cancellation of two pairs of terms, (8) yields
the differential equation (29.12) of the Laguerre function Lm(r), if one sets there
The last value agrees with the imaginary principal quantum number n of (30.7b). Thus,
and, finally,
Hereby, the representation (30.7) as been proved by the shortest route.
Plane and spherical waves in unlimited space of any number of dimensions
After speaking so often about plane and spherical waves in three dimensions as well as about plane and cylindrical waves in two dimensions, we cannot resist the temptation of transferring these formulae to poly-dimensions. In the process, we will encounter remarkable generalizations of the ordinary spherical functions - the Gegenbauer polynomials - and of the addition theorems of the Bessel functions. Once again, our earlier theorem regarding the Green function leads us to a systematic approach to these generalizations via the eigen-functions of the space under consideration.
A5.4.1 Co-ordinate System and
Notation:
Let the order of
the dimension be denoted by p + 2, so that p
= 0 and p = 1 are two- and three-dimensional spaces,
respectively. We will employ, on the one hand, Cartesian
co-ordinates x1, x2,
··· , xp+2 and, on the other
hand, co-ordinates 
Let them be linked by the relations
In order to cover the entire
space 
, the 
must vary between the
limits
Formation of the sum of squares of (1) yields
The (p + 2)-dimensional line element is defined by
If one computes from (1) for every
direction of the co-ordinates 
the relevant ds,
one finds readily
We will denote the co-ordinates 
appearing here on the right hand side in sequence by g1,
g2, ··· , gp+2,
so that
(2) and (2a) yield for the (p+2)-dimension space element
Denote the surface element of the unit sphere in (p +2) dimensions by dw, its total surface by W and set
where 
are the two components
which result from the integration of dw with
respect to 
, j1,
j2, ··· , jp, respectively. From (2b,c), we
obtain for 
We call Dp the Laplace operator in our space (in three-dimensional space, it would be D1) and form for a function only depending on r (Vol. II (3.9b))
The potential equation Dpu = 0 then becomes
Its solution, apart from one additive and multiplicative integration constant, is
We will generalize it to
If we place the second point,
introduced here, especially on the axis 
and denote its distance
from 0 by r0, then, by (1), y1
= r0, y2 = y3
= ··· = yp+2 = 0
and 
whence
is also a solution of Dpu= 0.
As in (22.3), we expand (4b) in increasing or decreasing powers of r/r0 and call the coefficients p-dimensional zonal spherical functions
or also Gegenbauer polynomials.
Gegenbauer's initial notation
(for example, Wien, Akad. 70 (1875)) is 
where in our poly-dimensional approach
one has to set especially n = p/2.
The Legendre polynomials should then be denoted by
Thus, we set
and conclude immediately that
(5) is not restricted to integer p: it fails for p = 0, because the power series in (4b) must then be replaced by the two-dimensional logarithmic potential.
Thus, we find from the potential equation Dpu=
0 for particular solutions, which only depend on 
,
the partial differential equation for u
and from there, removing the factor 
, the
ordinary differential equation for 
The reader should convince himself for this and the following formulae of their link to the equations from the theory of ordinary spherical functions, well known to us.
Similarly to the Legendre polynomials P in (24.24a), the Gegenbauer polynomials can be expressed in terms of hypergeometric series:
5A.2 The Eigen-functions of the Unbounded Poly-dimensional Space: We now step across from the potential equation to the wave equation. For a solely on r depending state, the equation is, by (3),
If we set u = r -p/2, we obtain for u Bessel's equation with the index p/2, whence we integrate (6) by using
or also
(6b) behaves asymptotically like
and satisfies the radiation condition (28.7)
as well (6c) the irradiation condition. Hence (6b, c) represent the emerging or entering spherical wave in the p+2-dimensional space. The same applies for the generalized location of the source point
We can call the function u in (6a) the eigen-function with spherical symmetry. We now ask more generally regarding the eigen-functions with zonal symmetry. They will have the form
Equation (5c) yields for Pn readily the differential equation for vn
If one treats this equation like (6) by setting vn = r-p/2, one finds for w Bessel's equation with the subscript n + p/2 and therefore, as its for r = 0 finite solutions,
and hence its eigen-functions
According to 5.26, any two such
eigen-functions are mutually orthogonal in the
continuous spectrum 
as well as in the discrete one, i.e.,
inside the sequence n = 1, 2, ··· .
For two eigen-functions un, um of equal k, but different subscript, we compute from (2b, c) and (8):
where Wj
is the integral (2f) above. Since
neither this integral nor the integral over r vanishes,
the integral with respect to 
must vanish due to the orthogonality:
Note in (10) the characteristic factor 
,
which in the three-dimensional case (p = 1) becomes the
already familiar factor 
in the case of the Legendre polynomials.
While it could seem to be artificial in the ordinary analytical
derivation of (1), it enters on its own in our poly-dimensional
approach from the meaning of dt
We note yet the correspondingly formed normalization integral for m = n
as a generalization of the corresponding integral of the ordinary spherical functions N = 1(n + ˝) for p = 1; the proof of (11) would have to be linked to the definition equation (5) of the Gegenbauer polynomials.
With respect to (2e), we can also write instead of (11)
5A.3 The Spherical wave and Green's Function in the poly-dimensional space: We have already described the spherical wave with zonal symmetry by Equation (7). It becomes the Green function of the (p+2)-dimensional unbounded space by addition of a factor f so that the source point Q of U becomes the unit source. According to 10.3, this means
The integration must be taken over a sphere of
radius 
; ds is the surface element of this sphere, dw is, as in
(2d), that of the sphere of radius 1. Hence (12) yields
We can employ for H, in the case of odd p, Equation (19.31):
If we employ a known G-relation, we can also write
For even p, Equations (19.26) and (19.47) yield the same value. Hence, there follows from (12a)
and from (7), after multiplication by f,
On the other hand, we want to construct G(P, Q), following 5.28, out of the eigen-functions u(P) in (8a):
and the corresponding u(Q)
for a point Q with the co-ordinates 
In both the representations (13a, b), l is the continuous integration variable in the continuous part of the eigen-value spectrum (cf. (28.14)). Like there, we integrate with respect to l in the complex l-plane and obtain, corresponding to (28.15),
In order to be able to employ this formula for
the Green function, we must still normalize to 1 the two functions u(P)
and u(Q). Hence, the general term on the right
hand side of (14) must be divided: 1. due to its
dependence,
by the normalization factor N from (11a), 2.
due to its independence from the j1, j2,
··· , jp by Wj
from (2f) and 3.
due to its r-dependence, according to (4), we have to multiply by k. All
of this yields - cf. also (5) - the factor
which must be added under the S-sign. According to our general theorem of 5.28, there arises the Green function of the unbounded space. Its comparison with (13) yields, after cancellations and combinations,
This general addition theorem of the Bessel functions applies for H = H1 as well as H = H2 and therefore for any combination of them, so that one can replace in (15) H on the left hand and right hand sides by
and especially by
where in the latter case the
condition 
is of no consequence. Moreover, there applies beyond
our derivation, if one, as in an earlier remark, replaces p/2
by any number, say n.
5A.4 Transition from the
spherical to the plane wave: For 
, we will derive from
the lower row of (15) a representation of a plane wave in a
poly-dimensional space.
To start with, the right hand side becomes by the Hankel approximation (19.55)
Correspondingly, the left hand side becomes
However, for ,
whence
whereby the left hand side of (15) with corresponding approximation of the denominator becomes
After cancellation of the common factor on both sides, one finds
This is a plane wave, entering from the
positive side of the axis 
, i.e., advancing to the
negative side of this axis. The wave, which advances in the
positive direction, is obtained from (16), if one interchanges on
the left and right hand sides +i by -i. Show
yourself that the corresponding formulae for p = 1 are
identical with the three-dimensional representation (24.7). In the two-dimensional case, when our approach (4b)
fails, (cf. earlier
remark), the place of (16) is taken by (21.2b).
Equation (16) yields by suitable averaging or on the basis of an addition theorem* for the Gegenbauer polynomials strange relations between the Bessel functions with integer and fractional subscripts**.
* Cf. the clear presentation of Gegenbauer's results in Magnus and Oberhettinger: Formula and Theorems regarding the Special Functions of Mathematical Physics, Springer 1943, especially page 77.
** G. Bauer: Session report of Bavarian Academy 1875, p. 247; a poly-dimensional generalization by the author, Math. Ann. 119 (1943).