murnaghan the cauchy-heaviside expansion formula and the boltzmann-hopkinson principle of...

7/27/2019 Murnaghan the Cauchy-heaviside Expansion Formula and the Boltzmann-hopkinson Principle of Superposition

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1927] THE CAUCHY-HEAVISIDE EXPANSION 81

T H E CAUCHY-HEAVISIDE EXPANSI ON FORMULAAND T H E BOLT ZMANN-HOPKINSON

PRINCIPLE OF SUPERPOSITION*BY F. D. MURNAGHAN

1. Introduction. It is the object of the present note to pointout that the well known expansion theorem of Heaviside isan immediate corollary of the classical method given byCauchy for finding a particular solution of a linear, non-homogeneous, ordinary differential equation when the generalsolution of the corresponding homogeneous equation is known.Various proofs of Heaviside's theorem have been given byBromwich and Wagner, using contour integrals, by Carsonand others and it seems hardly possible that the direct connection with Cauchy's method can have escaped att ent ion . Thismethod is interesting, furthermore, since it furnishes a rigorousproof of the useful Boltzmann-Hopkinson principle of superposition, f The explicit form of the expansion theorem in thecase when the characteristic equation has multiple roots is alsogiven.

2 . Cauchy^s Method. Although Cauchy's method is classicaland to be found in the older books, it does not seem to find aplace in the more modern texts and it will, therefore, be convenient to give an indication of its derivation. Let us suppose

* Presented to the Society, under a different title, September 9, 1926.t The following citat ions may serve to give a proper orientation : Bromwich, T. J., Normal coordinates in dynamical systems. Proceedings of theLondon Society, vol. 15 (1916), pp.40 1-448 . Wagner ,K.*W.}ber eine Formelvon Heaviside zur Berechnung von Einschaltvor gangen, Archiv fr Elektro-technik, vol. 4 (1916), pp. 159-193. Carson, J. R., O n a general expansiontheorem for the transient oscillations of a connected system, Physical Review,vol. 10 (1917), pp . 217 -225. Carson, J. R., 7 heory of th e transient oscillationsof electrical networks and transmission lines, Transact ions A. I. E* E., vol.38 (1919), pp. 345-427.


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82 F. D. MURNAGHAN [Jan.^Feb.,t h a t w e w i s h t o f in d a p a r t i c u l a r s o l u t i o n o f t h e l in e a r d i ff e re n t i a l e q u a t i o n o f t h e nth o r d e r(1 ) 4>{D)u - f(f) ; (D) m a o D " + + a,w h e r e u i s t h e d e p e n d e n t v a r i a b l e , t t h e i n d e p e n d e n t v a r i a b l ea n d Dzzd/dt. The coe f f i c i en t s a a r e e i t h e r c o n s t a n t s o r f u n c t ions o f t. If (vi, ,v n) a r e d i s t i n c t s o l u t i o n s o f t h e c o r r e s p o n d i n g h o m o g e n e o u s e q u a t i o n(1 0 4>{D)v - 0 ,i t s g e n e r a l s o l u t i o n i s(2 ) V - C1V1+ +CnVn,w h e r e t h e C s a r e a r b i t r a r y c o n s t a n t s . W e c h o os e t h e s e s ot h a t v a n d i t s d e r i v a t i v e s w i t h r e s p e c t t o t u p t o t h e ( 2 )dv a n i s h , w h e n / = r t h e ( w * - l ) s t d e r i v a t i v e t a k i n g t h e v a l u ef{r)/a^{r) ; r b e i n g a f i x e d b u t a r b i t r a r y v a l u e o f / . T o f i n d t h ep r o p e r v a l u e s f o r t h e C s , w e h a v e m e r e l y t o s o l v e a s y s t e m o fn l in e a r e q u a t i o n s w h o s e d e t e r m i n a n t is t h e W r o n s k i a n\v\vi Vfln~ l) | a n d t h i s d o e s n o t v a n i s h o n a c c o u n t o f t h ed i s t i n c t n e s s o f t h e v's. W h e n t h e v a l u e s o f t h e C s a s t h u sd e t e r m i n e d a r e e n t e r e d i n ( 2 ) , v b e c o m e s a f u n c t i o n of t a n d r ,a n d t o s t r e s s t h i s w e m a y w r i t e i t a s v(t, r ) . T h e n C a u c h y ' sc o n t e n t i o n i s t h a t t h e i n t e g r a l(3 ) u(t) jv(tir)drJoi s a p a r t i c u l a r s o l u t i o n of ( 1 ) . W e h a v e , i n f a c t , f ro m ( 3)

Du = lDv(t fr)dT + v(t,t) = lDv(t tr)dr9s i n c e V(T, r) v a n i s h e s b y d e f i n i t i o n f o r e v e r y v a l u e o f r .S i m i l a r l y

D 2u = )D 2v(t,T)dT, , D n~ lu = \D n^ lv(t )T)dr t/o /ow h i l e C* C % (*)D nu J D nv(t,r)dT + {D^hit.r)}^ - I D nv{t,r)dr + ,J o JQ # O ( 0
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1 927 ] TH E CA U CH Y-H EA V ISID E EX PA N SIO N 8 3s o t h a t

{D)u = f {D)v(t,r)dT + f{t) = f(t).JoT h e p a r t i c u l a r s o l u t i o n u o f ( 1 ) o b t a i n e d i n t h i s w a y i sc h a r a c t e r i z e d b y t h e f a c t t h a t i t v a n i s h e s w i t h i t s d e r i v a t i v e su p t o t h e ( f t - l ) s t w h e n / = 0 ; t h e nth. d e r i v a t i v e h a v i n g ,c o n s e q u e n t l y , t h e v a l u e / ( 0 ) / a 0 ( 0 ) w h e n / = 0 . I t is a p p a r e n tf rom the de f in i t ion o f the C's o c c u r r i n g i n ( 2 ) t h a t t h e y m a yb e o b t a i n e d f ro m t h e v a l u e s t h e y w o u l d h a v e if ( r ) w e r e = 1b y m u l ti p li c a ti o n b y / ( r ) . H e n c e if w e d e n o t e b y \p{t, r) t h ep a r t i c u l a r f u n c t i o n v(t> r ) c o r r e s p o n d i n g t o t h e p a r t i c u l a rp r o b l e m w h e r e ( 0 = 1 w e h a v e v(t,r) f(t)4t(t,r), a n d ( 3 )m a y b e w r i t t e n i n t h e f o r m(3 bis) U(t ) = C / ( T ) * ( * , T ) ( * T .J oD e n o t i n g b y (, 0 ) t h e p a r t i c u l a r s o l u t i o n c o r r e s p o n d i n g t o(/) s= 1 w e h a v e

* ( ' , 0 ) = f t{t,T)dT,Jow h i c h s u g g e s t s t h e n o t a t i o n(4) $(*,r) **fW ts)ds,w i t h t h e r e l a t i o n

: = - W>r).or

A n i n t e g r a t i o n o f ( 3 b ls ) b y p a r t s t h e n y i e l d s(5)

( u{t) = [ - f{tW,t) + /(0){(*,0)] + fkt,r)j-dr= / ( 0 ) W , 0 ) + f { ( < , T r , s in ce (M ) - 0 .J o dr

T h i s is t h e m a t h e m a t i c a l s t a t e m e n t o f t h e B o l t z m a n n - H o p k i n -s o n p r i n c i p l e o f s u p e r p o s i t i o n a c c o r d i n g t o w h i c h w e a r e a b l e


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84 F. D. MURNAGHAN [Jan.-Feb.,t o b u i l d u p f r o m a p a r t i c u l a r s o l u t i o n %(t, r ) o f the d i f f e ren t ia le q u a t i o n (1 ) w h e n (t ) s= 1 , a p a r t i c u l a r s o l u t i o n of t h i s e q u a t i o nw h e n ( / ) i s a n a r b i t r a r y f u n c t i o n o f t. T h e p a r t i c u l a r s o l u t i o no b t a i n e d i n t h i s w a y i s c h a r a c t e r i z e d b y t h e f a c t t h a t i tv a n i s h e s , t o g e t h e r w i t h i t s d e r i v a t i v e s w i t h r e s p e c t t o / u pt o t h e (w l ) s t , w h e n t~Q.

3 . The Part icular Case wh en the Coeff ic ients of (D) areConstant . I n t h e f i r s t c a s e w e s h a l l c o n s i d e r t h e g e n e r a l c a s ew h e r e t h e z e r o s o f t h e p o l y n o m i a l (D) a r e a l l d i s t i n c t a n dw e s h a l l w r i t e{D) s a Q(D - n)(D - n ) (D - rn).

W e m a y c h o o s e a s o u r n d i s t i n c t s o l u t i o n s o f ( 1 ' ) t h e f u n c t i o n sa n d o u r e q u a t i o n s t o d e t e r m i n e t h e c o n s t a n t s C of (2) are

C i + +C n~ 0 ,(6)

fiCi+ +rnC n = 0 ,

rin-lCl + . . . + fnn- lC n - ^ 0T h e s e e q u a t i o n s s h o w t h a t ( C i , , Cn) a r e th e coeff ic ientso f t h e a n a l y s i s o f f(r)/(D) i n t o i t s s i m p l e f r a c t i o n s . I n f a c ti f w e w r i t e f(r) C 1 C n+ +(D ) D - n D-rna n d e x p a n d e a c h o f t h e f r a c t i o n s o n t h e r i g h t - h a n d s i d e n e a rD =: o o , w e f in d o n o b s e r v i n g t h a t D = oo is a zero of o rd ern o f t h e l e f t - h a n d s i d e , t h e f i r s t t e r m i n i t s e x p a n s i o n b e i n g/ ( r ) / ( a 0 Z ) n ) , e x a c t l y t h e e q u a t i o n s (6 ) t o d e t e r m i n e t h e C ' s . F u r t h e r m o r e o n o b s e r v i n g t h a t D = rp i s a s i m p l e p o l e w i t h r e s i d u ef(r)/ f(r p) o f t h e l e f t - h a n d s i d e w e o b t a i n t h e r e l a t i o n s

fir)(7) C * = T , 7 ~ T ' ( # - l , , * ) .(rp)


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Hence the function v(t, r) is

so that

and

The relation

J ? - I 0 ^pj

n er , ( l -r ) n JP ~ I rpr{rv) 1 rpr{rp)

TL - tzr, 'obtained above shows thatj i-J_-,

*(0) -1 M'(>p) 'so that we may write

0 ( 0 ) pi ^' (r* )and, in particular,

1 w erp( 9 ) 0-W,0)---+ E(0 ) p-1 ^'(fp)It is this particular form of Cauchy's result which yieldsimmediately the Heaviside expansion theorem. Before proceeding toshow this it will be well, however, tosee whatmodification isintroduced when two or more of the zeros of0(J9) coincide. For simplicity of presentation let us supposethat r2 = rh the other roots being all different from T\ and fromone another. Then we take

and the equations to determine the constants C areCi + 0 C2 + C3 + . + Cn - 0,

fiCi + 1C2 + rzCz + + rnCn - 0,rfCi + 2nC, + r | C 3 + + rn2Cn 0,

/to00


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8 6 F. D. MURNAGHAN [Jan.-Feb. ,the coeff ic ient of C2 in e a c h e q u a t i o n b e i n g the d e r i v a t i v e w i t hr e s p e c t to r\ of the coeffic ient of C i . T h e s e e q u a t i o n s s h o w , ju s ta s b e f o r e, t h a t (Ci, , Cn) are the coeffic ients in the a n a l y s i sof(r)/(D) D - n (D - r i ) 2 D - rz D - rnD e v e l o p i n g t h i s i d e n t i t y in the n e i g h b o r h o o d of the d o u b l eze ro i? = r i , and s e t t i n g

w e h a v eF(D),

Ci=/ ( r )F ' (n ) , C 2 / ( r )F (n ) .The part of the function () corresponding to the double zeror = Ti may be conveniently expressed as follows. WritingCpf{r)AP) ( = 1, , n), so that

1 Ax A% A n(10) = ^ - + - - - + +0(Z>) Z > - n (Z >-n ) 2 P - r nwe have, for the part of (/) in question, to evaluate the integral

f {A1 + A2(t- r)}eT*-*UT.JoIf we w r i t e tT*=S, t h i s b e c o m e s

( i l l + i l r f )evfc = e'*1+ -)e* - ) ;Jo fi V i fi 8 / \ f i f i 2 /a s Ai F'(ri) a n d 4 2 = ^ 1 ) , the v a r i a b l e p a r t of t h i s may bew r i t t e n in the fo rm

F(n)- - - - - - - > er xif\ \ dri \ ri / )

In an e x a c t l y s i m i l a r m a n n e r it can be s h o w n t h a t the v a r i a b l ep a r t of (t) c o r r e s p o n d i n g to a m u l t i p l e r o o t of o r d e r m is


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w h e r e F(D)z=(D ri)m /(D); t h e e x p r e s s i o n i n p a r e n t h e s i sb e i n g w r i t t e n w i t h b i n o m i a l c o e f fi c ie n t s. T h u s i n t h e c a s e ofa t r i p l e r o o t t h e c o n s t a n t s C a r e g i v e n b y t h e e q u a t i o n/ ( r ) s Ci C2 2 !C 3 ^(D) D-n (D - n ) (Z? - n ) 3

I t f o llo w s f ro m ( 1 0 ) , o n p u t t i n g D = 0 , t h a t t h e c o n s t a n t p a r tof (t) is 1/0(0) . T h e s e r e s u l t s f u r n i sh t h e g e n e r a l e x p r e s s i o nf o r t h e p a r t i c u l a r s o l u t i o n o f (D)u = 1 w h i c h v a n i s h e s w i t hi t s d e r i v a t i v e s u p t o t h e (n-l)st w h e n / = 0 .4 . The Heaviside Problem. In h i s s tudy o f e lec t r i c osc i l l a t i o n s H e a v i s i d e h a d t o c o n s i d e r a s y s t e m o f o r d i n a r y l i n e a rd i f f e r e n t i a l e q u a t i o n s o f t h e t y p e

k(11) X araX = E r(t), (f = 1 , , * ) .a = lH e r e t h e a ' s a r e p o l y n o m i a l s i n t h e o p e r a t o r D whose coeff i c i e n t s m a y i n v o l v e t h e i n d e p e n d e n t v a r i a b l e t ( a l t h o u g hH e a v i s i d e w a s p a r t i c u l a r l y i n t e r e s t e d i n t h e c a s e w h e r e t h e s ec o e f f i c i e n t s a r e c o n s t a n t s ) a n d e x p r e s s i o n s f o r t h e k d e p e n d e n tv a r i a b l e s a r e s o u g h t f o r , t h e f u n c t i o n s E(t) o n t h e r i g h t - h a n ds i d e s of t h e e q u a t i o n s ( 11 ) b e i n g g i v e n . A l l t h e d e p e n d e n tv a r i a b l e s b u t o n e , x s s a y , m a y b e e l i m i n a t e d b y o p e r a t i n g o nt h e e q u a t i o n s ( 1 1 ) w i t h t h e co f a c t o r s o f t h e 5 th c o l u m n o f t h ed e t e r m i n a n t | a r s | s = 0 ( Z } ) . D e n o t i n g t h e s e c o f a c t o r s b y AU , , Ak* we f ind

k{D)x8 = T,A r8E r(t).

L e t u s n o w c o n f i n e o u r a t t e n t i o n t o t h e c a s e w h e r e E p(t ) = 1,a l l o t h e r E 's b e i n g ze r o , a n d w h e r e , i n a d d i t i o n , t h e c o e ff ic ie n ts


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8 8 F. D. MURNAGHA N [Jan.-Feb.,of the polynomials a r 8 are constan ts. In this even t, theoperators 0(J3) and A p 8 ar e commutative and we first set upthat particular solution of

4>(D)y9 - 1which vanishes, together with its derivatives up to that of the( l) s t order, w hen / = 0 , n being the degree of the polynomial (D) . T h e n x8p A p 8yp is a particular solution of theequation< f> (D)x8~A p 8 and it is characterized by the fact thatits derivatives up to an order one less than the degree in Dof the elements a r 8 vanish when / = 0 . From the exponentialform already given for yP{tfi) in the case where the zeros of< t > ( D ) are simple, it follows that(12) x 8p (t,0) =AP8 y p(t,0) = - ^ T - T - + ~ ~ f ,a-i r aV(r a) 0 ( 0 )wh ich is H eav iside's expansion formula. If now we consider theequation < t> (D)up = E p ( t), a particular solution is

C'dEpu p(t ) - , ( 0 ) y , ( * f 0 ) + yp(t,T)dTfJo drthis particular solution being characterized by the fact thatit , together with all its derivatives up to the (n l )s t , vanishwhen / = 0 . Th e function

r* dE p8P A P8u p{t) = JS(0)*,p(*,0) + I % ap(t,r)dTJo dris , accordingly, a particular solution of the equation

(D)x8p - A p8E P(t ) .Adding with respect to p from 1 to fe, we find that the set offunctions

k ( c l dE \(13) x. 2 \ E p(0)x 9p (t ,0) + -x 8p (t 9r)dr }P I V Jo dr )furnish that particular solution of the equations(ll) which is


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1927I T H E CAUCHY-HEAVISIDE EXPANSION 89characterized by the fact that all the x's and their derivativesup to an order one less than the degree of the ar8 in D vanishwhen / = 0.

When the zeros of {D) are not distinct we avail ourselvesof the relations

D(tert) = rte ri + ert , D{tmert) = rtmer* + tn tm-lert ,D2(tme rt) rH mert + 2rmtm"le rt + m(m - l)tm"2ert ,D*(tmert) = rH me rt + Zr2m tm-le rt + 3rm(m - \)tm~2eTt

+ m{m l)(m 2^m~V,which show that

Apt(tmert) 4 p 8 ( r ) /m e r < + Ai9(r)mtm~ lertmint 1)+ 4,"(r) *m~V e + .

When, then, we are operating with 4P on a term such as1 fF(ri) d F(ri) )

(w1)! I ri dri ri ;which corresponds to a multiple root r\ of order m, we obtain

1 AMFjrQ d / AMFjn) \~TTH * + 0 - 1)-( j ^ - 2-1 ) ! I r\ drx \ t\ t(w (m - l)(m - 2)

murnaghan the cauchy-heaviside expansion formula and the boltzmann-hopkinson principle of...

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