This chapter serves more to complete our mathematical equipment than the solution of new physical problems.
Among the most necessary equipment of the mathematical physicist are the **cylindrical
and spherical functions**. We will not treat them here in a mathematically
abstract manner, but by means of simple physical concepts. For the spherical functions, we shall link the work to **potential
theory**, from which it originally arose, for the cylindrical functions to the **wave
equation** and its simplest solution, the **monochromatic
plane wave**.

**4.19 Bessel and Hankel functions:****
**We assume in the wave equation (2.7.4)
that the time dependence is purely periodic, most conveniently in the form

We write with

(2.7.4)
for one or two dimensions retaining the function symbol *u*

We integrate (3a) by

By our choice of the negative sign in (1), the first of these equations is a plane wave advancing
in the positive *x*-direction, the second one is a plane wave advancing in the negative *x*-direction. The fact that
it is more convenient to operate with the wave, advancing in the positive *x*-direction, was the main reason for the choice
of the sign in (1). In the two-dimensional case, (3b) with equal choice of sign in (1) yields

Introducing polar co-ordinates

we obtain

This is a plane wave, advancing in the direction *j *=*
a*, where naturally we obtain again (4) for *a**=*0.
Using these plane waves, we can construct the general solution by summation (integration) with respect to *a**
*with coefficients *A*, which then can depend on *a.*

In terms of *r*, *j, *Equation
(3b) becomes

or also, what is the same thing, with *r **=
kr*,

We now seek solutions of this equation in the form

where the the symbol *Z *denotes a cylinder
function. In order to obtain it, we set in (5)

and integrate with respect to *a *between
suitable limits *b *and *g*
:

This equation does not represent, like (5), a
plane wave with the direction of advance *a*, but a **bundle****
**of such waves which extends from *a*
= *b *to *a *= *g *and
as such obviously also satisfies the differential equation (6a). We will now arrange it so that (8) assumes the form (7), i.e.,
split off the factor *e*^{in}^{j},
whence we set

and transform (8) into

However, the factor *e*^{in}^{j}
is then and only then a pure function of *r*,
as is required
by (7), if we artificially again suppress the dependence on *j**
*in* w*_{0}* *and *w*_{1}. This
happens according to (8a) for an arbitrary *n* by letting *b
*and *g *,
i.e. also *w*_{0}* *and *w*_{1}, in some manner grow to infinity. However, in order to be
able to do so, we must first examine the convergence of the integral (9) at infinity (Fig. 18). Obviously, one must find those
regions of the complex *w*-plane, in which *i**r *cos
*w* in the exponents of (9) has a **negative real part**.
To start with, we assume that *r *is
real and positive and set

**w = p + iq, ****i.e.****,**

Thus, for *q* > 0, for the upper *w*-half-plane, one must have

and for *q* < 0, the lower half-plane,

(mod 2*p , *say **modulo
2****p**,
means displacement of the given interval by complete multiples of 2p
in the positive or negative direction).Since the conditions (10a, b) only concern the real part *p* of *w *and do
not depend on *q*, the domains under consideration are parallel to the imaginary axis. In Fig. 18, the strips, accessible
for our bounds *w*_{0}* *and *w*_{1}*, *are shaded and form a chess board pattern.

If *r *is not positive real, but complex, say |*r*|
*e*^{i}^{q}, then the pattern
as such is conserved and displaced only by ±*Q *to the real *w-*axis,
where the + and - sign applies to the positive or negative imaginary *w*-half-plane. In fact, one must then only replace
during the execution of these convergence considerations sin *p* in (10a, b) by
cf. start of the hint of Exercise 4.2.

For every choice of the bounds *w*_{0}, *w*_{1}, which satisfies these conditions,
the factor of *e*^{in}^{j}* *in
(9) is a possible form of the **general cylinder function ***Z*_{n}(*r*),
introduced in (6), and satisfies, as can be seen by substitution of (7) into (6a), the differential
equation

**4.19.1 The Bessel
function and its integral representation:**** **We
select as a first special possibility

In Fig. 18, the associated integration path is denoted by *W*_{0} ; the corresponding
cylinder function is the **Bessel function**,
if the yet undetermined factor *c*_{n} in (9) is normalized as follows:

With the standard notation
*I*_{n}, one thus has

In the English literature, *J*_{n} is used instead of *I*_{n}*
*and *I*_{n}(*r*) = *J*_{n}(*i**r*).
We will reserve here *J* for **Intensity** and
will not require s special symbol for *I*_{n}(*i**r*).

The normalization (13) has been chosen so that, on the one hand, *I*_{0}(*r*)
is equal to 1 for *r *=0 and, on the other hand, *I*_{n}(*r*)
is real for any real *n* and *r*. The first follows
directly from (14), when one ensures that on transition to the dashed lines with rectangular form of *W*_{0
}in Fig. 18 the two (for
otherwise divergent) partial integrals, taken along the imaginary *w*-axis, compensate each other. You recognize this by
making the also otherwise useful substitution *w* - *p*/2
= *b*. When you specialize the integration path *W*_{0}
again to the rectangle already referred to, i.e., in terms of *b,*

then this path lies symmetrically with respect to the imaginary *b*-axis:
*I*_{n}(*r*)
decomposes itself for real *r *and *n *into the two real
components

the second of which arises from the two paths by
the further substitution *b *= *p *± *i**g**
*and, in general, does not vanish as was the case for *n *= 0. Hence, *I*_{n}(*r*)
is indeed also real for the in (14) introduced normalization, when *r *and
*n *are real.

Since our integral representation (14) converges for all values of *r*,
it follows that *I*_{n}(*r*)
is a function which is **everywhere regular and transcendental **and
has at infinity an **essential singular point **and
for *r =* 0 a **branch
point** of order *n*, which for negative *n* is at the same time an
infinity of the same order.

Let *n* be an integer, which we did not have to assume so far, then there disappears in
(15) the second term on the right hand side and one has

If one goes here from the exponential to the trigonometric functions and takes into consideration that sin *b*
is an odd , cos *b *an even function, one obtains a
representation which already was given by Bessel himself, i.e.,

Naturally, one has also the corresponding result from the initial integral in terms of *w*:
On the path *W*_{0}, for integers *n*, the two
branches, parallel to the imaginary axis, cancel each other and there remains only the path along the real axis of the *w*-plane
from - *p*/2 to 3*p*/2,
which one can also, thanks to the now periodicity of the integrand,
replace by the integral from *-p *to *+p*.
Thus,

which agrees with (16). Compared with these real representations, our integral (14) with its complex path *W *has
the great advantage that it is not limited to real *n*, but remains valid for arbitrary *n*. The integral (14)
appeared first in Schläfli's work of 1871, but only in the rectangular shape of the path of integration.* The following integrals
(22) were published by the author in 1896.

* Details are given in G.N.Watson: A Treatise on the Theory of Bessel Functions, Cambridge 1922, pp. 176 and 178.

Since the differential equation (1) only depends on *n*², *I*_{n}*
*as well as *I*_{-n}* *are solutions. Its general
solution then becomes

which, however, applies only when *n *is **not an integer**.
When it is an integer, *I*_{n}* *and *I*_{-n}*
*are not linearly independent, in fact, then

This follows directly from (16) by substituting* **b
*=* p - b* ' in *I*_{-n}(*r*).

**4.19.2
The Hankel Functions and their Integral Representation:**** **As
second and third possibility, we select now for the integration limits in (9) instead of (12)

The corresponding paths, which again in their asymptotic progress are tied to the shaded fields and otherwise
are arbitrary, have been denoted in Fig. 19 by *W*_{1} and *W*_{2}. The fact that we have placed
them through the points *w* = 0 and *w* = *p*,
respectively, is also arbitrary., but convenient for later work. In fact, the paths *W*_{1}**, ***W*_{2}*
*could be, as it happened in the case of *W*_{0}, set in the finite region over the un-shaded part. Instead of
by (13), we give now the constants *c*_{n} by

The thus **generated cylinder functions**
are called **first and second**
**Hankel functions**

For the mathematical physicist, they are almost more important than the Bessel function *I*_{n}.
They differ from these by becoming infinite for *r* = 0 also for
positive *n. *Indeed, the two integral, which result in (22) for* **r*
= 0

diverge at infinity in the negative, imaginary *w*-half-plane.

We will discuss the singularities which arise in *H*_{1} and *H*_{2} for *r*
= 0 in 4.19.3. By their origin, *H*_{1}
and *H*_{2} are naturally again solutions of the differential equation (11),
whence we could write the general integral of (11) in the form

We will demonstrate now that it yields our particular integral *I*_{n }of the same
differential equation when we set

This requires only one look at Fig. 19: The two paths *W*_{1} and *W*_{2},
followed one after the other, cancel each other in their lower branches and contract therefore into the path *W*_{0},
which yields, in view of the present definition of *c*_{n }in (21), twice as much as the earlier
definition (13). Thus, indeed,

Side by side with this, we consider the difference
which, written in the integration variables *b *and *g*
of (15), turns out in the case of real *r *and *n* to be
purely imaginary. We call it 2*iN*_{n} and call *N*_{n}(*r*)
the **Neumann function**

Equations (24) and (25) yet yield

In this decomposition of *H*^{1,2}, we recognize a complete analogy to the following
decomposition of the exponential function into its trigonometric components:

We will see in 4.19.4, that this analogy is not only qualitative, but it is also asymptotically (for ) quantitative. Just as for all vibration processes, one prefers the exponential imaginary representation. We will, as a rule, prefer the representation by Hankel functions to that by Bessel or Neumann functions, especially our complex integrals yield for those an equally convenient expression as for these.

Apart from in the form (26), when we assume *n* to be not an integer, *H*^{1},
*H*^{2} must also be contained in the form (19). We will find now the corresponding coefficients *c*_{1},
*c*_{2}. We consider for this: By (14),

and, correspondingly, for *l *= -*n*,
if we immediately exchange *w* for* -w *and therefore *W*_{0}* *for - *W*_{0}*,*

In
Fig. 20, The two paths *W*_{0} and *-W*_{0} have been entered for the sake of clarity in rectangular
form with indication of their sense of coverage. They cancel each other in their central part between *w=*-*p/*2
and *w=*+*p/*2. The remainder are two rectangular paths,
which we have again, for the sake of clarity, deformed into two segments like the path *W*_{2}* *in Fig. 19.
The path on the right hand side leading from agrees
completely with *W*_{2}, the one on the left hand side, from ,
could be denoted by *W *'_{2}. Hence, addition of (27) and (28) yields

Here, by (22), the integral along *W*_{2} equals

the integral taken along *W *'_{2} differs from it only by its direction and
displacement of its path by -2*p*, whence, again by (22), it is

Substitution of (29 a, b) into (29) yields after suitable cancellations

By (24), the corresponding representation of
*H*^{1}* *is

Hence, the coefficients *c*_{1}, *c*_{2}* *for the two Hankel functions in
(19) have been determined. We note now that when *n *is real and *r *is
complex

The symbol * means, as elsewhere, the transition to the conjugate values.
During the derivation of (32) from (30) and (31), we have employed, for example, the relation *I*_{n}(*r**)
= [*I*_{n}(*r*)]*. Moreover, one
concludes from (30) and (31)

We also note the formula, which follows from (25), (30) and (31):

**4.19.3 Series expansions at zero:****
**We have seen that* I*_{n}(*r*)
is everywhere in the finite region a regular function, whence it can be developed in terms of increasing powers of *r**.
*Indeed, it is readily seen that the differential equation (11) is satisfied by the
series

It assumes for *n* = 0 the beautiful form

which was already known to Fourier. We will prove in Exercise 4.1 that these series agree with our intergral representation (14).

In order to obtain the series for the **general
cylinder function** *Z*_{n}* *and to study
simultaneously the **singularity** for *r*
= 0, we follow the rules, developed for ordinary linear differential equations : We
start from

and enter it into the differential equation (11),
when every single term of the resulting power series must vanish. The lowest power *r*^{l-2}
then yields
the equation for *l*, the general
term *r*^{l+}^{k-}^{2
}a recursion relation for *a*_{k}, whence

and hence

which, by (37), simplifies to

Repeated application of this recursion formula yields for *k *= 2*m*

,

If one sets, in order to achieve agreement with (34), *a*_{0} = 1/2^{n}*G*(*l**
+ *1) and, moreover, *a*_{1}* *= 0, one finds

This confirms the expansion (34). By (37), it also is valid for *l *=
-*n *as well as for *l *= +*n*. Thus, we obtain
besides *I*_{n}(*r*),
as has already been emphasized,
as second solution of (11) *I*_{-n}(*r*);
while that series for a positive real part of *n* vanishes for *r *=
0 like *r*^{n}*, *this
series becomes **infinite** like *r*^{-n}*
*for *r*=0. Thus, one has as general solution of (11)
as counter part of (23) the former formula (19).

However, everything said here so far is only correct for **non-integer**
*n. *If *n *is an** integer****,
**or, said more generally, if the difference of two roots of (37) is an integer*, then there arise in the solution,
belonging to the smaller value of *l*,
complications, known from the general theory of linear differential equations, namely there occur, apart from powers with negative
exponents, also **logarithmic terms**. We
demonstrate this in the present case quite simply as follows:

* This is the case
with Bessel functions when *n *is half an integer. We will explain in 4.21.3
that here nevertheless the complications discussed above do not arise.

In (37b), set *l *= -*n* and *k = *2*n*,
so that the first term vanishes and we find: *a*_{2n-2} = 0. Hence, if we follow the recursion backwards
from *a*_{2n}, then there vanish in the series (36) for *Z*_{n}*= I*_{-n}*
*all terms ahead of *a*_{k}* = a*_{2n}*, *whence follows, as from the
series (34), the already known link (19a) between* I*_{n}*
*and *I*_{-n}* *.

Our concern is now to find a second solution of the Bessel equation (11)
which is really different from *I*_{n}. We are aided in this endeavour by a limit argument in which *n*
is taken to be arbitrarily less than a positive integer. We shall rather apply this instead of to the Hankel function *H *immediately
to the Neumann function (33) in which we are mainly
interested for the determination of the singularity of concern. Prior to the execution of the limit, it obeys (33), in the limit,
due to (19a), it assumes the form 0/0. The limiting value itself is determined by the rule of de l'Hospital (1661 - 1704), which
yields, denoting temporarily the integer *n *by ,
for the denominator of (33)

and for the numerator

Here, the limit indicates that one must execute for non-integer *n* the differentiations
with respect to *n* prior to taking the limit. Since we are in this case interested above all in *r
*= 0, one employs naturally the series (34),
which we know to be valid not only for* I*_{n}, but (for non-integer *n*) also for *I*_{-n}
. We will evaluate individually the two components of the right hand side of (39).

Since we know that

the first term of the series (27) yields

where we have indicated by + · · · the contributions from the following terms of the series, which vanish at higher order than . Moreover, we employ Gauss' abbreviation

where *C* is Euler's constant

By setting

we can, by (41) and (41a), write the content in {} in (40) as

On the other hand, the factor ahead of {} in (40), apart from terms of higher order than , equals Now, due to (41b), we can rewrite (40)

Here · · · indicates that (42) only claims to be accurate in the terms including *r*^{
n}.

Somewhat different is the computation for the second term on the right hand side of (39). We start from

which yields, differentiating first only the factor (*r*/2)^{-n}
with respect to *n*, as with (40),

Since all the *G *become infinite except in the
last term, this becomes for

Otherwise, differentiation of {} in (43) yields

In (44a), two minus signs have compensated each other. In fact, *z* = *-n *+ 1, -*n* + 2

Now,*Y*(*z*)
like *G*(*z* + 1) has simple poles
at *z* = -1, -2, -3, ···. For the neighbourhood of the *n*-th
of these, we have, by (41),

The expansion of *G*(*z* +
1) there is

whence

Regarding this work and the earlier formulae, cf. Jahnke-Emde, pp. 11, 10 and 18.

In contrast, in the neighbourhood of *z* = 0, due to *G*(1)
= 1 and *Y*(0) = -*C,* one has

After these preparations, we can perform the limiting process in (44a). All factors, apart from the last factor *Y*/*G
, *have, by (45) and (45a), the form and are to
be replaced, following (45b), by (-1)^{n}(*n
*- 1)! with

while in the last written out term (45c) takes over and yields

Thus, we obtain the limiting value of (44a) (instead of
we can write from now on *n*, where it is in future an integer):

The sum of (46) and (44) yields now as second term in {} in (39)

Together with (42), one obtains finally
as value of (39) for *n *> 0

The terms on the right hand side have been arranged according to their magnitude, the term with (*r*/2)^{-n
}is the term of highest, the logarithmic term of lowest order. Hence, for *n* = 0, one has a simple logarithmic
singularity; in fact, then

or, as we tell without proof, written out more fully

By (26), this singularity appears, as in *N*,* *in the imaginary part of the two *H*. Hence
we conclude that the *H*_{n}* *do* *not have only branching for non-integer, but also for
integer *n* at the origin of the complex *r**-*plane.
In fact, during one circuit about the origin, they increase, according to (26) and (47), by .
More details are given in Exercise 4.2. In Exercise
4.3, you will derive the logarithmic singularity of *H*_{0}* *by a more direct,
bat mathematically less satisfying method.

**4.19.4 Recursion Formulae:**
Just as the *Z*_{n}(*r*)
in their dependence on *r* satisfy
a **differential equation**, in their
dependence on *n, *they obey a **difference
equation**, both for any, also non-integer *n*. We draw this conclusion
from our general integral representation of *H,* and from it the validity of the corresponding formulae for arbitrary
linear combinations of the *H*, especially also for the *I *and *N*.

With reference to the fact that in (22)
the integration paths *W*_{1} and *W*_{2} did not depend on *n*, we form

where

Hence, we can write the integrals on the right hand sides of (49) and (50)

and convert (49a) by integration by parts into

With the aid of these transformations, we can now express the right hand sides of (49) and (50) directly by the
Hankel function with subscript *n*. However, we will write at once the corresponding formulae, which, depending on the
choice of the integration path, apply to *H*^{1} and *H*^{2} for the general cylinder function *Z*
as a combination of these two functions. These recursion formulae
are

They apply independently of whether *n *is an integer or not, positive or negative.

For *n = *0, we have then the special cases

and, by further specialization of (52a), the relation, which can be taken directly from the series (27) and (27a)

**4.19.5 Asymptotic
Representation of the Hankel Functions:**** **The
integrands of our representations (14) and (22) oscillate with increasing amplitude ever faster in the not shaded regions of the *w*-plane,
while their amplitudes there tend to zero. For real *r*,
the paths *W*_{1} and *W*_{2}* *of the functions *H*^{1} and *H*^{2}*
*can be executed solely in the shaded region, as has already been pointed out with Fig.
19. The fact that this is no longer the case for complex *r,*
show the figures of Exercise 4.2.
We also see from Fig. 19 that the points *w *= 0 and *w *= *p*,
which the paths *W*_{1} and *W*_{2}, respectively, pass between two unshaded regions, have a special
role for the asymptotic computation of *H*^{1} and *H*^{2}*, *respectively.

We shall demonstrate here the **saddle
point method**** **by our
example as clearly as possible, so to say, topographically, and consider the analytic refinements and generalizations in 4.21.
We will assume that

For *H*^{1}*, t*he path *W*_{1}
begins and ends in the **shaded low land **just
as for *H*^{2}* *the path *W*_{2}.
The decisive exponent *i**r*
cos *w* has an extreme value at

The extreme value, as is always the case with the real or imaginary part of a complex function, is not a real
maximum or minimum, but a **saddle point**. To the right
and left hand side rise near there rapidly mountain ranges. In between them, *W*_{1}
and *W*_{2} run as passes between the mountains. The height of the mountain pass at *w* = 0 and *w* =
*p* is

How does a mountaineer choose his way in order to get over the pass as quickly as possible? On the
steepest incline and decline, the so-called **fall lines**.
However, this prescription is not by itself obligatory and a detour can be followed for comfort's sake (analytic or mountaineering
considerations, cf. G. Faber, Bayr. Akad., 1922, p. 285). This is why Sommerfeld does not like the English term **method
of ****steepest descent**
instead of **pass method**.

We consider a short section of the path *W*_{1} in the neighbourhood
of the height of the pass; let *ds* be the arc element of the path, measured in the direction of *W*_{1}, *s
= *0 the height of the pass.
We write

Since the level lines of the real part are perpendicular to those of the imaginary part just as the fall lines to the level lines, the level line of the imaginary part is at the same time the required fall line of the real part, which determines the height of the passage. The level line of the imaginary part of (54), here of interest, is given by

with Const = 1, since the path through the pass height is to be *s* = 0; hence we conclude

For *H*^{1}* *, we must select the upper sign of *g*
(Fig, 19), whence (54)
becomes

We substitute this into (22) and set at the same time *s *=
0 in the **slowly changing **factor
exp{*in*(*w*-*p*/2)};
the integration can obviously be restricted to the immediate neighbourhood of the pass, say, to a distance s < *e*.
Thus, one finds

With , for which the integration limits become , the integral becomes Laplace's integral, whence we have the final result

In the case of *H*^{2}, where the saddle point lies at *w* = *p*,
one must use the path *W*_{2}* *and select in (54a)
the lower sign of *g* to obtain,
correspondingly,

Ir follows for half the sum of (55) and (56) that

While these asymptotic representations have been derived under the assumption of
real *r* , they can be extended
analytically into the complex regime of the *r*-plane,
whereby one has to imagine that in the representations of the two *H *functions, because of the branching at the end of the
section *C*, these must be cut along a suitable half-ray. We now state, on the basis of (55) and (56): *H*^{1}*
vanishes asymptotically in the positive imaginary , H*^{2}* in the negative imaginary**
r **-half-plane.* The
special suitability of the Hankel functions for problems of damped vibrations rest on this. In contrast, the Bessel functions *I
*(as well as the Neumann functions *N*) become infinite asymptotically in both half planes.

We will show in 4.21
how our asymptotic limits can be transformed into asymptotic series and how they change when we drop the condition (53). The
factor in (55) and (56) is linked to this, so that *H*^{1}_{0}^{
}(or in case of another choice of the time dependence *H*^{2}_{0}) speaking in terms of space, i.e.,
by inclusion of a *z*-co-ordinate, perpendicular to the *r*,*j*-plane,
represents an **advancing cylinder wave****
**with source at *r* = 0. That the energy, which flows through every *r*
cylinder, is measured by 2*p**
r*|*H*_{0}|² and (in the absence of absorption) must not depend on *r*, follows indeed: *The
proportionality of H*_{0 }*with r *^{-1/2}* or, what says the same thing, with **r*^{-1/2}*.*

In my lectures, I showed cardboard models of the real and imaginary parts of *H*^{1}_{0
}as well as of *I*_{0}, which had been built as surfaces over the complex *r*-plane
and showed the contours of suitable, inside each other located plane sections. The surface for osculates
the positive *r*-half-plane
exponentially and displays towards the negative half-plane exponentially rising mountains, intersected by correspondingly
themselves deepening valleys. The surface for displays a
similar behaviour and , besides, a small funnel at the origin, which corresponds to the logarithmic singularity of *H*_{0}
[ equal to that of *N* in (48)] as well as a discontinuity along the real negative axis [corresponding to the branching at
that location which has already been discussed].
consists of a flat, wavy valley, flanked on both sides by
rising mountains. The waviness of
the valley follows from the asymptotic equation (57) and indicates infinitely many roots of the equation *I*_{0} =
0 along the real axis. These roots are shown in Fig. 21. The surface of
has a quite similar appearance, except that in its valley the bottom is throughout horizontal, corresponding to the fact that *I*_{0
}is along the real axis.

**4.20 Heat Compensation in a Cylinder:****
**We will now consider another heat conduction problem which is an excellent example
of the application of Bessel functions and was already treated by Fourier. In fact, these functions for integer *n* already
occur in Fourier's work, for which reason one occasionally speaks of Fourier-Bessel
functions.

We will treat the problem in three stages:

A. An infinitely long cylinder and axial symmetric initial state *f*(*r*).

B. The initial state *f*(*r, **j*)
depends also on *j.
*C. A cylinder of finite length and a general initial state

For the sake of simplicity, we will impose throughout isothermal conditions

**(1) u = 0 ··· for ****r**** = ****a
= ****radius of cylinder mantle**

Moreover, one has for the complete cylinder as, so to say, a further **boundary
condition **that of finiteness along the cylinders axis:

**4.20.1 One-dimensional case ****f
= f ****(****r****): **The heat
conduction equation is

With

we obtain for *R* the differential equation

This is Bessel's equation (19.11)
with *n = *0* *and* **r* = *l**r*.
We write its general solution in the form

However, we must set *B *= 0 due to the finiteness condition (1a); moreover, due to (1), we must demand

As we know already, this equation has infinitely many roots, which follow each other asymptotically
at the distance *p *;* *the *m*-th root is, by (19.57),

This approximation applies down to *m *= 2 with an approximate
accuracy of 1%; for *m* = 1,

compared with 2.36 by (4a) (Fig. 21).

Hence (3a) places infinitely many solutions at our disposal:

Correspondingly, we have, instead of (3), as general solution of the problem

Now, we need only satisfy the initial condition

One way to the solution shows us the treatment of the an-harmonic series in 3.16. In order to make the complete analogy of Equation (6) there, we set

and write the present equation (3a) in the form

Hence, multiplication by *u*_{m}* *and *u*_{n},
respectively, and subtraction yields an equation. analogous to (16.5a),

Here, the left hand side is a complete differential quotient. Integration over the basic region
0<*r*<*a *yields therefore, as analogue,

This is simultaneously **Green's theorem****
for the two-dimensional region of the circle** *r* = *a*.

Now, in (7a), the right hand side vanishes; in fact, for *r* = *a*, due to (1),
for* r* = 0, due to the factor *r* and Equation (1a). Since

there follows now the **orthogonality
condition**

The **weight
factor**** ***r *obviously
points to the two-dimensional area element *rdrd**j**
*in Green's theorem.

However, we can also take from (7a) the **normalization
integral**

if we drop there the assumption that *l*_{n
}is to be a root of (4). Rather, we will view *l*_{n
}as being a continuous variable and let it coincide in the limit with *l*_{m}.
By (7a), *N*_{m}* *is then represented by a fraction, which assumes for
the form 0/0. Differentiating the fraction's numerator and denominator with respect to *l*_{n}*
*and substituting *r *= *a *and* r *= 0 yields
then, by (1):

However, for *r* = *a,*

On substitution, this yields

Equations (8) and (9) now yield the coefficients *A*_{m}* *of of the
series (6) in Fourier fashion

which we need only substitute in the series (5) in order to complete the solution of Problem A.

**4.20.2 Two-dimensional
case ****f = f****(****r, ****j****):****
**To start with , develop (best in complex form) the Fourier series (1.12)

By (2),

and the expanded substitution

the functions *R*_{n}(*r*) must satisfy

This is Bessel's equation (19.11)
with *r *= *l**r.
*Since, due to (1a),* *the only
solution of interest here has the form *A*_{n}*I*_{n}(*l**r*).
Due to (1), *l
*must satisfy *I*_{n}(*l**a*)
= 0, which, like *I*_{0}(*l**a*) = 0, has
infinitely many roots

Each of these yields a particular solution of the form (13)

each of which satisfies (12). The general solution of (14). which meets the boundary conditions, is composed of them by superposition:

We still must determine the coefficients *A*_{n,m} so that for *t *=
0 and every integer

where, by (11), the left hand side is a known function of *r*. The last equation demands development in
terms of the Bessel functions *I*_{n}. This is possible due to their **orthogonality**,
which follows as in (7) and (7a) from the Bessel equation (14) or from Green's theorem.

During the application of Green's theorem to the circle *r = a* in the *r*, *j*-plane,
in order to avoid the trivial result 0 = 0, one must employ the complex conjugate functions

With the abbreviations

one has the generalization of (7)

Here, as well, the right hand side vanished. For , we now have

At the same time, one obtains for by the limit (9)

With (18) and (19), we can now find *A*_{m} in (10) for a given *f*(*r*).
Substituting (11) for *C*_{n}(*r*), one finds

This completes the solution (16).

**4.20.3 The Three-dimensional
Case ****f ****= ****f****(****r, ****j****,
z****):**** **Let
the cylinder have finite length and 0<*z*<*h*. We first expand* f*(*r, **j**,
z*)* *in a Fourier series with respect to *z*, which, due to the boundary condition *u*=0 becomes for *z*=0
and *h* a pure sin series:

next, we expand *B*_{m}* =
B*_{m}(*r, **j*)
in a series involving *e*^{in}^{j }:

Finally, we must expand *C*_{m, }_{n}*=
C*_{m, }_{n}(*r*) in a
series of the Bessel functions *I*_{n}(*l**r*),
which advances according to the roots of

Based on the once more extended heat conduction equation

,

the time factor is

Hence, the complete solution is a threefold infinite sum

The coefficients *A* are determined by (10) from *N *by the correspondingly expanded scheme (10),
(20), respectively:

This concludes the solution (23).

For a hollow cylinder, due to the disappearance of the finiteness
condition (1a) for *r* = *a*, there can appear beside the functions *I*_{n}
and *N*_{n}* *(or written symmetrically instead of *I*_{n} and *N*_{n}*
*the two functions *H*^{1}_{n} and *H*^{2}_{n}); as an
application, one could deal with the heat flux through a **heating
tube**.

**4.21. More about Bessel functions:**

**4.21.1 Generating Function
and Addition Theorems:**** **In
4.19,
we have started from the two-dimensional wave equation *D**u
+ k*²*u* = 0 and its simplest solution, the plane wave

If we expand this in a Fourier series, the coefficients, corresponding to the origin of Bessel's equation (19.11),
must be Bessel functions, where due to the regular nature of (1), only the *I*-function can enter for *r* = 0. Thus,
we set the coefficient of *e*^{in}^{j}*
*equal to *c*_{n}*I*_{n}* *and have, by (1.12),

If we compare this with (19.18),
where one can obviously interchange *w* and -*w*, we find

Thus, we have the Fourier expansion

or also, with *y *= *j *+ *p*/2,

In older treatments, you may find instead of (2) the less symmetric form

Accordingly, one calls the left hand sides of (2) or (2a) the **generating
function of the Bessel functions with integer subscript****.**

We now proceed from the plane wave to the **cylindrical
wave** with its logarithmic source at the origin, which, by 19.5,
is represented by *H*_{0}(*r*).
(We omit here the superscript, because the following applies to both *H *functions, i.e., the radiating and absorbing wave
type). However, we will shift immediately the zero point from *r *= 0 to
*r* = *r*_{0}, j = j_{0}, whence *H*_{0}(*r*)
becomes

If we expand again in a Fourier series of multiple values of *j
- j*_{0}, its coefficients must again be cylinder
functions and, indeed, *H*_{n}(*r*)
for *r *< *r*_{0}.
The latter follows from the fact that the point *r*=0 is
a regular point, the former that an individual term of the series for
must have the same radiation- or absorption-type as *H*_{0}(*r*)
itself. The same argument also applies for reasons of symmetry to the dependence of the coefficients on the variable *r*_{0},
only with interchange of the functions *I*_{n }and *H*_{n},
since the condition is the same as
The *n*-th Fourier coefficient must then be

The added factor *c*_{n}* *does not depend on *r**
*and *r*_{0}* *and
is the same in both expansions, because both of them must become each other continuously at *r**
*= *r*_{0} (unless
simultaneously *j *= *j*_{0}*,
*when both series diverge); it turns out directly to equal 1, if one performs for *r*
< *r*_{0} the
transition to the plane wave , sets *j*_{0}=
*p *and compares the arising
asymptotic formula with (2). Thus we arrive at the **addition
theorem**

If we imagine the same written down for **both **Hankel
functions and take half the sum, we obtain the **Addition theorem for the Bessel
function**

In the same way, one obtains the correspondingly formed half difference of the addition theorem for the Neumann
function, where once again, as in (3), one must distinguish between *r**
> **r*_{0}*
*and* r**<**r*_{0}*.*

With (3), note that the series advancing with *I*_{n} corresponds in the theory of
complex functions to the **Taylor expansion**,
that with *H*_{n} to the **Laurent
series**.* This becomes clearer with the
following example, where one can imagine *z* and *z*_{0} being
replaced by :

We will derive the corresponding addition theorems for spherical waves in space in 4.24; also the representation (2) of the plane wave will find there a match in space.

* In Section 2 of the already cited author's
work, this is also extended with respect to the question of convergence. We also refer the reader to a
large paper of H. Weber , Math. Ann. Vol. I, p. 1, with which this journal was started. It is concerned with the transfer of the
methods of Riemann's dissertation, i.e., of the theory of the differential equation *D**u
*+ *k*²*u = *0

**4.21.2
Integral Representations in Terms of Bessel Functions:**** **We
will derive an expansion, analogous to the Fourier integral,
of a given function *f*(*r*). Following (12.11),
a function of two variables can be represented by the Fourier integral

We introduce polar co-ordinates

For *f*(*x, y*), we introduce the dependence on the angle *j*

With the aid of

(5) transforms into

In order to compensate the integrals with respect to *a *and
*y *of (19.18)

we multiply under the last and last but one integral of (7) by the factors

and compensate for their reciprocal product by adding

Then *e*^{in}^{j }cancels
with the same factor on the left hand side and *e*^{-in}^{y }compensates
against *e*^{+in}^{j} on the
right hand side. Thus, we arrive at the simple representation

Analogous to the formulation 4.13 of Fourier's integral theorem, we can also give this relation a symmetric form

During the transition from (5) to (7), the transformation of the double integrals from rectangular to polar
co-ordinates is linked to certain conditions regarding the behaviour of *f* at infinity, which, however, we will not
consider here. Equation (7) will become useful in 4.24
during the discussion of the spherical wave; we have already used it in Volume II, 27 during the presentation of hydrodynamic ring
waves.

A further application for (8) occurs when we let there *f*(*r*) degenerate into a *d*
-function and, indeed, in the following sense: Let

where, however,

We specially draw attention to the weight factor in the integral in (9), which is characteristic for two-dimensional cylinder problems. It has the consequence that that now, not as before,but, as in (9), .

This equation expresses the **orthogonality **of
the two functions *I*_{n}* *at two locations *r* and *s *of the continuous range of
values ; it takes a place side by side with (20.18),
which deals with two locations *m* and *l *of the discrete *l*-series.
We will return to this important relation (9a) in 6.36.