**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 *y*_{n}

_{}

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 *y*_{0}:

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 **d**r*),
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 *I*_{n}(*r*)
and (*r*²/2)^{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 *z*_{n},
corresponding to *y*_{n}, by the equation, analogous
to (11),

(you must insert here in your mind the superscripts 1 and 2). We are especially interested in *z*_{0},
which follow from *H*_{½} which itself is computed from (19.31)
and (19.30) as well as (11):

The following remarks relate to the notation: Our notation agrees with the original definition of *y*_{n
}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 *y*_{n
}defined in (16) is denoted in technical acoustics by *S*_{n}; the function *C*_{n}*
*also used there would in the present terminology be called a **Neumann Function**;
it is proportional to *z*_{m}^{1}*-
z*_{m}^{2} **.**

Corresponding to Equations (14a) for *y*_{n},
one can express the *z*_{n }in *elementary
form*** **by *e*^{(±i}^{r}^{)}.
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**
+ i**h*, 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 *F*_{0}*,F*_{1 }to cos *r*,
sin*r*. 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*_{-½}*, *the second to *I*_{+½}, 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 *a*_{0} = 1,

If we employ Hankel's symbol

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

Hence, the series for *H*^{1} and *H*^{2}, 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 *z*_{n}*,
**y*_{n}*.*

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** '*(

it lies for *H*^{1}* *and

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*=-i*a*.
For this one, *F*"(*b*_{0}) = *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*"(*b*_{0}),
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*
- *b*_{0}). 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 H_{1/3} by its limiting value for small argument, which, according to (19.31) and (19.30), is (*I*_{1/3}
can here be neglected compared with *I*_{-1/3})

Due to *z* = (*n*/3)tan^{3}*a *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 *H*_{½} (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 *H*_{n}^{1,2}(*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 *H*^{1}* *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 *H*^{2} 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 *H*^{1}_{n}* *(*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 - *a*²/2,
yields

These roots of *H*^{1}_{n}(*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 *e*^{F(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 *H*_{1/3}
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 *x*_{0} , *y*_{0}
, *z*_{0}

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 *x*_{0}
, *y*_{0} , *z*_{0}. Then *x*_{0} = *y*_{0} = 0, *z*_{0 }=
*r*_{0}, according to the general relation

One can expand (1) in ascending or descending powers of *r*, depending on whether *r*<*r*_{0
}or *r*>*r*_{0}.
We call *P*_{n }the coefficient belonging to
the ascending or descending powers and
have

The *P*_{n }must be the same
in both expansions, since both must become each other for *r* = *r*_{0 }and
. The point *r* = *r*_{0},
= 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 *P*_{n}, 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 *P*_{n
}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 *D**u
= *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 *r*^{n-2}/*r*_{0}^{n-1}

the second line
yields the same, but with the factor *r*^{n}_{0}/*r*^{n-3}. We now replace
the argument of *P*_{n}* *by *z, *writing

and noting that

Then (5) yields

Imagine (6) written down for a second subscript *l* and form, according to the scheme of
Green's theorem, by cross multiplication with *P*_{1}* *and *P*_{2}

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 *P*_{n }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 *P*_{n }there are completely determined by their orthogonality (8) and normalization (9). Thus,
our present definition by the generating function leads to the same functions *P*_{n }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 *P*_{n }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

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 *a*^{n}
, 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 *a*^{n+1
}to zero

^{}

If one multiplies this equation by 2*n* + 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 ****P**^{m}_{n},
associated with *P*_{n}, 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 *P*_{n
}**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 *P*^{m}_{n},
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 *P*_{n}.
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, *P*^{m}_{n}, just as *P*_{n}, 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 2*n*+1 values
Also for negative *m*, *P*^{m}_{n}* *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-*z*²)^{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 *m*²).
Consequently, *P*^{m}_{n}* *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*^{-m}_{n}* *and *P*^{+m}_{n}*
*. 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 *P*^{m}_{n},
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 *P*^{m}_{n}
is always viewed to be positive and the dependence on *j *is
given by cos *m**j *or sin
*m**j*. 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 *P*_{n
}*= P*^{0}_{n }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 *P*^{m}_{n}
in (18) the factor (-1)^{m}, which, however, is of no
consequence for our present work.

Hence, we have exactly 2*n *+ 1 possible associated *P*^{m}_{n},
one coinciding with *P*_{n}, the others in
pairs apart from the constant factor *C are *equal to each other and, moreover, differ by the factor *e*^{im}^{j}*,
*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 2*n* + 1arbitrary constants

Multiplied by *r*^{n} (or *r*^{-n-1}),
*Y*_{n} 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 *u*_{nm
}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 *A*_{n,m}* *in
place of *A*_{m }in (19), we have indicated that the constants available in every *Y*_{m }are,
of course, independent and differ from the constants of every other *Y*_{n}. 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 *r*^{n} 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 *r*^{n }in (21) is therefore a surface spherical harmonic *Y*_{n}.
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 *Y*_{n}, 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 2*p
*by setting *z*=1;
in fact, then the integrand equals 1 and therefore indeed *P*_{n}(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*^{-im}^{j}*
, *reduces to

During the binomial expansion of this expression, one need only take account of the term with *e*^{im}^{j},
since all other terms vanish during the integration with respect to *j*.
The factor *e*^{-im}^{p/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 to*z*,
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 *P*^{m}_{n}*
*in Equation (18) and recompute, using (18b), *P*^{m}_{n }into *P*^{-m}_{n}.

**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 *N*_{n}*
*or also *N*^{0}_{n}. Its
evaluation in 1.5
was based on the symbol *D*_{k,l} in (1.5.9).
First of all, consider as generalized form of the normalizing integral

Written with the symbol *D*_{k,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 *N*^{0}_{n} 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 *P*^{m}_{n},
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 *Q*_{0}
= 0 (its *F*_{0}*
*becomes indefinite), while the *Q*-co-ordinate
of the former point is now given by

However, with *Q*_{0}
= 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 *P*_{n}(cos
*Q*) and *P*_{n}(1).
Thanks to the assumed invariance
of *Y*_{n}, 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 *P*^{m}_{n}*
*in terms of *P*^{m}_{n}. 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.,