**4.23 The Green Function of
Potential Theory for the sphere. Sphere and Circle Problems for
other Differential Equations:**** **We superimpose two principal solutions *u*,
*u' *of the potential equation *D**u* = 0

and want to find the level
surfaces of the function *G* = *u* - *u*',
especially the shape of the surface *G*=0, which, by (1),
is given by the equation *R*'² = (*e*'/*e*)²,
i.e.,

This is the equation of a sphere; the position
of its centre and the size of its radius are readily determined
from (2): The centre *O* lies on the line joining the two **source points ***Q = **x, h, z *and *Q'= **x**'**, h**'**, z**'*;
the radius *a* turns out to be the central proportion of *OQ
*= *r *and *OQ' *= *r**'*

so that one of the source points
lies inside, the other outside the sphere with radius *a*.

We will not link the further considerations to the clumsy formula (2), but rather to the elementary geometrical Fig. 22.

**4.23.1 The Geometry of
Reciprocal Radii:**** **** **Fig.
22 demonstrates the relationship (3) between the **reciprocal radii**.* We call *Q*' the mirror image
of *Q* with respect to the sphere of radius *a *or
also the **electric
image point **(Maxwell);
the electric viewpoint is linked to the notation *e* and *e*'
in Equation (1). Obviously, this relation between *Q* and *Q*'
is reciprocal: *Q* is also the the reflected image of *Q*'.
It follows from (3) in a known fashion that the pair of points *Q*,
*Q*' lies harmonically to the pair of the two points of
intersection *P*_{1}*, P*_{2},
determined by the ray *OQQ*' on the sphere of radius *a*.

* The term **reciprocal **radii arises from the (bad)
habit of setting *a* = 1, when directly *r* ' would be equal to 1/*r *. We consider it for dimensional reasons
better to treat the radius of the sphere as a length.

The **method
of reciprocal radii** has been
treated masterly in William Thomson's first publications*, where
he applies it kaleidoscopically to the different problems of
electro- and magneto-statics. Instead of speaking of
transformation by reciprocal radii, one also talks of **inversion**.

* Journal de Mathem. **10 **(1845),
**12** (1847). Maxwell cites in his treatise Vol. I,
Chapter XI, a paper in the Cambridge and Dublin Math. Journ, of
1848.

Moreover, (3) yields that the triangle *OP*_{k}*Q
*in Fig. 22 is similar to the triangle *OQ'P*_{k},
i.e.,

We have used here *P*_{k }as
the symbol for a special point on the sphere, while we reserve *P*,
*R *and *R*' for an arbitrarily given point *P =
x y z *and its distances from *Q* and *Q*'. In
order to determine now the **reflected
charge ***e'*,
introduced in (1), we compare (4) with the relation

which follows for every point *P*_{k
}from (1) and conclude that

( the last with respect to (3)).

**4.23.1 The Boundary Value
Problem of Potential Theory for the Sphere, Poisson's Integral:**** **With Equation (5), we have returned to
our starting point - the demand *G* = *u - u' = *0
- and can at last justify the notation *G *there. It
points at **Green's
function**. Indeed,

is the Green
function for the **inner
boundary value problem**: Find the potential *U*, free from singularities
inside the sphere, for given boundary values on the
surface of the sphere. Just so is

the Green function
for the corresponding **outer
boundary value problem**. Among the three conditions a) b) c), which served for the definition of the Green function,
the conditions a) and b) are fulfilled b) thanks to Equation (5),
a) because the potential equation is self-adjoint, i.e., the
differential equation for *G* concides with that for *U*.
In order to fulfil also the condition c) of the unit source,
according to the table in 2.10.6, one only needs set for the inner, outer potential
problem,

We write (6) in detail by introducing spherical
co-ordinates for *P *(*r* = 0 at the centre of the
sphere, directed arbitrarily). Let the corresponding
co-ordinates for *Q* and *Q*' be

moreover, as in (4.33), let

use for *e'*/*e* the
first of the values in (5), for *e* the value -1/4*p*. Equation
(6) then becomes

The solution of the inner boundary value problem is then, by the scheme in 2.10.5

The integration on the right hand
side concerns the point *P,* sweeping over the surface of
the sphere, is as much as for *r *= *a *and he
point *Q *has an arbitrary position inside the sphere. To
start with, we compute generally from (7) with the former
abbreviations *R* and *R*'

i.e., for the surface of the
sphere, where, by (4), *R*'_{k} = *aR*_{k}*/**r**,*

With , Equation (8) now yields

This representation was derived by Simon-Denis Baron Poisson (1781 - 1840) using the long bypass over the development of in spherical functions. We see here that the straight path leads via the Green function (7).

The corresponding formula for the
outer boundary value problem, using (6a) with *e*' = -1/4*p *and *e*
taken from the second value in (5), is

It is strange that the so-called **second boundary value problem**, in which one would have on the
sphere's surface , cannot be solved by the method or
reciprocal radii.

We should still, for example in
the case of the inner potential problem, clarify geometrically
how the throughout analytical formula (9) succeeds in
representing point by point on the surface the arbitrary, i.e.,
in general, not analytic function . For this purpose, we
must focus our attention on the limit . During this limit,
the factor 1 - *r*²/*a*²
in front of the integral (9) vanishes and with it the
contributions of all area elements *d**s* for which not simultaneously vanishes
the denominator *R*². Hence it depends only on the
environment of that area element, towards which *Q *moves;
thus it is already clear that in the limit only the special value
on this area element is decisive. We emphasize this by
rewriting (9) into

In the numerator
of the last integral, we were able to replace by *Q **d**Q** d**F*,
whence the integral can be executed immediately:

A simplified approach, similar to (9) and (9a), can be performed in the two-dimensional case ( circle instead of sphere), when instead of (9) one has

**4.23.3 General remarks regarding
the transformation by reciprocal radii:**** **Returning to three dimensions, we will
now study our transformation by reciprocal radii from a more
general point of view. For this purpose, we select an arbitrary
point *O *as **inversion
centre** and also as
origin of a spherical polar co-ordinate system and form about it
as **inversion sphere** a sphere with arbitrary radius *a*.
An arbitrary point transforms into a point * *these two points are interrelated as
follows:

For the sake of completeness, we add also the corresponding transformation in rectangular co-ordinates: . By (22.2) , (12) yields immediately

Hence, written altogether, one has

and conversely

We now ask for the transformation of the line element in polar co-ordinates By (12), we have

Hence, the transformation by
reciprocal radii is **conformal**: Every infinitesimal triangle with
sides *ds* is changed into a **similar **triangle with sides *ds*' (the enlargement of
each side is (*a/r'*)² = (*r*/*a*)²). At
the same time, this mapping is, according to a theorem of Joseph
Liouville (1809 - 1882), in three-dimensional space the only
non-trivial conformal transformation.

The geometrical
mark of our transformation, generally speaking, is that it
changes spheres into spheres, where the plane should be viewed to
be a sphere with infinite radius. A **sphere**, which passes through the inversion centre, becomes a **plane**, since the inversion centre itself is
transformed to infinity. (In this **sphere geometry**, infinity is a **point**, not, as in **projective geometry**,**
a plane**.
Conversely, a plane, which does not pass through the inversion
centre, becomes a sphere.

**4.23.4 Spherical reflection in
potential theory:**** **The
next topic of interest to us is the transformation of the
differential parameter *D**u*.
We start with a function and transform the product *ru *by
reciprocal radii (*r* = distance from the inversion
centre). We denote the resulting function by

Thus, as this equation expresses,
we shift the value *ru* from the initial point to the
point *P*' with the co-ordinates

and want to show that then the differential
parameter *D**'v*, evaluated in the co-ordinates , is given by

The reason for this relationship
lies again with the conformity of the mapping, as is indicated by
the square (*r*/*a*)² in the enlargement ratio
(13). A brief elementary calculation proves (15): We define the
operators *D* and *D, *corresponding to (22.4), by

Then, if *D*'
and *D**' *are the same expressions in , (14) and (14a) yield

hence, by (15 a, d, c),

However, the
expression in the last brackets is, by (22.4) used already above,
nothing else but *r**D**u,
*whence (15c) states

in agreement with our statement (15).

The usefulness of
the dimensional way of writing with retaining *a* is now
clear: With *a* = 1, the factor 1/*r*'^{4}
in (16) could not be understood, while in our way of writing the
dimensional consequence of (16) is clearly expressed.

If we start from a solution *u*
of the differential equation *D**u *= 0 in the co-ordinates *x* *y*
*z*, the function *v=ru*, transformed by reciprocal
radii, satisfies the differential equation *D**v *= 0, written in the
co-ordinates *x' y' z'*.

This theorem is due to William Thomson. It
enables us to transfer solutions of potential problems, obtained
for a certain domain *S* of space to the transformed space
domain *S* '. This applies, in particular, to Green's
function: If this function is known for a region *S*,
bounded by planes, with the boundary condition *G* = 0,
then the theorem yields Green's function *G*' for the, in
general, by spherical surfaces bounded region *S*' and
assures also for this region the satisfaction of the boundary
condition *G*' = 0. The region *S*' can in this
process have different shapes, depending on the position of the
inversion centre. The totality of the regions, treated in 3.17 by elementary reflection,
becomes a great variety of partial spherical regions, which are
accessible to our generalized inversion reflection and lead in
the process like those earlier domains to a simple, gapless
coverage of space. The former condition, that the edge angles
could only be sub-multiples of *p*, remains in force
also for the spherically bounded regions due to the conformity of
the mapping. Where there arose formerly infinitely many image
points (e.g., plane parallel plates), also now infinitely many
image points are required (e.g., the space between two spheres,
touching each other at the inversion centre, as image of the
plane parallel plate). Where we formerly managed with a finite
number of image points (e.g., the wedge with 60º opening in Fig. 17), the reflection process ends also for the
corresponding spherical problem after a finite number of steps.

Exercises 4.6 and 4.7 are examples, where especially we also deal with the appropriate choice of inversion centres.

Obviously, all these considerations apply in **two-dimensional potential theory**, where the reflection in the sphere becomes one in an
arbitrary circle. Simultaneously, the mapping possibilities
spread immensely, since in two dimensions every transformation *z*'
= *f*(*z*) of the complex variable *z = x + iy *leads
to a conformal mapping. The enlargement condition of the line
element then becomes |*df* / *dz*| and the place of
(16) is taken by

**4.23.5 Failure of spherical
reflection for the wave equation:**** **Unfortunately, these mapping methods in
two and three dimensions are **restricted to potential theory**. If you wanted to perform in the simple
wave equation

a transformation
with reciprocal radii, then there would appear in the
differential equation, according to (16), the factor (*a*/*r*)^{4}*r
*and one would have to rewrite it into

Only in the
potential equation (*k* = 0) drops the disturbing factor (*a*/*r*')^{4}
out. In the wave equation, it means that the initially
homogeneous medium (*k* constant) appears converted into a
high degree inhomogeneous medium, which in the neighbourhood of
the point *r*' = 0 has a lens-type singularity of the
refraction index. Naturally, the same applies to the heat
conduction equation, which, starting from the mode of writing
familiar to us, with *u* as temperature and *k *as
thermal conductivity would change to

Surely, this form of equation cannot serve for the simplification of a boundary value problem for the sphere. We rather must then rely on the much more inconvenient expansion in series, as we have employed in the corresponding two-dimensional case of 4.20.1.

**4.24 More about Spherical
Functions:****
**

**4.24.1 Plane Wave and Spherical
Wave in Space:****
**The simplest
solution of the three-dimensional wave equation

is the plane wave, for example,
a purely periodic sound wave, progressing in the *z*-direction,

If we expand it in zonal spherical
functions , the coefficients are the *y*_{n}(*r*) of 4.21.3. This follows, on the
one hand, from the wave equation, on the other hand, from the
differential equation for *P*_{n}. In the
case of independence of the polar co-ordinate *j*, due to
the left hand side of (4.22.4), Equation (1) becomes

After this, *u *can be separated into
the product of *P*_{n} and a function *R*(*r*)
which only depends on *r*. Due to *P*_{n}'s
equation (4.22.5), *R *must
satisfy the equation

which we can rewrite with *r*, introduced above,
in the form

However, this is the same equation
as (21.11a), by which we have defined the *y*_{n}
as its continuous solution for *r *=0. Apart from a
constant factor, we find from (21.11)

In the same manner, one find the solutions of
(3), discontinuous on *r*. as linear combination of the functions *z*^{1,2}(*r*), defined
in (21.15). Since this solution is not of interest for the
expansion of the plane wave under consideration, we must use

We must still determine the coefficients *c*_{n}.
They follow from the orthogonality of the *P*_{n}*.
*In fact, by (22.8) and (22.10a), if we denote the integration variable as before by , then

Compare the asymptotic values on
the two sides for Due to the relation between *y*_{n}* *and *I*_{n+½}
one has, by (19.57),

The integral on the right hand side can be
expanded by integration by parts in a power series in 1/*r*.
Neglecting all higher order powers of 1/*r*, we find

The dependence of this expression
on *r *is the same as that of (6a). Hence, substitution from
(6a, b) into (6) yields

The expansion (5) of the the plane wave thus assumes the definite form

It is to be compared with the Fourier expansion
(21.2b) of the two-dimensional plane wave. Just as we conceived
this as the generating function of the *I*_{n},
we can call the three-dimensional plane wave the **generator **of the *y*_{n}. At the same time, (6) with the value
(6c) of *c*_{n }yields the integral representation of
the *y*_{n}

which clearly says nothing new regarding the expansion (7).

The next simple solution of the wave equation
(1) is the **spherical
wave**

It represents a wave which advances **radiates** in the positive *r*-direction,
if we give the time dependence *e*^{-i}^{w}^{t}.
According to (21.15a),
Equation (8) is identical to the solution of (3), singular at the
point *r=*0,

However, we want to shift at once its source
point *r* = 0 to the arbitrary point

Hence, Equation (8) yields

However, we can also expand this function in
terms of spherical functions *P*_{n}(cos *Q*), when
again solutions of the differential equation (3) must appear as
coefficients, i.e.,

the first, because the point *r*
= 0 is now a regular point of the spherical wave, the second,
because the type of the **radiating** wave must be preserved in every term of
the expansion. Due to the symmetry of *R* with respect to *r*
and *r*_{0}, the inverse applies for the
dependence of *r*_{0}. Hence
there must appear as factors in the coefficients of *P*_{n}(cos
*Q*)

so that the expansion becomes

The numerical factors *c*_{n} must be the same in both series, because for *r*_{ }= *r*_{0
}both of them must transcend into each
other (apart from the point *Q*, where simultaneously *Q* = 0 and
both series diverge). The situation here is just as in the case
of the cylinder waves (21.4). Inside the sphere, we
have a **Taylor
series**, outside it
a series of the **Laurent
type**. The still
lacking determination of the *c*_{n }can
again occur by the limit , when

Thus, there arises on the left hand side of
(9), employing -*i**r*_{0
}cos *q* instead of +*i**r*_{0 }cos *q*,

Due to (21.15) and (19.55), the second row on the right hand side of (9) becomes during the same limiting process

This agrees term by term with the left hand side when we set

The representation of the
spherical wave, completed by (9a), could also be called the **addition theorem of the function**

If we go on the left hand side of (9) from the radiating to the entering spherical wave

there also takes
on the right hand side *z*^{ 2} the place of *z *^{1}. Half the sum of the two
representations yields still the **addition theorem **of the singularity free **standing wave**

for which the difference between disappears.

**4.24.2 Asymptotic Matters****: **If we perform in the differential
equation 4.22.13 of the associate spherical functions the limit

we find

This is the differential equation (19.11) of the cylinder function *Z*_{m}.
Since *P*^{m}_{n }and
therefore also *O*_{m}* *are
finite for , only the Bessel function *I*_{m}
can be a solution of (11a), whence

This follows from the fact that for *m*
= 0 and *h **= *0 , on the one hand,* I*_{0}(*h*) = 1, on
the other hand, by (11), since *h* = 0 also i.e., and
therefore also *O*_{0}(*h*) = 1. In order to
determine *C*_{m} also for *m*
> 0, we go to (22.18), which yields for

We rewrite the function, which is to be differentiated, in the form

We have written here from the executed binomial
expansion only the term with (*z
- *1)^{n+m}, because
the terms with smaller exponents drop out during the
differentiation, those with larger exponents vanish in the limit . The
written term yields now after (*n + m*) differentiations

Its substitution into (12a) yields

The last fraction here contains in the
numerator 2*m* factors more than the denominator; since
one can identify all of these due to with *n*, one
finds instead of (12b)

The comparison with (12), where
one might replace *I*_{m}(*h*) by the
first term of its power series (19.34), yields

Thus, if *m* > 0, one
must divide *P*^{m}_{n }prior
to the execution of the limit by *n*^{m},
in order to obtain directly *I*_{m}*.*

Obviously, in geometrical terms,
this result means: The surface of the sphere can be replaced for
the neighbourhood of the North pole by its tangential
plane. The solution of the three- dimensional wave equation, the
behaviour of which is determined by *P*^{m}_{n
}*e*^{im}^{j}*, *changes
in the process into the solution of the wave equation for the
tangential plane, namely, into* I*_{m }(*h*)*e*^{im}^{j},
in as far one subjects at *m* > *n *not *P*^{m}_{n
}itself, but *P*^{m}_{n}/*n*^{m}
to the limiting process. Obviously, the same is true for the
South pole of the sphere .

After having dealt here first with
the special cases and , we will now study, in
general, the asymptotic limiting value of *P*^{m}_{n
}for .

For this purpose, we apply the **saddle point method** to the integral (4.22.23), which we rewrite in the complex form:

the last term coming from (22.23a). The integration is to be extended over the unit circle
in the *w*-plane in the positive sense
(counter-clockwise); *f*(*w*) is

whence

Hence there are two saddle points *w*_{0},
which under the assumptions lie on the unit circle,
i.e.,

and

With the same assumption, one has

Corresponding to (19.54), we set up the one and the other saddle point and obtain, using (15a),

If we set here

then becomes real This choice of *g *means
that we integrate at both saddle points along the respective fall
line, the direction of which, according to (15d), depends on . Hence
both integrals assume the common value

which due to *dw = e*^{i}^{g}*ds*,
following (15d), must still be multiplied by for both saddle
points by the different factors

The integral (16) is reduced by a simple substitution in the limit to Laplace's integral and becomes

Thus, one obtains altogether for (14) with respect to (15c) and (16a, b)

Under the assumption , made throughout, the
value (14) of *C* , in the numerator of which *m*
there are more factors than in the denominator, similarly as in
(12b), reduces to *C* = *n*^{m} *e*^{-im}^{p}^{/2}*,
*i.e., the value of *C*/*i *to* n*^{m}
*e*^{-i(m}^{+1)p}^{/2}.*
*Combining this with the two exponential functions in (16c),
one finds after simple manipulations

Hence *P*^{m}_{n
}is for real *n *a rapidly oscillating function
with varying amplitude which is small in the vicinity of and grows symmetrically
on both sides from there with decreasing or increasing . For and *p *fails
(17), because, by (15a), *f*"(*w*_{0})
vanishes there, i.e., the series for *f*(*w*) - *f*(*w*_{0})
also only starts with the third term, as in the limiting case
which led to Airy's
integral. Here the place if (17) is
taken by the limit (12c).

The representation (17) will be employed in the
Appendix to
Chapter VI and, indeed, for complex *n*
with positive real part, in which case, as our derivation shows,
it is the same for positive real *n*.

**4.24.3 The spherical function as
electrical multi-pole****:**** **We return now to potential theory. Since
we were in 4.22.5 able to define the spherical surface functions of *n-*th
order generally as homogeneous potentials of degree *n*
or, as we rather say now, of degree *n* - 1, we must now
be able to generate them by repeated differentiation of the
elementary potential 1/*R* **in ****n****
directions*** . *This is Maxwell's point of view in
Chapter IX of his treatise. We express this by Maxwell's
prescription:

Thus, the **observation
point** *P=**x, h, z *is
to lie on the sphere of radius 1, the **source point** *Q*= *x y z* near the
origin. We can apply equally the **directional differentiations ***h*_{1}, *h*_{2},
··· , *h*_{n}* *to the
co-ordinates of *P *as to those of *Q*. We do the
latter and after the differentiation go, has already been
indicated in (18), to the limit . There arises in this
way in *Q* a multi-pole of the higher order the more
frequently we have differentiated.

We will start with the simplest
case in which the directions *h*_{1}, *h*_{2},
··· coincide between them and, say, with the *z*-direction.
The thus created spherical surface function is symmetric about
the *z*-axis and therefore a **zonal** spherical of the Legendre type *P*_{n}.
We will follow step by step its formal generation from one line
to the next, where we indicate the limiting process, defined by
(18) by the symbol :

This sequence *P*_{1}, ··· , *P*_{4},
which can yet be completed by the zero-th derivative of 1/*R *,
i.e., by *P*_{0}=1, agrees completely with the
values, arising from our original definition in 1.5 ( *x* there is
identical to the later and present *z*). This must be so
according to the connection between spherical functions and
homogeneous polynomials, so that in (18) above actually only the
normalization factor 1/*n*! was to be controlled. We also
draw attention to the connection of this prescription with the
second equation
(22.3), which we can write with the
change of notation *r*_{0} = *z *(the point
*Q* on the *z*-axis) and *r* = 1 (the point *P
*on the unit sphere)

whence indeed for

We will combine the notation and imagery of
sequential multi-poles, where, in order to avoid the limiting
process of the combination of *Q* and *O, *we
replace the differentiation with respect to *z *(source-point
co-ordinate) that with respect to *z*
(observer-co-ordinate), but with inverted sign and can interpret *r*
as the distance .

There arises by combination of two **quadru-poles** of opposite lay-out an **octu-pole** with
the potential

(The reader should draw the associated charge
distribution!) By *n*-fold differentiation, we obtain the

In wireless telegraphy, one says,
as a rule instead of bi-pole di-pole. Quadru-poles and octu-poles
occur in **atomic
physics**.

However, we must now still
consider examples of differentiations in **different directions** by adding to those in the *z*-direction
those in the *xy*-plane. In order to make sure that there
occurs hereby a certain symmetry, we imagine the last ones, say *m
*differentiations, arranged as stars under the mutual angle *p* /*m *between two neighbouring
ones, while the *n* - *m *remaining
differentiations are in the *z*-direction. We thus arrive
at the tesseral spherical area functions

where naturally
instead of the here stated two exponential dependencies there can
also occur a linear combination of them, for example, Among
(19), one has especially the so-called sectorial spherical area
functions (also this terminology belongs to Maxwell) with *m =
n*, the representation of which by (4.22.18) is

We will show this in detail for *n* = 2.
The required star-shaped layout follows here as we place *h*_{1}
and *h*_{2}* *in the *x-* and *y-*directions.
Equations (18), *et sequ.* then yield

which is indeed of the type (19a). Also in this
case, one speaks of a **quadru-pole**(cf. the right image of the following scheme); the one on the left hand side, differently placed in
space, belongs to *P*^{1}_{2}.

The fact that (18) yields the complete system
of the 2*n* + 1 spherical area functions of degree *n*
follows from the number of constants involved in (18): Two
direction elements at a time for every differentiation *h*
and a multiplicative factor.

**4.24.4 Details of hypergeometric
functions****:
**We define these
functions best by their differential equation

and draw from it conclusions regarding their
representation by Gauss' series (2.11.10a) using the method of 4.19.3. Hence we start with unknown exponents *l* and
unknown coefficients *a*_{k}

substitute this in (20) and set the arising
factor of the starting term *z*^{l-1}* *as
well as of the general term *z*^{l}^{+k}
equal to zero, whence we obtain, on the one hand,

on the other hand

Equation (21a) has the two solutions

we will first investigate the first of these and find by substituting it into (21b)

In this way, we obtain indeed, setting still *a*_{0}
= 1, Gauss' series

The other solution (22a) yields, as is easily seen,

In addition, there are still a large number of
related representations for changed parameters *a**, **b**, **g *and
linearly transformed *z*, which range-wise agree with (23)
and were arranged by Gauss as * relationes inter contigua* with great care.

If we compare the differential equation (22.6a) of the zonal spherical functions

with Equation (20), we see that it arises from the last by the substitution

Hence we conclude that *P*_{n}*
*must agree with *y*_{1} as well as *y*_{2}*
*apart
from a factor, i.e., that

which also yields the correct normalization,
namely *P*_{n}(1) = 1 for *z **= *+1.
*P*_{n}. Like every hyper-geometric
series with negative integer *a* or *b* stops *P*_{n}: Because *a *= - *n,
P*_{n} is a polynomial of degree *n*
(the coefficient of and all the following powers would have
in the numerator the factor *a
*+ *n* = -*n + n = *0).
We also note that the series *P*_{n }advancing
with 1 - *z *is simpler ( because it is hyper-geometric) than the
series advancing with *z *, whence we have not given it here in its general form.
In fact, that series is

Also the associated *P*^{m}_{n}*
*can be represented hyper-geometrically. One need only
observe for this purpose the general relation, which follows
directly from (23) by term by term differentiation:

Hence, one finds from (22.18) for positive *m *the representation

At first, it fails for integer negative *m*,
but it can be extended to these by a limiting process; it then
agrees with our general definition in (22.18).

We now proceed still from Gauss'
hyper-geometric function to the **confluent hyper-geometric function**, which is extremely important for **wave mechanics**. It only depends on two parameters *a* and* g*, since one submits the third one *b *to the limiting process:

Hence, it follows from (23) that

and from (20) as associated differential equation, if one takes account of

and once divides by *b*

We will meet this equation again
with the **hydrogen-eigen-functions
of wave mechanics**.

**4.24.5 Spherical functions with
non-integer subscripts:**** **In
two respects, our representation requires additions. We have
limited ourselves to **integers*** n* and *m *and to
throughout finite functions* **P*^{m}_{n}. By the way, both were suggested to us
from the start through the relationship to potential theory.

As far as the
first point is to be regarded, one has to say that for
non-integer *n *the hypergeometric series to be set to the
characteristic points in terms of powers of do not
**end** as for integer *n, *but that the
at the North pole *z *= +1 regular solution diverges at the South pole *z *= -1 and *vice versa*. The two
hyper-geometric series, compared in (24a), then represent
different solutions of the differential equation, so that their
combination by the factor (-1)^{n} is only justified for integer *n*.
The **demand of
throughout finiteness on the entire sphere** can thus be only met for integer *n*.
On the other hand, non-integer *m* are excluded by the
finiteness demand with respect to the co-ordinates.

We could extract
the type of becoming infinite of *P*_{n}(*z*)
for non-integer *n* from the general theory of the
hyper-geometric series, but we prefer to derive it directly.

As a consequence
of its initial definition (22.3), *P*_{n }were defined as the coefficients of a
Taylor series advancing with *t* = *r*/*r*_{0}* *,
i.e., for integer* n*, as the *n*-th differential
quotient

Hence, one knows that one can write according to Cauchy's theorem

However, the same representation is also valid
for non-integer *n*, except that one has to apply then,
due to the multi-valuedness of *t*^{-n-1},
a branching cut, for example, from , and that the
integration path, which in (30a) was anti-clockwise along an
arbitrary route about *t* = 0, now becomes a loop around
the branching cut, starting at the negative shore of it at runs
around the point *t* = 0 in the same sense and ends on the
positive shore at . In fact, it is clear that for this
definition the differential equation of the *P*_{n}
is met irrespectively of whether *n* is an integer, and
(30a) defines just that particular solution of the differential
equation which is regular at *z=*1 and meets the normalization condition *P*_{n}(1)
= 1. Thus, for *z *= 1, it follows from (30a) that

The integrand has now at *t*
= 1 a simple pole. One can deform the loop, described for (30a),
into a circuit of the pole, which now is followed in the
clockwise direction. This circuit yields, again according to
Cauchy's theorem, -2*p **i*, i.e., for the right hand side of (30b)
indeed the required value +1.

However, for , the
integrand of (30a) has, apart from the already mentioned
branch-point, yet two more such points, which arise from the
square root in the denominator and lie on the unit circle of the *t*-plane
at

We again connect them by a
branch-cut which we will take along the unit circle. Again, it
may not be crossed by the integration path. For move
now its starting- and end-point to the negative real axis of the *t*-plane
and squeeze the loop together. One understands from this that
(30a) for has a singularity.

For its discussion, we set for the
neighbourhood of *t *= -1 on the upper and lower shore of
our branching cut

so that the square root in (30a), apart from
terms of higher order in *t
*and*
d *,* *becomes

Hence, for small *d*, only the
neighbourhood of *t* = 0 contributes to the limit . Hence, we can limit
the integration over the upper and lower shore of the small
region between

and set at the same time with respect to the sense of travel along the two shores

so that

Now, by a known formula, one has

in the case of indeterminate upper and lower boundaries, whence one finds for the definite integral in (31)

and for

In the limit , only
the term *d* ² contributes, whence (31) yields

where the terms* represented by
··· reduce for to a finite constant, which, of course,
does not depend on *e .*

* They are evaluated in Hobson's text, equation (53) on page 225.

**4.24.6 Spherical functions of
the second kind:**** **As
has been shown at the start of 4.24.5, there exist for non-integer *n* two different
solutions *P*_{n}* *of the
hyper-geometric differential equation. They coincide only when *n
*is an integer. However, there must also exist here for
integer *n*, beside our for finite solution, a second solution which
then will be singular for . It is called the **spherical function of the second
kind** and denoted
by *Q*_{n}.

The type of
singularity can then be predicted following general theorems. As
we have seen in the case of (21a), the quadratic equation for the
exponent *l
*in the case of the
spherical functions (*g *=
1) has the **double
root***
l *= 1, which by a
limiting process already points towards a logarithmic
discontinuity at . We learn more details regarding the thus
arising **spherical
functions of the second kind** *Q*_{n} as in the case of the
functions of the first kind by a **generating function*** (C. Neumann)

Accordingly, the** ***Q*_{n}(*h*)** **are defined as
expansion coefficients of 1/(*h
- z*) for the *P*_{n}, i.e., by the integral representation (F. Neumann)

This formula has the meaning that one should
escape the discontinuity location *z *=* h *of
the integrand into the complex regime, to the right as well as to
the left, where the placed ahead, but in the sequel omitted
symbol *A*.*M.* point out that one has to take the
arithmetic mean of the values of the integral obtained (identical
its so-called **principal
value**). The fact
that this side-tracking at the borders is not possible, causes
the logarithmic singularity, already referred to earlier.

In view of the symmetrical form
in *h, z *and *Q, P *of the definition (33)
is to be expected that *Q*_{n}(*h*) satisfies the (in *h *rewritten) equation (24); however, this
can also be shown directly as follows: Write (24) in the form

and understand by *L*_{h}{*Q*} the analogous expression,
formed with *h* and *Q*; moreover confirm the
identity

Then one concludes from (34) that

and by two
integrations by parts, during which the terms, arising from the
limits , vanish due to the factor 1 - *z *² in *L*_{x }:

It is now easy to
compute, from (34), the first *Q*_{n}(*h*) from the known *P*_{n}(*h*), whence, for |*h*| < 1

The general rule (due to Christoffel

where *P *is a polynomial of
degree *n *- 1, which is composed by addition out of all
those *P*_{n-2k-1 }with
non-negative subscripts. Finally, there arises from (34) by *m*-fold
differentiation with respect to *h *and multiplication
with the factors (1 - *h*²)^{m/2}, analogous to as generalization of (34)