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A method for finding the asymptotic behavior of a function from its Laplace transform Froese Fischer, Charlotte 1954

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A METHOD FOR FINDING THE ASYMPTOTIC BEHAVIOUR OF A FUNCTION FROM ITS LAPLACE TRANSFORM  CHARLOTTE FROESE  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n t h e Department of MATHEMATICS  We accept t h i s t h e s i s as conforming t o t h e standard r e q u i r e d from c a n d i d a t e s f o r t h e degree o f MASTER OF ARTS.  Members of t h e Department o f Mathematics.  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1954.  Abstract  In many p r a c t i c a l problems, p a r t i c u l a r l y i n c i r c u i t a n a l y s i s , the L a p l a c e Transform method i s used to  solve l i n e a r d i f f e r e n t i a l equations.  When o n l y t h e  asymptotic behaviour at i n f i n i t y o f the s o l u t i o n i s o f i n t e r e s t , -it i s not n e c e s s a r y t o f i n d t h e exact We have developed a method f o r f i n d i n g  solution.  the asymptotic  behaviour o f a f u n c t i o n d i r e c t l y from i t s L a p l a c e t r a n s f o r m . The method i s a g e n e r a l i z a t i o n o f one g i v e n by Doetsch f_ 5,6], The behaviour of a f u n c t i o n F ( t ) f o r l a r g e t depends upon t h e s i n g u l a r i t i e s o f i t s t r a n s f o r m f ( s ) on the l i n e t o t h e r i g h t o f which f ( s ) i s r e g u l a r .  The  asymptotic behaviour o f F ( t ) i s expressed i n terms of comparison f u n c t i o n s ^ ( t ) whose t r a n s f o r m s have the same s i n g u l a r i t i e s as f ( s ) .  We have c o n s i d e r e d s i n g u l a r i t i e s  such as l / s v + l ,  (ins) /s  (i ns)V^/s  or e ~ V s  ,  v + 1  ,  n  K/  v + 1  V+1  , l/s  ins.  s t u d i e d e x t e n s i v e l y by Doetsch,  v + 1  /ns,  e'^/s^ , 1  The f i r s t two have been  Acknowledgement s The author wishes t o express her thanks t o Dr, T,E, H u l l f o r s u g g e s t i n g t h e t o p i c o f t h i s t h e s i s , and f o r i n v a l u a b l e a s s i s t a n c e i n i t s p r e p a r a t i o n , and a l s o to Dr. F, Goodspeed f o r h i s helpful criticisms.  She g l a d l y acknowledges her  indebtedness t o the N a t i o n a l Research C o u n c i l o f Canada whose f i n a n c i a l a s s i s t a n c e has made t h i s study p o s s i b l e .  Contents  I  Introduction Chapter 1.  H i s t o r y o f the L a p l a c e Transform  1  Chapter 2«  Development o f t h e Theory  &  1.  I n d i r e c t A b e l i a n Theorems  12  2.  Comparison F u n c t i o n s and t h e i r transforms  15  3«  Chapter 3 . 1.  Asymptotic behaviour o f a f u n c t i o n whose t r a n s f o r m has one s i n g u l a r i t y on R(s) - o. G e n e r a l i z a t i o n s o f t h e Theory  20 23  The t r a n s f o r m has a f i n i t e number of s i n g u l a r i t i e s on the l i n e R(s)=o  24  2.  The t r a n s f o r m has an asymptotic exp a n s i o n about a s i n g u l a r i t y  25  3.  The asymptotic behaviour f o r v a r i o u s arguments o f t 26  Chapter 4.  Applications  28  1.  The asymptotic expansion o f \ 3t\.t\Ax, <  2.  The asymptotic expansion o f a f u n c t i o n which o s c i l l a t e s  29  3.  The asymptotic expansion o f a f u n c t i o n whose t r a n s f o r m has p o l e s on t h e l i n e R(s) = o.  31  2$  Introduction  The  Laplace t r a n s f o r m o f a f u n c t i o n F ( t ) i s  d e f i n e d t o be f(s) provided  - J  o  e" F(t)dt s t  the i n t e g r a l e x i s t s i n some r i g h t  half-plane  R(s) > s . 0  The  t r a n s f o r m i s used e x t e n s i v e l y i n c i r c u i t  a n a l y s i s and i n t h e study o f automatic c o n t r o l systems t o solve l i n e a r d i f f e r e n t i a l equations or Volterra'type i n t e g r a l equations.  I n such cases one f i r s t f i n d s t h e  transform of the desired s o l u t i o n , then t h e s o l u t i o n i t s e l f can u s u a l l y be found by means o f t h e i n v e r s i o n formula F(t) - l/2rri  r x+i«> \ e* f(s)ds J x-i«> s  ;x > s  a  Unless f ( s ) has o n l y p o l e s , t h e e v a l u a t i o n o f t h e i n v e r s i o n i n t e g r a l can be q u i t e l a b o r i o u s .  I n problems  where o n l y t h e asymptotic behaviour of t h e s o l u t i o n i s of i n t e r e s t , i t i s not necessary t o f i n d t h e exact solution for a l l t .  I t i s t h e purpose o f t h i s t h e s i s  t o g i v e a method whereby t h e asymptotic behaviour o f F ( t ) can be determined d i r e c t l y from i t s t r a n s f o r m f ( s ) without c a l c u l a t i n g t h e i n v e r s i o n i n t e g r a l s Before c o n s i d e r i n g  any t h e o r y , we s h a l l o u t l i n e  t h e h i s t o r y o f t h e L a p l a c e Transform.  Then we s h a l l prove  - II -  s e v e r a l theorems which g i v e the asymptotic F ( t ) f o r p a r t i c u l a r transforms f ( s ) .  behaviour of  L a t e r we  extend these theorems t o more g e n e r a l c a s e s . a p p l i c a t i o n s w i l l be g i v e n i n t h e l a s t  shall A few  chapter.  Chapter 1 History of the Laplace Transform The r e s u l t s of applying the Laplace Transform to l i n e a r d i f f e r e n t i a l equations with constant c o e f f i c i e n t s are i n many respects similar to those of applying an operational calculus.  Consequently, the development of  the Laplace Transform i s c l o s e l y connected with that of operational calculus. The idea of replacing an operator by a symbol goes as f a r back as the time of L e i b n i t z (1710) who noticed the resemblance between the formulas f o r the nth derivative of a product and the nth power of a sum.  Later Lagrange  (1772)  expressed the Taylor series i n the purely symbolic form, f(x+h) » e f ( x ) h D  where D =_d dx  Calculations were carried out exactly as though D were an algebraic quantity^ not an operator, and questions of v a l i d i t y were just avoided. Between 17^2 and 16*12^ Laplace developed the new idea that to a function y(k) defined f o r equidistant arguments ( i . e . , k = 0,1,2...) as object function,, there i s a resultant function (1.1)  u(t) » 21 y ( k ) t * k=o  I f the summation extends to negative powers then to y(k+l) corresponds the function u ( t ) / t .  The process of  forming f i n i t e differences of the object  function  r e s u l t s i n a purely algebraic process on the resultant function, namely m u l t i p l i c a t i o n by ( l / t - 1). saw  Laplace  i n t h i s the r e a l basis f o r the Lagrange formula  when t « e"*"*. Given a difference equation f o r y(k), he formed the corresponding algebraic equation and Then he t r i e d to express y(k) i n terms of u ( t ) . introduced the problem of how  solved i t . This  the c o e f f i c i e n t s of a  power series are determined by the function, formula f o r the c o e f f i c i e n t s was  Gauchy-s  not known u n t i l lo"31,  but Euler (1793) had derived an expression f o r the c o e f f i c i e n t s of what was  l a t e r c a l l e d a Fourier Series,  Through the s u b s t i t u t i o n t » e**", Laplace formed the 40  series (1.2)  u(e  ) = U(co) -  2-  y(k)e  k=o to which he applied Euler-s formula and obtained (1.3)  fir ] U(co)e" dco  y(k) » 1/2TT  By t h i s method he was  kico  able to express the unknown f u n c t i o n  i n terms of a d e f i n i t e i n t e g r a l .  The method was  purely  formal since the s e r i e s expansion of u(t) might not converge f o r t on the unit c i r c l e and the true sense of complex integration was (1S14).  not known u n t i l the time of Gauchy  - 3 D i s s a t i s f i e d t h a t h i s formula should c o n t a i n imaginary q u a n t i t i e s , he set (1.4)  y(k)  = [  ^"^(tjdt  where the path of i n t e g r a t i o n was to be determined* Except f o r the factor. l / 2 f r i , t h i s i s t h e Cauchy formula f o r the c o e f f i c i e n t s and i s i n the form of a M e l l i n Transform^ which can be d e r i v e d from t h e Laplace t r a n s f o r m by the s u b s t i t u t i o n z » e .  Then he went back and set  - t  the unknown f u n c t i o n (1.5)  y(s)—  J  e" U(t)dt s t  which was suggested by the form of (1.3) but continued t o use  still  (1.4)•  The f i r s t research on the transform (1.6)  A(t,cp) = ^ e q>(x)dx tx  i s due t o E u l e r (1737) who a p p l i e d i t t o the i n t e g r a t i o n of d i f f e r e n t i a l equations.  But because Laplace's r e s u l t s  were b e t t e r known, the transform ^ e - ^ F f x J d x was l a t e r c a l l e d a f t e r him. Guided by the work of Laplace, Abel (18*24) set up the f u n c t i o n a l r e l a t i o n s h i p (1.7)  cp(x,y,x,...) - i e  3 t u + y v + z p +  *-*f(u,v,p...)dudvdp...  - g f f } , say, which i s e x a c t l y the r e v e r s e of Laplace-s d e f i n i t i o n .  He  derived several fundamental properties, such as (l.d)  g|u f(u)[ = d /dx n  n  g{f(u)J.  n  The Laplace transform (1.9)  T°e- F(t)dt -  ( e-  st  e'  i v t  xt  F(t)dt  can be considered as the Fourier transform of e ^ ^ F t t ) where F ( t ) = o f o r t < o.  With Cauchy's work on complex  integration and the development of the Fourier Integral Theorem (16*20), the r e s u l t s could be applied to the Laplace transform. Apparently not much more was done u n t i l Heaviside (16*50-1925) developed an operational calculus independently of what had been done previously.  He  was an e l e c t r i c a l engineer without a University education but h i s curious methods l e d him to important  results.  Most of h i s work was i n electro-magnetic theory  9  where i t was necessary to solve l i n e a r d i f f e r e n t i a l equations with constant c o e f f i c i e n t s . substituting p  n  f o r d /dt n  n  , p~  By formally  for J ( d t )  n  n  n  the l i n e a r  d i f f e r e n t i a l equation was reduced to an algebraic equation.  Then he applied various r u l e s which he had  developed f o r obtaining solutions^ the most important ones being the Shift Rule, the Expansion Theorem, and the i n t e r p r e t a t i o n of the I r r a t i o n a l  Operator.  In some cases, Heaviside s r u l e s are a r b i t r a r y . T  For example, p/p-a  1 can be expanded i n two ways,' i n  e i t h e r an ascending o r a descending power s e r i e s . f i r s t corresponds t o t a k i n g d e r i v a t i v e s o f a and was t h e r e f o r e the  i n t e r p r e t e d as b e i n g  The  constant  equal t o zero;  second, t o s u c c e s s i v e .-integrations o f a  constant at  and was i n t e r p r e t e d by H e a v i s i d e In [8]  Heaviside  rather d i f f i c u l t  as being  equal t o e  •  states " I t i s i n i t s generality a and obscure m a t t e r .  I have not  succeeded i n determining'the amount o f l a t i t u d e t h a t i s permissible operators.'"  i n t h e p u r e l y a l g e b r a i c treatment o f  Consequently, he used h i s method  subject  t o independent t e s t s f o r guidance but never gave any reasons f o r what he d i d . him,  To t h o s e t r y i n g t o f o l l o w  h i s a t t a c k seemed f u l l o f i n c o n s i s t e n c i e s . Most w r i t e r s r e f e r t o h i s method as t h e  •Heaviside  o p e r a t i o n a l c a l c u l u s , except f o r Gardner and 1  Barnes Q?], along with Murnaghan  , who c a l l i t t h e  •Cauchy-Heaviside o p e r a t i o n a l c a l c u l u s " . 1  According  to  them, Cauchy about 1&L2 developed an o p e r a t i o n a l c a l c u l u s based on t h e Laplace and F o u r i e r t r a n s f o r m s , formally i d e n t i c a l with parts of Heaviside*s Consequently, they s t a t e had  n  method.  I t i s now c l e a r t h a t Cauchy  not o n l y s u p p l i e d t h e o r i g i n a l o p e r a t i o n a l  of the type c o n s i d e r e d  calculus  but had d e r i v e d i t u s i n g t h e  Laplace transformation.  He had t h e r e b y s u p p l i e d a  b a s i s f o r i t s r i g o r o u s treatment." c r e d i t f o r t h e extensive  Heaviside  applications only.  i s given  6 ..-  Heaviside was followed by many who t r i e d t o make h i s r u l e s consistent and r i g o r o u s .  One of the f i r s t  of these was Carson (1919) £ 3 j who reduced t h e problem of i n v e r s i o n t o the p u r e l y mathematical problem of s o l v i n g the i n t e g r a l equation (1.10) for h(t).  f(p) =  r°e" h(t)dt pt  But then, according t o Bush ( 1 9 2 4 ) f 2 ] t h e  method was no longer s u i t a b l e f o r an engineer, so he attempted t o define a system of operating r u l e s which would avoid.the Heaviside i n c o n s i s t e n c i e s but which would keep h i s s i m p l i c i t y of method. Bromwich (1916), £l] took an e n t i r e l y d i f f e r e n t point o f view.  He assumed that t h e s o l u t i o n of a  d i f f e r e n t i a l equation could be expressed as a contour i n t e g r a l i n the complex plane of the form (1.11)  0(t)»l/2rri(  p(X)e dX Xt  where t h e path of i n t e g r a t i o n was t o the r i g h t of a l l the s i n g u l a r i t i e s of p ( M and p(X) was t o be determined from the d i f f e r e n t a i l equation and i n i t i a l c o n d i t i o n s . This approach had been suggested e a r l i e r by Cauchy who had solved d i f f e r e n t i a l equations by means of contour integrals. I t was not u n t i l t h e time of Levy (1926), [JLO] that the two points o f view were r e c o n c i l e d .  He  - 7 -  showed t h a t , under c e r t a i n c o n d i t i o n s , t h e s o l u t i o n o f r  (1.12)  e + «cc  h(t) = l/2tTi \  p  t  e  f(p)dp  was (1.13)  f(p) -  (  e" h(t)dt p t  and v i c e v e r s a  Jo Of t h e many o t h e r s who a l s o t r i e d t o p l a c e H e a v i s i d e s method on a f i r m b a s i s , some of t h e more f  prominent ones a r e Cohen, Berg," Carslow, Jaeger, and  Marsh,  Josephs. Those who have c o n s i d e r e d t h e L a p l a c e  t r a n s f o r m from a more t h e o r e t i c a l p o i n t o f view a r e J e f f r i e s , van der P o l , Doetsch, McLachlan, and C h u r c h i l l . The  Laplace transform i s u s u a l l y considered  as a Riemann o r a Lebesque i n t e g r a l but i t c a n a l s o be c o n s i d e r e d as a S t i e l t j e ' s i n t e g r a l ,  C  e" d<3?(t). st  A complete development o f the t h e o r y f o r t h i s i s g i v e n by Widder  transform  £l$3  #  An e x t e n s i v e b i b l i o g r a p h y of t h e h i s t o r i c a l development can be found  \  i n Gardner and Barnes £ 7 J .  Chapter 2 Development of t h e  Theory  The asymptotic behaviour of a f u n c t i o n can be expressed i n s e v e r a l ways. I f F ( t ) - G(t) - o ( l ) i as t -> °°,  (i) F(t)  then  i s d e f i n e d t o b e i d i f f e r e n c e asymptotic t o G ( t ) . (ii)  F(t)  I f F(t)/G(t) - 1 + o ( l ) ,  as t * o o ,  then  i s d e f i n e d t o be q u o t i e n t asymptotic t o G ( t ) . In  some i n s t a n c e s both d e f i n i t i o n s are  but t h i s i s not t r u e g e n e r a l l y . the f u n c t i o n [t]j sense o n l y .  satisfied  For example, c o n s i d e r  here [t] ~ t i n the q u o t i e n t asymptotic  Finally,  (iii) if t [F(t) - a n  Q  F(t) ^ a  Q  + a-j/t + a / t  - a /t - a^/t x  + ...  2  2  2  .... - a / t ] -> o as t -> 00. n  n  T h i s i s t h e P o i n c a r e £12] d e f i n i t i o n of t h e asymptotic expansion o f F ( t ) about may  infinity.  The s e r i e s  itself  d i v e r g e everywhere but i f t i s l a r g e , the terms  decrease very r a p i d l y at f i r s t  and the e r r o r i n  approximating to F ( t ) by the sum o f the f i r s t  n terms  i s o f lower order t h a n t h e l a s t term c o n s i d e r e d as t -> 00. An o p e r a t i o n a l method f o r d e t e r m i n i n g t h e asymptotic behaviour of a f u n c t i o n was  first  developed  by H e a v i s i d e but h i s method d i d not always l e a d t o correct r e s u l t s . method.  The f o l l o w i n g example i l l u s t r a t e s h i s  To f i n d the c u r r e n t I e n t e r i n g a c a b l e o f d i s t r i b u t e d r e s i s t a n c e R and C a p a c i t y C, i f an emf —itcfc e  i s impressed  a t time t • o, he s e t up t h e  l i n e a r d i f f e r e n t i a l equation, r e p l a c e d d/dt by p, and  obtained  0)  + p  Expanding i n a power s e r i e s o f p  (2.2)  h(p) = ^cTFp /u (1 - p / ^ + P A > . . . ) 1 2  2  4  4  and u s i n g t h e o p e r a t i o n a l r u l e s  he  obtained  (2.3)  I  ^-jC/Rnt (l/2o*  - 1.3.5/(2wt)  3  + ... )  H e a v i s i d e knew t h a t t h e steady s t a t e s o l u t i o n was /"toc72R(cos cat + s i n wt) so he came t o t h e c o n c l u s i o n t h a t t h e method had g i v e n t h e t r a n s i e n t d i s t o r t i o n . Obviously t h e H e a v i s i d e approach i s inadequate and a t h e o r y must be developed which w i l l g i v e both t h e steady s t a t e s o l u t i o n and the t r a n s i e n t d i s t o r t i o n .  For  t h i s t h e Laplace t r a n s f o r m method i s more s a t i s f a c t o r y , s i n c e the c o n d i t i o n s are p r e c i s e l y known under which  10  various.operations Heaviside s  are v a l i d .  Also, i t includes  method as a s p e c i a l c a s e .  f  We  r e c a l l t h a t the f u n c t i o n f ( s ) i s s a i d t o  be t h e L a p l a c e  transform _g  (2.4)  -  f(s) - )  o  e  of the o b j e c t f u n c t i o n F ( t ) i f  t  F ( t ) d t , Re(  s)>s  Q  I n what f o l l o w s we w i l l c o n s i d e r o n l y Riemann i n t e g r a l s , improper i f necessary.  The  function f ( s ) i s  determined u n i q u e l y by F ( t ) but t o a g i v e n f ( s ) t h e r e correspond an i n f i n i t e number o f o b j e c t f u n c t i o n s which, however, d i f f e r from each other o n l y by n u l l f u n c t i o n s . The  p a r t i c u l a r o b j e c t f u n c t i o n which i s c o n t i n u o u s  •f, throughout o r r i g h t continuous ( i . e . , l i m f(x+h) e x i s t o ) h*o i s u n i q u e l y determined.  We  s h a l l consider only  such  f u n c t i o n s , and t h e n a one-to-one correspondence e x i s t s between the c l a s s of a l l o b j e c t f u n c t i o n s F ( t ) whose Laplace  t r a n s f o r m s e x i s t and t h e c l a s s of a l l r e s u l t a n t  functions f ( s ) . The  r e l a t i o n s h i p between the  behaviour of the two here.  Two  c l a s s e s i s of p a r t i c u l a r i n t e r e s t  b a s i c t y p e s of theorem d e s c r i b i n g p r o p e r t i e s  of the t r a n s f o r m and  asymptotic  can be g i v e n ; t h e s e are the  Abelian  Tauberian theorems. The  A b e l i a n theorems p r e d i c t t h e  asymptotic  behaviour of f ( s ) at p a r t i c u l a r p o i n t s from the behaviour  ]  - 11 -  of F ( t ) .  An example would be t h e theorem which s t a t e s as t -** 00, t h e n f (s) -> JL, as s -*• o  that i f  i n s i d e the sector S: a r g s  < TT/2.  The T a u b e r i a n theorems p r e d i c t t h e asymptotic behaviour o f F ( t ) at e i t h e r zero or i n f i n i t y from t h e behaviour o f f ( s ) , but  they require that c e r t a i n  r e s t r i c t i o n s be placed on F ( t ) .  The r e a s o n f o r t h i s  can be seen from the f o l l o w i n g example. the l i m i t A a s , t -*• , then, a c c o r d i n g 0 0  theorem, f (s) ^  A/s,  I f F ( t ) has  t o an A b e l i a n  as s -*• o i n s i d e t h e s e c t o r S.  I f F ( t ) i t s e l f has no l i m i t but the Cesaro mean o f order k  has t h e l i m i t  A as t •> »  then by means o f t h e c o n v o l u t i o n  i n t e g r a l i t can be shown t h a t f (s) i n s i d e t h e s e c t o r S.  A/s, as s -*• 0  The f u n c t i o n f ( s ) i s t h e r e f o r e  i n s e n s i t i v e to whether o r not the l i m i t I f the only information  i s t h a t f ( s ) ^ A/s, as s -»- o  i n s i d e a s e c t o r S, t h e n i t i s i m p o s s i b l e t h a t F ( t ) ^ A, as t -*• 00.  to predict  For reasons l i k e t h i s , the  T a u b e r i a n theorems a r e more complicated corresponding A b e l i a n  of F(t) e x i s t s .  than the  theorems.  From now on we s h a l l  assume t h a t f ( s ) i s  known and the asymptotic behaviour f o r l a r g e t o f F ( t ) i s t o be determined.  I t would seem t h a t a T a u b e r i a n  theorem c o u l d be a p p l i e d but because F ( t ) i s unknown,  12  we do not know whether o r not i t w i l l necessary r e s t r i c t i o n s . we c o n s i d e r  To a v o i d t h i s  s a t i s f y the difficulty  an i n v e r s i o n f o r m u l a such t h a t F ( t ) = L~"*"|f(s)  Then the r o l e s o f the two c l a s s e s o f f u n c t i o n s - t h e c l a s s o f o b j e c t f u n c t i o n s and the c l a s s of r e s u l t a n t f u n c t i o n s - are i n t e r c h a n g e d .  I f c o n d i t i o n s a r e imposed  on f (s) t o ensure t h a t t h e r e e x i s t s a f u n c t i o n F ( t ) such t h a t L J F ( t ) ^ • f ( s ) , i t i s p o s s i b l e t o a p p l y a theorem, l i k e a £5  n  A b e l i a n theorem, to f ( s ) which Doetsch  , p. 2 2 4 J c a l l s an ' I n d i r e c t A b e l i a n theorem' and  o b t a i n t h e asymptotic behaviour o f F ( t ) without any r e s t r i c t i o n s on F ( t )  itself.  For t h e L a p l a c e t r a n s f o r m  the i n v e r s i o n formula  t h a t we s h a l l c o n s i d e r i s r F(t) = l/2TTil  (2.5)  Conditions  x+i x + i °° oo . e* f(s)ds s  x - i oo  .  a r e known which ensure t h a t L $ F ( t ) J  f o r t h e F ( t ) d e f i n e d by ( 2 . 5 ) [ V ,  p. 1 2 6 ] .  =f(  F o r our  purpose i t i s s u f f i c i e n t to assume an F ( t ) e x i s t s such that L)JF(t)^  • f(s).  The f o l l o w i n g theorem i s s t a t e d  without p r o o f |j>, P« 1 0 7 ] • Theorem  2 - 1 Let L ^ F ( t ) | - f ( s ) and l e t f ( s ) be r e g u l a r f o r  R(s)>s . t  If \ e  i v t  f(x+iy)dy  converges u n i f o r m l y f o r  T and a f i x e d x > s , where s •» x + i y , t h e n Q  - 13 «* X + l oo  r  (2.6)  F(t) » l/2tri \  J  e x  t s  f(s)ds  for t ^ T .  - i oo  Under t h e r a t h e r m i l d c o n d i t i o n o f t h e theorem it  can be shown t h a t r I  T  9 * dt l / 2 n i  3  J  for  T  2  rx+i« \ e x - i oo  To f (s)ds » \ T-j_ r  t s  F(t)dt  J  > Tj » T  from which the r e s u l t f o l l o w s . Without l o s s o f g e n e r a l i t y we may assume t h a t f (s) i s r e g u l a r  i n the h a l f•••plane R ( s ) > o and has a t  l e a s t one s i n g u l a r i t y on t h e l i n e R ( s ) = o. always be done by c o n s i d e r i n g  T h i s can i n s t e a d of  LJF(t)J. We s t a t e t h e f o l l o w i n g theorem without p r o o f . Theorem 2.2. Let f ( s ) s a t i s f y t h e f o l l o w i n g (i)  F ( t ) » l/2iri J  e  i s such t h a t L f F ( t ) } (ii)  t s  conditions:  f ( s ) d s e x i s t s f o r x > o and  - f(s).  f ( s ) i s a n a l y t i c and r e g u l a r f o r R(s)  some a > o, except f o r a f i n i t e  -a,  number of s i n g u l a r i t i e s  on R(s) = o. (iii) (iv)  f(s)  o as | s | •><» f o r R(s) ^ - a.  e 3 f(-a  I  1  r c  + ty)dy  converges u n i f o r m l y f o r  t > T then (2.7)  F ( t ) = l/2ni  \  e  t s  f(s)ds  — 14 •»  (where C i s any  contour from (-a-i°°) t o  (-a+i°°) such  t h a t a l l s i n g u l a r i t i e s of f ( s ) are t o t h e l e f t of it)} and  the c o n t r i b u t i o n from p o r t i o n s a l o n g R(s)  are o{t )  * -a  as t -> °°.  mat  From now  on we  s h a l l say t h a t f (s) belongs t o  the c l a s s A i f i t s a t i s f i e s the c o n d i t i o n s  of  this  theorem. We  s h a l l assume f i r s t t h a t f ( s ) has  s i n g u l a r i t y on the l i n e R(s)  only  one  = o, namely at s =  o.  Then C i s a contour of the type i n F i g . 1.  S - pMne, FIG. The  1.  s i n g u l a r i t y t h a t f ( s ) may  be e i t h e r  have at the o r i g i n  a p o l e , a branch p o i n t  singularity.  We  or an  essential  s h a l l show t h a t each type o f  of the t r a n s f o r m at the o r i g i n l e a d s to a behaviour of the o b j e c t f u n c t i o n at Consider G(t,v) » t / (v+1). v  the t r a n s f o r m l / s  v  +  1  can  singularity  certain  infinity. This function  f o r R ( v ) ^ - 1 , and R(s) >  o.  has For  a l l o t h e r v the i n t e g r a l d e f i n i n g the t r a n s f o r m does  - 15  -  not converge at the o r i g i n . Define (2.8")  r o . o  <  t  t t/r(v+l)  G(t) =  Then L J o ( t ) | =  ;  t  1  >  1  ^ - t /r(v+D e  st  v  st  = l/s  v +  ^  , v / -1,-2, . • dt.  t / r ( v + l ) dt - j e - H 7 f ( v + l ) d t  For R(v) > - 1 , L{G(t)f = ^ e~  For R(v) < - 1 ,  <  s  v  + integral, function.  we can choose an i n t e g e r n such t h a t  - 1 < R(v+n) < o.  I n t e g r a t i n g by p a r t s n times we  obtain  LfG(t)} = e - [ l / r ( v + 2 ) - . s  s -Vr( n  v+n+1  )}  +  s ien  s t  t ' v  + n  /r(  ) t  v + n + 1  d  The f i r s t term i s an i n t e g r a l f u n c t i o n of s; the i n t e g r a l can be t r e a t e d the same as b e f o r e I» \ G ( t ) \ = l / s  v  +  1  and so  + integral function, R(s)>o,  Furthermore, L-(G(t)| s a t i s f i e s c o n d i t i o n s of Theorem 2.2. transform  9  (i)-(iii)  Because the i n t e g r a l d e f i n i n g the  converges a b s o l u t e l y and G(t) i s of bounded  v a r i a t i o n , the i n v e r s i o n f o r m u l a h o l d s £_6, p. 3 6 0 J • a p p l y i n g the same procedure as Doetsch £ 6 , p. show t h a t  v ^ -1,-2, .  By we  can  :  - 16 (2.9)  L{G(t)f- e  i ( v + 1 )  o  * p  e  -  s  e  l  t  r  r / r ( v + l ) dr v  JL  v+1  = l/s  + integral function  which d e f i n e s t h e a n a l y t i c c o n t i n u a t i o n i n t o the h a l f - p l a n e R U e ^ ) > o.  The f i r s t  1  zero as K s e ^ )  i n t e g r a l tends t o  -* °° f o r each f i x e d R ( s e ^ ) > o;  second i n t e g r a l tends t o zero as y  °° f o r a l l x  provided -rr/2 < \jr< o and tends t o zero as y a l l x provided o < t  < TT/ * 2  as / s| -*• PO f o r R(s)  -a.  T h e r e f o r e LfG(t)j.  for o  We have not been able t o show  t h a t i t s a t i s f i e s c o n d i t i o n ( i v ) of Theorem 2.2 assume i t does so.  the  but  Then L^G(t)j belongs t o t h e c l a s s  shall A.  The e s s e n t i a l p r o p e r t y of G(t,v) used t o get t h e above r e s u l t s i s t h a t i t s a t i s f i e s t h e  (2.10)  d/dt G(t,v+1) =  equation  G(t,v)  Many s p e c i a l f u n c t i o n s are known t o s a t i s f y t h i s [l4j  and the same method may  be a p p l i e d t o them.  example w i l l serve t o i l l u s t r a t e what new  equation Another  r e s u l t s can be  obtained. I t i s known t h a t  (£fc)^J  (.2fkt ) s a t i s f i e s 1  (2.10) and has the t r a n s f o r m  f o r R ( v ) > - 1 , R(s)  Define o (  2  -  1  1  }  G  (  T  )  and proceed  "  J  o < t <  ^/KV^JV  as before*  equation  (2ye)  1 , t  >1  >o.  - 17 -  Then (2.12)  L$G(t)j  + integral function, R(s)>o  f o r a l l v and furthermore i t belongs t o the c l a s s A i f we assume c o n d i t i o n ( i v ) o f Theorem 2.2. I n these two examples the above arguments a l s o apply when G(t,v) i s r e p l a c e d by d / n  d v I 1  G(t,v) o r  Thus we have generated a number o f f u n c t i o n s whose t r a n s f o r m s have the t y p e s o f s i n g u l a r i t y at the o r i g i n t h a t were mentioned e a r l i e r .  The f u n c t i o n s G(t) and  t h e i r t r a n s f o r m s (except f o r t h e i n t e g r a l f u n c t i o n s ) are g i v e n i n t h e t a b l e on t h e f o l l o w i n g page.  - IS -  i  G  -! 1  , 0  O £  t  *  <-  '  r o  , ofet t ;  1 1 1  0  , O fe -fc 4. »  1  -Or  !  !  none  0  j  , 0 A-  t  <- /  >-  -  >•  none.  i j t  O  (  0fert  <- '  -fe  -* none  - 19 The  asymptotic behaviours o f the f i r s t  four  f u n c t i o n s i n t h i s t a b l e are w e l l known, but a few remarks should  be made about t h e l a s t two.  Function  G(v) t /r(v+l)dv v  which have been s t u d i e d e x t e n s i v e l y by Colombo £4]« I f L^Gft)} = g ( s ) e x i s t s f o r R ( s ) > o, t h e n t h e L a p l a c e transforms o f these f u n c t i o n s e x i s t s and LJ  I°G(V)  t /r(v+l)dv] V  a g(>£ns)/s  1  R(s) >  In p a r t i c u l a r  Lf J "  ^ £ *A > i  "R(^)?-',^'s>> For m = o, i t c a n be shown by means o f a c o n t o u r integral representation that  (2.13)ft /n ^ l) «---  -t /jtntr(v+l), t  ^  (-l)" (-v-l)!t /(^nt)  v+u  v+  +  d  oo,  v  o  V  v  2  v/-l,-2  ,  t -»- oo, v = -1,-2,-; Function  (7) i s r e l a t e d t o t h i s c l a s s o f f u n c t i o n s i n  that  j ^ t S j ^ U f t W - j*J ,(2fw)da}j a /r(t+l)dr ,P  5:  <  >  A g a i n by means of a contour i n t e g r a l I t c a n be shown that f o r k  o  -  (** (2.14)  ^  J  ( % ) - J  The  20  u  l * ^ ) ^ * - - *  f u n c t i o n s g i v e n i n t h i s t a b l e a l l belong t o  t h e c l a s s A and  may  be used t o express the  behaviour-of our unknown f u n c t i o n F ( t ) . we  s u b t r a c t one  transform  of the f u n c t i o n s G(t)  has the  to Theorem  I f from F ( t )  of Table I whose  same s i n g u l a r i t y at s  f ( s ) , then according  asymptotic  B  o as t h a t  of  2.2.  (2.1$) F ( t ) - G ( t ) - l / 2 n i ^ e $ f ( s ) * g ( s ) - i n t e g r a l i d s . £ ' function i of s S t  Because the the  integrand  i s now  regular f o r R ( s ) ^ -a,  i n t e g r a t i o n can be taken along t h e l i n e R(s)  and t h e  = -a  difference F ( t ) - G(t)  » o(e"  a t  ) , as  I f none of the f u n c t i o n s G(t) transform  t*«, of T a b l e 1 has  w i t h the same s i n g u l a r i t y at s=o  as  that  a of  f ( s ) , then L J F ( t ) - G ( t ) j w i l l not be r e g u l a r at t h e origin.  Consequently we  Theorem  2.3.  need the f o l l o w i n g theorem.  Let f ( s ) belong t o the c l a s s A and have s i n g u l a r i t y on R(s)  one  = o at s - o; l e t L (G(t)|= g(s)  i n t e g r a l f u n c t i o n , where G(t)  +  i s a f u n c t i o n from T a b l e  I f f ( s ) - g ( s ) i s not r e g u l a r at s  8  o  and  1.  21  f ( s ) - g(s) (2.16)  • o|/ty(s)/j as  JF[t)  - G(t)/ < A  |s/ -*• o,  then  / e H s ) / ds t s  j  ,  £-></>  *-1 where  i s the p o r t i o n of t h e  1  contour C of FIG.  l y i n g i n s i d e a s u f f i c i e n t l y small c i r c l e about the  origin.  Proof. Given an e>o,  there  R such t h a t i f / s / < R, Choose a < R. i n t o two  p a r t s ; C^ the p o r t i o n l y i n g i n s i d e the C  2  circle circle.  belongs t o the c l a s s A, the i n t e g r a l  i s o ( e " ) as t •><»,  Therefore,  a t  2  divided  the p o r t i o n l y i n g o u t s i d e t h i s  Because L J F ( t ) - G(t)}  small  | f (s) - g ( s ) | < e | ^ ( s ) | .  Then the contour i n t e g r a l can be  o f r a d i u s R and  along C  exists a sufficiently  ( 2 . 1 7 ) / F ( t ) - G ( t ) | =|l/2TTi J  by  (2.15)  e j f ( s ) - g ( s ) - i n t e g r a l functionjd t s  **i + But and  \  e J i n t e g r a l f u n c t i o n ^ ds • o ( e " t s  a t  ),  o(e" ) a t  as t -*-«>,  so, ;  |F(t) - G ( t ) | = | l / 2 t i i j < e/2-nr ^ j e  e * J f ( s ) - g ( s ) ] ds + oU' ) S  t 5  t ( s ) I ds  ai  j0  u4±->c*>  which i s the r e s u l t of our theorem. From t h i s theorem i t f o l l o w s t h a t i f f ( s ) - ©{|s/kj, \ r e a l , then  - 22  (2ol^)  F(t) = o ( t " " ) X  -  •-  1  v  , c^>  t  -  o  The t h e o r y developed i n t h i s chapter g i v e s d i f f e r e n c e asymptotic r e l a t i o n s h i p s ,  A function  which f o r a l l T has d i s c o n t i n u i t i e s f o r t  T  cannot be d i f f e r e n c e asymptotic t o a continuous f u n c t i o n as was  seen e a r l i e r f o r £t].  Since our  comparison f u n c t i o n s are a l l continuous f o r t > l , F ( t ) must be continuous f o r l a r g e t .  • 23 -  Chapter 2 G e n e r a l i z a t i o n s of t h e Theory We  s h a l l now  extend Theorem 2,3 to i n c l u d e the  case when f ( s ) has a f i n i t e number o f  singularities  on t h e l i n e R(s) => o, and t h e case when f (s) has an asymptotic expansion about t h e s i n g u l a r i t y a t s = o. we  s h a l l show how  Lastly,  the asymptotic behaviour of F ( t ) can  be obtained f o r v a r i o u s arguments o f t . Suppose f ( s ) has a f i n i t e number of  singularities  on the l i n e R(s) « o at s = s^, k <s 1 , 2 , . . . , n , suppose that f ( s J ^ g ^ C s ) , as s •> s^. t o t h e c l a s s A, then F (t) * l / 2 t T i j (  where C  w  and  I f f (s) belongs e f(s)ds, t t s  ';  i s a contour o f the type i n FIG, 2 . 9  .4  J  ——/•  cr' -  .  ../  V  .S  — — V  )  V,  s FIG, 2 The comparison f u n c t i o n s i n Table 1 a l l have transforms whose s i n g u l a r i t i e s a r e at the o r i g i n .  If  -  2 4 -•.  s i s replaced by s - s^, then gts-s^) has a s i n g u l a r i t y at s  5  s^, and the corresponding  comparison function w i l l be e * G ( t ) .  We  shall  say g(s) i s a function from Table 1 even though i t s s i n g u l a r i t y occurs at s^ instead of the With the a i d of Theorem 2 . 3 the following  origin.  theorem i s r e a d i l y proved. Theorem 3.1 Let f ( s ) belong to the c l a s s A and l e t f ( s ) have s i n g u l a r i t i e s at s , k • 1,2,...n with &(s ) k  If f ( s ) / ^ Sjj.(s) as s +  k  * o.  s , where g^s i s a function k  from Table 1, then F(t) - £ QAt) *, * s  = l/2ni\  c"  e^jfts)  £ { I e* c  where C  fc  k  i s the portion.of C  5  - £ g ( s ) - integralV function H  1  gjs)|dsj,  *  I  a*  ±-»«o  l y i n g inside a  n  s u f f i c i e n t l y small c i r c l e about s^. Because we can neglect at most a f i n i t e number of terms that are o ( e " ) , f ( s ) can have only a t  a f i n i t e number of s i n g u l a r i t i e s on the l i n e R(s) = o. It may  happen that f ( s ) has an asymptotic  expansion about the s i n g u l a r i t y at s * o. case ve can prove Theorem 3 . 2 .  In t h i s  r 25 -  Theorem 3,2 L e t f ( s ) belong t o t h e c l a s s A, Z~ g-?(s) + oi  f (s) »  j=l  g (s) \ M  If s-* o  ,> s  -  3  where g.s(s) i s a f u n c t i o n from T a b l e 1 t h e n F ( t ) - f G,(t) ^ 4 I / 2 T T (  |e g t S  0^i^<*>  (s)|dsL  -A combination o f t h e above two cases i s a l s o possible. Theorem 3 ,3 Let f ( s ) belong t o t h e c l a s s A and l e t f ( s ) have s i n g u l a r i t i e s at s^, k = +1,2,.,.n, w i t h R ( s ) « o. I f k  f(s) -  ^k  ;  g  J k  r 7 ( s ) + a ^ g ^ ( s ) \ , as a - s K  k  where g ^ ( s ) i s a f u n c t i o n from Table 1, t h e n x  F(t)  where  Z  G  -n>> <  v \ l g ^ (s)e lds K A d ^  \Z-  i s the p o r t i o n of C  t s  k  K  n  lying inside a  s u f f i c i e n t l y small c i r c l e about t h e s i n g u l a r i t y  T h i s l a s t theorem has been proved e a r l i e r by Sutton, [ifj , f o r p a r t i c u l a r f u n c t i o n s f ( s ) .  He  assumed t h a t about each s i n g u l a r i t y , f ( s ) / s could be  - 26 -  expanded i n a convergent power s e r i e s such t h a t t h e contour i n t e g r a l along t h e c i r c l e s about t h e s i n g u l a r i t i e s tended t o zero as t h e r a d i u s o f t h e s e c i r c l e s approached zero. So f a r we have c o n s i d e r e d v a l u e s o f t , but t h e theory i n c l u d e complex v a l u e s  F ( t ) only f o r r e a l  can be g e n e r a l i z e d t o  as has been done by Doetsch.  I f t h e path o f i n t e g r a t i o n o f J changed t o a l i n e p a s s i n g  e"" F(t)dt i s st  through t h e o r i g i n and making  an angle q> w i t h t h e r e a l a x i s , and the i n t e g r a l converges f o r some's, we w i l l o b t a i n a f u n c t i o n which depends on cp, namely L^F{  - e f> \ i(  o  e-  e l 9  s F(re r  I (  P)dr. .  I t c a n be shown t h a t the i n t e g r a l has a h a l f - p l a n e o f convergence and t h a t , i f L 9 , ( F } and L<P«-}FJ have a common r e g i o n o f convergence, t h e n L ^ ' ^ F f « i / ^ F J i n that region. the  Therefore,  by l e t t i n g 9 v a r y we may generate  a n a l y t i c continuation of f ( s ) . Except i n t h e t r i v i a l case, F ( t ) • o, f ( s ) w i l l  have a s e t o f s i n g u l a r i t i e s which c a n be enclosed convex r e g i o n .  in a  The convex h u l l i s d e f i n e d t o be t h e  i n t e r s e c t i o n of a l l these r e g i o n s .  A l i n e tangent t o  the convex h u l l i s c a l l e d a " s u p p o r t i n g  l i n e " and forms  the boundary l i n e o f t h e r e g i o n i n which L^^FJ- i s r e g u l a r , where 9 i s t h e angle t h a t the normal t o t h i s  "supporting  - 27-  l i n e " makes w i t h t h e r e a l a x i s . behaviour o f F ( t ) f o r t = r e  1 < } >  ,  The asymptotic r -»• °° a n t h e n be C  found from t h e s i n g u l a r i t i e s on t h i s l i n e . Starting  from an F ( t ) d e f i n e d f o r t > o, we  therefore obtain f ( s ) = L ^ F ( t ) J .  In p a r t i c u l a r ,  i f f ( s ) has a f i n i t e number o f s i n g u l a r i t i e s , we may o b t a i n the a n a l y t i c  continuation of F(t) f o r  a l l arguments o f t by l e t t i n g t h e " s u p p o r t i n g r o t a t e about t h e convex h u l l .  line"  - 2a Chapter l± Applications  We s h a l l conclude w i t h a d i s c u s s i o n o f t h r e e examples which i l l u s t r a t e t h e method we have developed.  The examples are chosen t o i l l u s t r a t e  c e r t a i n obvious g e n e r a l i z a t i o n s o f the method. Consider f i r s t t h e f u n c t i o n  (4.1)  F(t) =  f°°  J ( y ) / y dy G  which has t h e t r a n s f o r m  (4.2)  L jF(t)}f(s)  «  JU  ( i T ^ - S ) /  S  T h i s f u n c t i o n belongs t o t h e c l a s s A and has branch p o i n t s a t s = + i .  Since  s + l - s i s never 2  zero -£n(/s*"+l - s) may be expanded i n a power s e r i e s . Near s =  (4.3)  -i^  Us)  ~> ****  Near s • 1,  that  (i 4  f ( s ) behaves l i k e t h e complex conjugate o f  t h e above expansion. follows  WM^-^^S^I)^  From 2.IS  and Theorem 3.1  it  29  (4.4)  Ftti - ^ ^ ( t - % ) $ ,  In  g £  the above example the remainder term was of  the form 6,j/s'^jr.  To show that our method a p p l i e s  even i n more complicated cases, consider the f o l l o w i n g transform belonging t o t h e c l a s s A.  (4.5)  f(s) =  ^  ,  Z  s  * * ° ^ / S ^ ' x ^ / j ^ as l - ? o  5  By Theorem 3.2 and case (3) of Table 1  (4.6) Ftfi}Put  (4.7)  U ^ s ^ ' l M s ^ - t ^  Ze>fe)'*X;l*™  s = ufk/tT, ds = f k/t* du,  J c  and o b t a i n  - ftf* $ |**'-*4>^lH*l c  »  Assume Re(/lc) > o.  For C*" take the contour of  FIG. 3, and then the greatest c o n t r i b u t i o n t o the i n t e g r a l w i l l come from a small i n t e r v a l near u =• 1. Divide C* i n t o two p a r t s , I and I I ,  as shown.  - 30 -  I t - j"'«ht FIG. 3 On I , u = e*  ff  I  X  Choose 6 ° t VT o  ^4  Z  ** , then  i s s m a l l but  i s l a r g e and our i n t e g r a l i s equal t o  /«•«  «-*/•  3  l/ft  /  -31 -  (4.3).  which i s o f lower order than  Similarly  i t can  be shown t h a t t h e i n t e g r a l along t h e remainder o f C* i s o f lower o r d e r .  (4.7), (4.9)  and  (4.6)  we have  f(t) ^  This result  Combining these r e s u l t s w i t h (4.S),  £ tn*/£\'^Ji%f^)  +*  i s s t i l l t r u e i f R(/l?) - o.  An example o f a f u n c t i o n which has p o l e s on the l i n e R(s) = o i s  (4.10)  f ( s ) = /ns/l+s . 2  The f u n c t i o n i s r e g u l a r f o r R(s) > o, but has p o l e s a t s » + i and a l o g a r i t h m i c s i n g u l a r i t y at t h e o r i g i n . The convex h u l l i n t h i s case i s simply a s t r a i g h t  line  F o r o < a r g t < TT o n l y the s i n g u l a r i t y  from i t o - i .  at s • - i c o n t r i b u t e s t o t h e asymptotic  behaviour and  F ( t ) /v T £ e"* ; s i m i l a r l y f o r - T T < a r g t < o , o n l y xt  F{t)^%e^;  the s i n g u l a r i t y at s = i c o n t r i b u t e s and  but f o r r e a l v a l u e s o f t a l l t h r e e s i n g u l a r i t i e s contribute. Near s = o, f ( s ) »  JU% 2-60* s  l  * -r  oS/s/ ^7 T  - 32 -  Then by Theorem 3 . 1 (4.11)  F(t)  The i n t e g r a n d  and  -  TTjt  7  5>J***£  -7  A».o  i s now r e g u l a r at s = + i , so we  obtain  so  (4.12)  F(fc) ~  <^t  - >  In a p r a c t i c a l problem  ^  -ai  (TT/2^  £f  cos t would be  i n t e r p r e t e d as t h e steady s t a t e s o l u t i o n and t h e s e r i e s as t h e t r a n s i e n t d i s t o r t i o n . We a r e now i n a p o s i t i o n t o see why method breaks down i n the case c o n s i d e r e d  Heaviside s  earlier.  our n o t a t i o n f ( s ) i s e q u i v a l e n t t o k(p)/p; t h e r e f o r e  res)«  yj>  fs ™JI  T  In  33  T h i s f u n c t i o n has a branch p o i n t at s = o and at s = lico.  By expanding i n a power s e r i e s H e a v i s i d e  completely n e g l e c t e d the p o l e s . i n t o account  poles  the c o r r e c t r e s u l t  I f these are  taken  i s obtained.  Under s p e c i a l c o n d i t i o n s H e a v i s i d e " s o p e r a t i o n a l method g i v e s the c o r r e c t r e s u l t .  The method g i v e n here  i s more g e n e r a l i n t h a t i t can d e a l w i t h f u n c t i o n s l i k e Jbaz or e  5  which do not have an expansion about t h e  o r i g i n and i t removes the apparent  c o n t r a d i c t i o n s and  a r b i t r a r i n e s s o f H e a v i s i d e s method. f  Bibliography 1  Bromwich, T . J . , Proc. Lon. Math. Soc.  ( 2 ) , 15  (1916) 401-44S. 2  Bush, V., J . Math. Phys. Mass. I n s t , o f Tech.  3 (1924) 95. 3  Carson,  J.R., B e l l System Tech. J o u r n a l , Nov. 1922,  1-13. 4  Colombo, S., B u l l e t i n des Sciences Mathematique ( 2 )  77 (1953 ) 6*9-104. 5  Doetsch,  G., T h e o r i e und Anwendiing der Laplace  Transform,  1943, Dover P u b l i c a t i o n .  6  Handbuch der L a p l a c e T r a n s f o r m a t i o n I , 1950^  7  Birkhauser Basel.  Gardner, M.F. and Barnes, J.L., T r a n s i e n t s i n L i n e a r Systems.  £  1942, New York, W i l e y .  H e a v i s i d e , 0 . , Roy. Soc. Proc. 5 2 (1893) 504.  9  E l e c t r o m a g n e t i c Theory, 1 (1893), 2 (1899),  3(1912).  New York, Van  Nostrand.  10  Levy, P., B u l l . Soc. Math. ( 2 ) , 5 0 (1926) 174-192.  11  Murnaghan, F.D., Amer. Math. Soc. B u l l . , 33 (1927)  81-89. 12  P o i n c a r e , H., A c t a Math., g (1926) 2 9 5 .  13  Sutton, W.G.,  14  Truesdell, C ,  J . Lon. Math. Soc. 9 (1934) 131-137. A U n i f i e d Theory of S p e c i a l F u n c t i o n s ,  Annals o f Mathematics S t u d i e s Number 18, 1948, P r i n c e t o n Univ. 15  Press.  Widder, D.V., The L a p l a c e Transform, Press.  1941, P r i n c e t o n Univ.  

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