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 Du = 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' = (e'/e), i.e., by

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 relationship 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 P1, P2, 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 OPkQ in Fig. 22 is similar to the triangle OQ'Pk, i.e.,

We have used here Pk 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 from (1) follows for every point Pk and conclude that

(the last with respect to (3)).

4.23.2 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, 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 -1/4p . 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 The 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 = aRk/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' = -1/4p 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/a in front of Integral (9) vanishes and with it the contributions of all area elements ds for which there does not not vanish simultaneously the denominator R. Hence, it depends only on the environment of the 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) in the form

In the numerator of the last integral, we were able to replace by Q dQ dF, 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 will now seek 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') = (r/a)). 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 Du. 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 tells us, 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) 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 rDu, 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 Du = 0 in the co-ordinates x y z, the function v=ru, transformed by reciprocal radii, satisfies the differential equation Dv = 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 space S 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 60 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 we also deal especially 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 expand 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 for the simple wave equation

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

Only in the potential equation (k = 0) drops out the disturbing factor (a/r')4. In the wave equation, it means that the initially homogeneous medium (k constant) appears converted into a high degree non-homogeneous 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 yn(r) of 4.21.3. This follows, on the one hand, from the wave equation, on the other hand, from the differential equation for Pn. 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 Pn and a function R(r) which only depends on r. Due to Pn'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 yn 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 a linear combination of the functions z1,2(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 cn. They follow from the orthogonality of the Pn. 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 relationship between yn and In+, 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 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 In, we can call the three-dimensional plane wave the generator of the yn. At the same time, (6) with the value (6c) of cn yields the integral representation of the yn

which obviously 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 in the positive r-direction and radiates, if we give it the time dependence e-iwt. 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 Pn(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 r0, the inverse applies for the dependence of r0. Hence there must appear as factors in the coefficients of Pn(cos Q)

so that the expansion becomes

The numerical factors cn must be the same in both series, because both of them must transcend for r = r0 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 cn can again occur by the limit , when

Thus, there arises on the left hand side of (9), employing -ir0 cos q instead of +ir0 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 z 2 on the right hand side 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 go in the differential equation 4.22.13 of the associate spherical functions to the limit

we find

This is the differential equation (19.11) of the cylinder function Zm. Since Pmn, and therefore also Om are finite for , only the Bessel function Im can be a solution of (11a), whence

This follows from the fact that for m = 0 and h = 0 , on the one hand, I0(h) = 1, on the other hand, by (11), since h = 0 also i.e., and therefore also O0(h) = 1. In order to determine Cm 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 taken 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 2m 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 Im(h) by the first term of its power series (19.34), yields

Thus, if m > 0, one must divide Pmn prior to the execution of the limit by nm, in order to obtain directly Im.

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 Pmn eimj, changes in the process into the solution of the wave equation for the tangential plane, namely, into Im (h)eimj, in as far one subjects at m > n not Pmn itself, but Pmn/nm 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 Pmn 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


Hence, there are two saddle points w0, which under the assumptions lie on the unit circle, i.e.,


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 = eigds following from (15d), must still be multiplied 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 there are m more factors than in the denominator, similarly as in (12b), reduces to C = nm e-imp/2, i.e., the value of C/i to nm e-i(m+1)p/2. Combining this with the two exponential functions in (16c), one finds, after simple manipulations,

Hence, Pmn 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 , (17) fails, because, by (15a), f"(w0) vanishes there, i.e., the series for f(w) - f(w0) also only starts with the third term, as in the limiting case which led to Airy's integral. Here, the place of (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 electric multi-pole: We return now to the 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 h1, h2, , hn to the co-ordinates of P as to those of Q. We do the latter and after the differentiation go, as 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 h1, h2, 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 Pn. 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 P1, , P4, which can yet be completed by the zero-th derivative of 1/R , i.e., by P0=1, agrees completely with the values, arising from our original definition in 1.5 ( x is there 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 r0 = 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) by 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 speaks as a rule instead of bi-pole of 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, Included in (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 h1 and h2 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 P12.

The fact that (18) yields the complete system of the 2n + 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 the unknown exponents l and unknown coefficients ak

substitute this in (20) and set the arising factor of the starting term zl-1 as well as of the general term zl+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 a0 = 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 Pn must agree with y1 as well as y2 apart from a factor, i.e., that

which also yields the correct normalization, namely Pn(1) = 1 for z = +1. Pn. Like every hyper-geometric series with negative integer a or b Pn stops: Because a = - n, Pn 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 Pn 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 Pmn 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, yet 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 as associated differential equation from (20), 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 Pmn. By the way, both were suggested to us from the start through their relationship to potential theory.

As far as the first point is concerned, 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 Pn(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), the Pn were defined as the coefficients of a Taylor series advancing with t = r/r0 , 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 running around the point t = 0 in the same sense and ending on the positive shore at . In fact, it is clear that for this definition the differential equation of the Pn 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 Pn(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, -2p 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 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 Pn of the hyper-geometric differential equation. They coincide only when n is an integer. However, there must also exist 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 Qn.

The type of singularity can then be predicted according to 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 Qn, as in the case of the functions of the first kind, by a generating function*(C. Neumann)

Accordingly, the Qn(h) are defined as expansion coefficients of 1/(h - z) for the Pn, 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. points out that one has to take the arithmetic mean of the values of the integral obtained (identical to its so-called principal value). The fact that this side-tracking at the borders is impossible causes the logarithmic singularity, already referred to earlier.

In view of the symmetrical form in h, z and Q, P of the definition (33), it is to be expected that Qn(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 Lh{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 in Lx :

It is now easy to compute, from (34), the first Qn(h) from the known Pn(h), whence, for |h| < 1,

The general rule (due to Christoffel) is

where P is a polynomial of degree n - 1, which is composed by addition out of all those Pn-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)m/2, analogous to as generalization of (34),

last next