Ztschr. f . angew. 234 G o 1 d s t e i n , The application of Heavisi'dek operational method Math. und Mech.

The application of Heaviside's operational method to the solution of a problem in heat conduction.

Ijy Sydney Goldstein, Cambridge (England).

1. Introduction. Many problems in wave.motion and in diffusion, including the conduction of heat, are most conveniently solved by a method due originally to Oliver H e a v i s i d e I). The method has been clarified and extended by B r o m w ich2) and others, and a general account of it given by J e f f r e y s 7, in whose tract references to other publications dealing with the subject will be found.

A summary of tlie method, and reasons for recomniending it, will be given below; and it will then be illustrated by consideration of three related problems in the conduction of heat. In all three problems it is required to find the temperature a t any time a t any point of a given solid body, which a t a given instant ( t = 0) is at a given constant temperature and is placed in a medium at another given. lower, constant temperature, into which radiation takes place from the surface of the solid body. First, we shall consider the solid body to be an infinite slab, of thickness 2 1 ; second, an infinitely long circular cylinder of radius R; third, a circular cylinder of length 2 1 and radius R. The last problem formed the subject of a paper by Dr. Franz B e r g e r in the Z. f . angew. Math. u. Mech. for February, 1931 (Bd. 11, pp. 46-58), and that is why 1 have chosen to discuss it here; the first two problems lead up to this one.

2. Summary of Heaviside's Operational Method. This summary is not be regarded as complete; nor will any justification be given here for the rules quoted. The justification, and other rules and formulae, are to be found in the publications previously mentioned.

Let a variable v satisfy one of the differential equations of mathematical physics. Write the equation with all the terms on the left-hand side of the sign of equality and zero on the right. Write down the ,,auxiliary equation'' as follows. If a term t3 u/d t , where t is the time, occurs on the left, replace it by pv, and add p v o to the right side, where vo is the initial value of v. If a term d2 u/d tZ occurs on tlie left, replace it by p'v, and add p2vo +pvVo' to the right, where vo' is the initial value of dv/d t . It is assumed that sufficient initial values are given to ensure that the differential equation and the initial conditions determine the value of v for all subsequent time.

This gives a solution

And so on.

Solve the auxiliary equation for 2) as if p were a number.

Interpret the solution according to one or more of the following rules and methods. (1) If ~ ( p ) is single-valued, and is equal to f ( p ) / P ( p ) , where f ( p ) has no poles, then

9 = ( P ( p ) , say.

where the K are the zeros of F(p) , and it is assumed that they are all simple and none of them zero. The dash denotes differentiation, and the summation is over all the zeros of F ( p ) . This rule is known as the partial-fraction rule, and gives the solution as a series of normal functions.

(2) The interpretations of several of the most commonly occurring forms of ~ ( p ) are known. A list is given by J e f f r e y s (loc. cit.). Other interpretations are given by v a n d e r P o l (Phil. Mag. VIII, 1929, 861-898) and in a fortlicoming paper by G o l d s t e i n .

(3) The initial value of v is found by makingp infinite in ~ ( ( p ) . To find a form suitable for computation for s m a l l values of t , expand p(p) in the form which would be most suitable for computation of y ( p ) for l a r g e values of p if p were a number, and interpret term by term.

(4) If the system approaches a steady state, i. e. if v tends to a limit as t tends to infinity, this limit is found by letting y tend to zero in ( p ) . A form suitable for computation for l a r g e values of t can sometimes be found by expanding ~ ( p ) in a form which would be suitable for the computation of q ( p ) for s i n a l l values of 1' if p were a number, and inter- preting term by term. Care is necessary in this, however. This method of interpretation is usually unnecessary i f (1) has been successful; and this method is successful when ~ ( p ) , as a function of p , has a branch point and no poles.

1) Proc. Roy. Sor., A., X, 1893, 504-529; 54, 1884, 105-143; Electromagnetic Theory; Electrical Papers. 2) Proc. Loridon Math. Soa. (2), 15, 1916, 401-448. and la ter papers. 3) Operational methodh i n mathematical physics; no. 23 of the Cambridge tracts in mathematics arid mathe-

matical physics, Firs t Edition, Mi. A second edition is in tlie ~irebs.

G o 1 d s t ,e i n , The application of Heaviside’ls operational method 235 Rand 12, Heft 4 Angnst 1932

(5) In any circumstances, the interpretation of cp(p) is given by

where 1 arg 1 1 < TC , and L is a curve froni c - i co to c + i a, where c is finite and positive, and all the singularities of the integrarid are on the same side of the curve as -co. For positive values of t , I, may usually4) be taken to be any curve beginning a t infinity in the third quadrant, passing round the origin, and proceeding to infinity again in the second quadrant, provided that all the singularities of the integrand remain on the same side of the curve as --CQ.

This rule provides a basis for a justification of the method; and if in doubt, we can discover from this rule if we are justified in interpreting in tlie other ways.

The rule is due to B r o m w i c h (loc. cit.) and tlie line integral will be referred to as B r o m w i c 11’s integral6).

(6) If f ( t ) is the interpretation of cp(p), then f ( t ) is the solution of the integral equation,

m p(p) = p 1 e - ” t f ( t ) d t .

0

The integral is due to Car son” ) . It is known that Bromwich’s integral is the solution of C a r s o n’s integral equation.

(7) If f ( t ) is the interpretation of ~ ( p ) , then tlie interpretation of q(p) - is \ f i t ) at7). More generally, if f,(t) and f ,( t) are the interpretations of cp,(y) and cy,(p) respectively, then

the interpretation of p1 ( p ) q2 (11) y is

l + Y 0

1

t t

(8) There are certain other subsidiary rules to be found in the literature which are

(a) If f ( t ) is the interpretation of y ( p ) , tlie interpretation of q ( p + h ) , where h is a

sometimes useful, e. g.

positive parameter, is t

0 e - ’ l t f ( t ) + h J e - h t f ( t ) d t .

(b) If cp (y) can be expanded in powers of p-’ , the result of operating on r a t F(t) with ~ ( y ) is given by the equation

q(p) eAUt F ( t ) =,-( I t cp (P -- a ) F ( t ) . (c) If f ( t ) is the interpretation of q ( p ) , tlie interpretation of pp7 (y) is f ’ ( t ) , where, if

cy(z), as a function of z, has an infinite number of singularities in a strip parallel to the imaginary axis, the condition f(0) = 0 is necessary.

(d) If f ( t ) is the interpretation of rp ( p ) , the interpretation of e--hp y (11) is 0 for t < h , and f ( t - I / ) for t>h, where h is a real parameter.

We are now in a position to make the following small addition to the discussion of the auxiliary equation. If a known function of t occurs as a term in the original differential equation, then in tlie auxiliary equation it should, if possible, be replaced by its ,,operational representation“. [If f ( t ) is the interpretation of y ( p ) , then p7 ( p ) is called the ,,operational representation“ of f ( t ) .]

4 ) A sufficient condition is that fp (A)/,? should tend to zero uniforinly with regard to a r p 1 (for tlie values of a r g I concerned) as 11 1 tends to infinity. Soine such condition is also ticressary for the t ruth of the partial fraction rille (1 ) . It is always satisfied in physical exaniplos.

6 ) The statenients of this rule and the partial fraction i ~ i l e are quoted with slight changes f r r m J e f f r e y s (loc. cit.). ”) Electric Circuit Thoery and the Operatictrial t’alenlus, New York, 19Sfi. 7 ) J e f f r e g s ’ s exposition of the solntion of a system of ordinary differential equations, snrii as occurs i n the

theory of the oscillations of a mechanical system with a fiiiite rrurnber of degrees of freedoin, is based entirely on defiriite integration.

Ztsehr. f . angew. 236 G G 1 d s t ,e i n , The application of Heaviside'ls operational method Math. und Mech.

3. Advantages of Heaviside's Operational Method. The most usual method of solving the problems for which the operational method is recommended is by use of a series of normal functions. Compared with this method, the operational method has the advantages that it is easier to find the coefficients in the series by using the partial fraction rule than by the usual methods, it is not so usual to make mistakes in the working, and errors due to assuming a wrong form for the expansion of an arbitrary function - errors which have certainly been made in the past -- are avoided. Finally, and in some ways most important of all, the solution by use of a series of normal functions is unsuitable for computation or for exhibiting the character of the solution when t is small. Thus B e r g e r (loc. cit.) says ,,Es ist auch hier darauf hinzuweisen, dab fur sehr kleine Werte von t die Berechnung der Temperatur dadurch unsicher wird, dah die Konvergenz der Reihen sich mit abnehmendem t verlangsamt." By use of the operational method, it is easy to find a form of the solution especially suitable for small values of t .

Since B r o ni w i c h's integral gives the interpretation of an operational solution 9 (p ) , the operational method is equivalent to the use of complex integrals. But there is an enormous gain in conciseness of expression, and many steps suggest themselves in the course of the working which would be missed when using the more cumbrous notation.

Somewhat similar remarks apply to the use of Green's functions and integral equations, though in tlie latter case numerical solutions are occasionally obtainable which are very difficult to obtain by other methods.

In the solution of many problems, it is sufficient to use rules (1) and (3) of 5 2, and to make use of known results. This will be the case in the problems solved below. But in obtaining the ,,known results", use has been made of rules (5) or (6). Examples of the use of all the rules given, arid of others, will be found in the literature.

4. Conduction of Heat in a Plane Slab with Radiation at the Surface. At time t = O an infinite slab, of thickness 2 1 , a t constant temperature, is placed in a medium a t another lower constant temperature. As B e r g e r (loc. eit.) points out, there is no loss in generality in taking the initial temperature of the slab as unity, and that of the medium as zero, and we shall therefore do so. We shall use the same notation as B e r g e r ; 6 denotes the temperature, a the thermometric conductivity, 1% tlie relative eniissivity, t the time, and z distance perpendicular to the face of the slab measured from the middle of the slab. Then 6 is a function of z and t only, and satisfies the equations

a I7 328 a f ' U S . . . . . . . (11,

. . . . . . . . . 8 1 when t = o (21, a $9 - + h ?9 = 0 a z

when x =;; 1 . . . . . . . . . . .

when z = I , . . . . . . . .

The ,,auxiliary equation" is

. . . . . . . . . . . . 3'0 (32

a 7 - $1 19 = --1' (5) t which, if we write

becomes p=nq2 . . . . . . . . . . . . . . . .

From (3) and (4) it is clear that I9 is an even function of x , and the solution of (7) required is, therefore,

. . . (8). 1 2 [cosh Q Z = 7 (eQX + c-~')] , . . 6 = 1 + A cosh q z .

From (3) [or (4)] it follows that

and, therefore, qAsinh q l + h + h A c o s h q l = O ,

237 Band 12, Heft 4 Aiiizrirt 1932 G o 1 d s t e i n , The application of Heaviside's operational method

Apply the ,,partial fraction" rule. single-valued function of p . Q = f i I<, where

or

The value --iK of g gives the same value, - - -aK2 , of p as the value +iK, and, therefore, in applying the partial fraction rule only the values + i K need' be considered. If we take

The solution is an even function of Q , and, therefore, a The nunicrator has no poles; the denominator lins zeros when

. . . . . . . . . . . . . . - K s i n K l + h c o s K l = O (11)

KZ tan KZ = hZ *) (12). . . . . . . . . . . . . . . . . .

. . . . . . . f ( p ) = h c o s h g z , F(p)=ysinliyZ+I.LcosliyZ (1%

(14)

we have F(0)=l ~~ f (0) . . . . . . . . . . . . . . .

cl 1 d 1 1) ell) F ( p ) = q dq F ( p ) = q [( 1 + ?I E ) sinli y 1 + (i Z cosh y I ] (16).

and when g is equal to i K this is equal to

1 -K[(1 +h7)sin liZ+Kl cos l i t ] . . . . . . . . . (17). 2

The partial fraction rule therefore gives

(18).

Now from (11) arid (12),

and therefore hlK=tanl<Z, k l= l \ rZ tanKl ,

tan KZ - h K { ( I + h ~ ) sin K Z + K E G 1\'1 - ( I+ [\:I tan KZ) sin rcz + k ' z c o s ~ z

-

. . (1% tan K l sin k ' l

so that

wliicli is the usual form of the solution. For t = O , we find

sin l i l K l + sin h ' l cos li 1 cos K z . . . . . . .

K

@l).

The summation is over all the positive roots of (12). For large values of at/,?', one or two t e r m of (20) are all that are necessary to provide

a satisfactory approximation to the value of 8 . But the solution (20) is unsuitable for computation or for exhibiting the nature of the solution when at/P is small. To find a form suitable in such circumstances, we expand (10) as follows.

8 ) The roots of this equation arc all real, I ( beiiig positive.

ZtRchr. f . angew. Math . urrd Mech. 238

Now

G o 1 d s t e i n , The application of Heavisisde's operational method

(23)

( J e f f r e y s , loc. cit. p. 6O), where

. . . . . . . . . . . . . (24)

and is tabulated by J a h n k e and E m d e , who call it the ,,Fehlerintegral", @ (2). For large values of x,

1 - erf x - .

If we neglect all but the first three terms of (22) , we find

When 1 - z or I + B is not small, u-e inay use (25) to approximate to these expressions, But (26) as it stands applies whether 2 ~ z or 1 + x is small or not.

When an expression such as (26) is available, it is scarcely ever necessary to consider more than four terms of such a series of normal functions as (20) for any value of t .

5. Conduction of heat in an infinite cylinder with radiation at the surface. At time t=O an infinite circular cyliiider of radius R , a t unit temperature, is placed in a medium at temperature zero. If r is distance from the axis of the cylinder, the temperature B satisfies the equations

==a d @ (t;;." - I + - 1 a r a ") . . . . . . . . . . . . (1).

. . . . . . . . . . . . B = 1 w l i e n t = O (2).

(3). d B - - f h 9 = 0 when r = R . . . . . . . . . . . . dr

The ,,auxiliary equation" is

where . . . . . . . . . . . . . . . &=at12 (5)

a = l + A T o ( y . r ) [ = l + A J o ( i q r ) ] . (6).

as before. B must remain finite when r = 0 , and, therefore, with a usual notation for B e s s e l functions,

. . . . . . . . From (3),

ynr"'(qn) +h + h A I , ( q I t ) = o . ~ . . . . . . .

and, therefore,

since d

IOt (x) (= f i To(%)) = T,(x) . . . . . . . . . .

. (9)

. (10).

Apply the ,,partial fraction rule'. Again the solution is an even function of q , and, therefore, a single-valued function of p. The numerator has no poles; the denominator has zeros when q = f i n , where

i n I , ( i n R) + h 1, ( i + z R) = 0 ,

G o 1 d s t e i n , The application of Heavisi,de's operational method 239 Band 12, Heft 4 August 193%

that is,

As before, only tlie values + i n of q need be considered in applying the partial fraction rule. If we t,ake

. . . . . . . . . . . . . n R J , (nR) = f t f iJ,(nR)') (11).

f ( p ) = h I , ( q r ) , E'(p)=ql , (qn)+hIo(qR) . . . . . . (H), we have

f ( 0 ) ~ 1 . . . . . . . . . . . . . . . . E' (0)

. . (13)

. . . . . . . . . . . f ( p ) = h J , (nH) when y = in (141,

and wl1cn q is equal to i n this is equal to

1 2 - - - ~ [ ( ~ + I I ~ ~ ) , T , ( ~ R ) + ~ R J , ' ( I ~ R ) ] . . . . . . (16).

The partial fraction rule, therefore, gives

Now by a well-known recurrence formula

. . . . . . . . . . . . (1% n RJ,' (n R) = n R J , (nR) - J , (n R)

and, therefore,

and on substituting for h from (11), this becomes

Again using (ll), we see that this is also equal to

r , 1 hus

For t = 0 we find

. . . (22).

The summat,ion is over all the positive roots of (11) I").

0 ) The roots of this equation are all real, sirice / I is positive. See, for exalnple, W a t s o I I ' S I3 e s s v 1

10) If iiiitially the temperature of the cylinder is zero, and that of the niediunn ontside is unity, the teniye- Functions, p. 4x2.

rature at any time would be 1-9, where 3 is given by (22). A similar remark applies to the other problerlls.

2 t s c h r . f . angew. Math, und Meoh.

Either (21) or (22) is suitable for large values of ut/R2. For small values, we return to (9), and use the asymptotic expansions of the functions I,, and I, for large positive values of the argument. (See, for example, W a t s o n ' s Bessel Funktions, p. 203). We thus obtain, as an asymptotic equality for small ut/r*,

240 G o 1 d s t e i n , The application of Heavisi'de's operational method

1 e- ( R - 7 1

(i

1 p - C r ( H - t ) x I - h ( R / r ) ' - ~

3 = 1 - h (R/ r )2

Now it is known") that

qme-qx = 12/n)l(2at)-i m e - x * 1 8 ( ~ t n ~ ~ - ~ ( i J / - k ) . . . . . . .

where D , (a) satisifes the differential equation

Dm"(z) + ( m + D,,t(z) = O . . . . . . . . .

For large positive values of 2, Dm(z ) has the asymptotic expansion

It also satisfies the recurrence relation

. . (24).

. . (25)

. . (26).

. . (27).

' * (2%

. . (2%

so that D-*, D-, etc. are easily found, and could, in fact, be easily tabulated, from the re- currence relation (28).

From (24) it follows that 1 3 R - r

6 = 1 - 2 h (R ut/n r)l j , ~ ) + (-- - + -- h) (2 at): 1)- ' { r:r 8r 8 2

When R - r is not small, we can approximate to the Z I functions by using (27); but (30) as it stands applies whether R - r is small or not, so long as r is not small.

In obtaining (24) or (30) we used not only the asymptotic expansions of I, (y R) and I, (q R), but also that of I, (y r). We thus assumed not only that B / l / z is large, but also that r/@ is l a r r . When R / l / a is large, but r/1/x is small, we can approximate to the result by using t ie asymptotic expansions of Il ( q R) and I, (y R) , and the series for I , (y r ) in a s c e n d i n g powers of qr . We thus obtain

We can then interpret term by term by (251, and use the approximation (27) for the B functions. Thus the first approximation for points on or very near the axis is

11) See a forthcoming paper by G o 1 d 8 t e in .

24 1 Band 12, Heft 4 A iignst 1Y32 G o 1 (1 s t e i n , The application of Heaviside's operational method

I 1 1 if = 1 ~~ h ( ~ T c B ) ~ q - 2 e--U

In these wags we can find satisfactory approximations when R/(at)$ is large, and r/(nt)a is either large or small, but when r/ (at): is moderate, the approximations are more complicated, and are probably too complicated to be of much use. In practice, the expressions already found are generally sufficient for computation.

6. Conduction of heat in a finite circular cylinder with radiation at the surface. At time t = 0 a circular cylinder of length 2 2 and radius It?, at unit temperature, is placed in a medium at temperature zero. If r is distance from the axis of the cylinder, and B distance parallel to the axis measured from its inid point, the temperature 8 satisfies the equations

. . . . . . . . . . . . . (11,

(2) 1 CY = 1 when t = 0 . . . . . . . . . . . . . . . . a 1'1

a if a z

. . . . . . a-+h~Y=O r when r = R . . . . . . - (3),

-+hB=O w h e n z = l . . . . . . . . . . . . . (A),

. . . .

The equations are linear, and we can build u.1) the complete solution by superposition. Suppose that instead of (2), we have

. . . . . . . . . . . . . . 6 = c o s K z when t = O (6) 1

K 1 tan R1= h 1 . . . . . . . . . . . . . . . (7).

(8) 1

where K is a root of the equation 5 4 (12). namely

Then

where v is a function of r and t only, satisfying the equations

. . . . . . . . . . . . . . . . 8 = v (r, t ) cos K s

Wl v = l w h e n t = O . . . . . . . . . . . . . . . .

(11). . . . . . . . . . . . . . c ) U - - t h v = O when r = B dr

The auxiliary equation is

. . . . . . . . . . . (12).

(1% If we write

this becomes cc K 2 + y = (1 (12 . . . . . . . . . . . . . . . . . .

and the solution which remains finite on the axis is

u=p/ag"AAI,(q?.) . . . . . . . . . . . . . . . (15).

q A I , ' ( q R ) + h p / n q * $ . h A T,(qR)=O (Wl

From (11) . . . . . . . . . . . .

16

Ztschr. f . nngew. Math. und Mech. 242

and, therefore, since I,’ (q I?) = I, (c/ R) ,

G o 1 d s t e i n , The application of Heavisitle’ls operational method

Now apply the partial fraction rule. First

. . . . . (18).

. . . . . . . . . (19).

Next, the sccond term is, as before, an even function of q, and, therefore, a single-valued lun.ctim of 2). The numerator has no poles; the denominator has zeros when

f = O , p = a R 2 . . . . . . . . . . . . . . . (ZO),

(21)

and when Q : t i n , p =: -- n (li + n2) , where

. . . . . . . . . . . . . . n R J , (72 l?) = h E J I , (n Ft)

as in 5 (11). Again only the values + i n of q need be considered. If we take

and . . . . . . . . . . . . (W

(251,

f ( p ) = h p when q = 0

=h,pJ , (mr) when q = i n . . . . . . . . . d 1 df) ZJ ~ F ( P ) =-’-- 2 a q d q F ( P ) -=Z>[q I l (q R) + h L ( q 741 + % p q [ ( l + h R) r1 (cfR)+qRI,’(qR)] (26).

When q = O , this is e c p 1 to p h ; when q is equal to i n , it is equal to

1 2 - - 1’ f i [(I + h R) J , (m R) + r~ R J,’ (n R)] . . . . . . . . (27)

Hence the partial fraction rule gives

T’he coefficients in this series arc the sanie as those in 6. (17), and tlierefore, as in 5. (19) and f20),

Hence, if B = cos K z initially,

lf, then, initially 2 sin l<l

B=1= LII<l + sill Ii 1 cos - c o s I ~ z li 2 * . . K

by the superposition of solutions we find

243 Band U , Heft 4 Angitst 1932 C: o 1 d s t e i n , The application of Heaviside's operational method

The solution required, therefore, is the product of the series found in 4. 120) and 6. c21). The values of these series for small values of t have already been discussed, so that further discussion is unnecessary.

Zusammenfassung. Heavisides Operatoren-Methode wird angewandt, um die folgenden drei Probleme der Warmeleitung zu behandeln : Bestimmung der Ternperatur fur jedrn Zeitpunkt und an jedem Punkte 1) in einer unendlich ausgedehnten Platte von der Starke 2 I, 2) in einem unendlich langen Kreiszylinder vom Radius R und 3) in einem endlichen Kreis- zylinder von der Lange 2 1 und vom Radius R. Der Korper besitzt eine bestimmte konstante Anfangs-Teniperatur (welche gleich 1 gesetzt wurde) und befindet sich innerhalb eines Mediums von niedrigerer Temperatur (welche gleich 0 gesetzt wurde), so dab die Oberflfiche des Korpers Warme ausstrahlt.

Die Operatoren-Methode last sich folgendermaken zusamnienfassen : Es genuge eine Variable u irgendeiner Differentialgleicliung der Physik ; man bringe alle Glieder der Gleichung auf die linke Seite des Gleichheitszeichens. Darunter schreibt man die nach folgenclen Regeln aufgebaute ,,Hilfsgleicliung". Hezeichnet t die Zeit, so ersetzt man jedesmal

durch pw und fugt auf der rechten Seite p u u o hirizu, wenn v, den Anfangswert +on u

bezeichnet ; ebenso ersetzt man 7 durch p2 u und fugt rechts p2 w, + p iio' hinzu, wobei u,'

a ?I

a t a2 2, a t

d V a t der Anfangswert von - ist usf. Man lost nun die ,,Hilfsgleichung", innem man w als

Funktion von p berechnet: u = y (p) . Diese Losung lakt sich nach verschiedenen Regeln und Methoden interpretieren, wie - vor allem in 6 2 - ausgefuhrt wird.

Bezeichnet 9 die Temperatur, a die Warmeleitfahigkeit, F, die relative Warmeubergangs- zahl, t die Zeit, z den senkrechten Abstand der Oberflache der Platte, geniessen von der Mitte der Platte, und setzt man y = a q', dann ist die Losung von Problem 1) gegeben durch GI. (lo), 8 4, S. 236. Nun ist eine der Regeln zur Interpretation einer derartigen Operatoren-

Losung, die ,,Partialbruchregel": Hat die Funktion f f p ) von p keine Pole, und ist

eine eindeutige Funktion von y , so gilt:

f (21) P ( p j

Dabei sind die K die Nullstellen von F ( p ) , die alle als einfach (und von Null verschieden) vorausgesetzt werden, der Strich in F' ( K ) bezeichnet Differentiation, und die Summation ist zu erstrecken uber alle Nullstellen von B'(p).

Die Anwendung dieser Regel zur Bestimmung von 6 in 4. (10) S. 236 fuhrt nun zu 4. (20), wobei die Summation uber alle positiven Wurzeln von 4. (12) zu erstrecken ist. Diese ubliche

at Form der Losung ist ungeeignet, falls 7 klein ist. Uni auch fur diesen Fall eine geeignete 1 Losung anzugeben, benutzt man 4. (10) wie in 4. (22), und nach der bekannten Interpretation von he- (h+ q) ist 6 dann annahernd durch 4. (26) gegeben, und zwar fur kleine Werte

at von 1. 1

Bezeichnet r den Abstand von der Zylinderachse, so ist die Operatoren-Losung des zweiten Problems in 5. (9) S. 238 gegeben, wobei I wie iiblich die B esse lsche Funktion be- zeichnet. Die Interpretation nach der ,,PartialbruchregelL' gibt 5. (21) und (22), wobei die

Summation sicli uber alle positiven Wurzeln von 5. (11) erstreckt. 1st 5 klein, so erhalt man, wenn man die asymptotischen Ausdrucke fur die Funktionen 1, und I, fur groke positive Werte des Arguments benutzt und Glied fur Glied interpretiert, einen Naherungsausdruck fur 8, der in 5. (30) gegeben wird. Dabei sind die D-Funktionen in 5. (26) bis 5 . (29) kurz

ut at definiert. Wenn 7 klein, aber - rob ist, so benutzt man die asymptotisclien Entwicklungen R r2 von I , (y R) und Il (q R) sowie die Reihenentwicklung fur I , (y r ) ; man findet dann, dab fur Pnnkte auf oder sehr nahe der Achse 8 naherungsweise durch 5. (32) gegeben ist.

Schlieklich findet man die Losung des 3. Problems, indem man 19 gleichsetzt dem Produkt der Reilien 4. (20) und 5. (22), welche Losungen des ersten bzw. zweiten Problems sind. Die Werte dieser Reihen fur kleine Werte von t sind bereits diskutiert worden. 190

at r

16"