A Clarification of Heaviside's Operational Calculus Citation Widess, Moses B. (1936) A Clarification of Heaviside's Operational Calculus. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4jkh-wk78. https://resolver.caltech.edu/CaltechTHESIS:11192024-180047006 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Electrical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1936 Record Number: CaltechTHESIS:11192024-180047006 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11192024-180047006 DOI: 10.7907/4jkh-wk78 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16865 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 20 Nov 2024 18:49 Last Modified: 20 Nov 2024 19:09 Thesis Files PDF - Final Version See Usage Policy. 17MB Repository Staff Only: item control page

A CLARIFICATION OF HEAVISIDE'S OPERATIONAL CALCULUS Thesis by M. B. Widess In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy, California Institute of Technology Pasadena, California 1 9 3 6 CONTENTS Page 1. Introduction 1 2. The Unit Function 3 3. Multiple Order Impulses 9 4. Algebraic Properties of the Operator II p" 11 5. Inverse Polynomial Operators 13 6. Linear Differential Equations with Variable Coefficients 18 7. Fractional Order Derivatives 21 8. Infinite Non-Inductive Cable 24 9. System Initially Not in Equilibrium 33 Swnmary 41 Appendix: Non-Inductive Cable Terminated by Inductance 44 References 53 10. - 1 - 1. INTRODUCTION Heaviside's operational calculus 1 is fast becoming one of the most useful tools in the solution of differential equations, its power -- becoming generally realized and its methods expanded. However the procedures now employed, far different from those initially developed by Heaviside, involve contour integration and integral equation solution~ rather than .the much simpler algebraic manipulations of the original treatment. The supplantation was necessitated since Heaviside's treatment, making no pretense to rigor, involves unexplained rules and occasionally leads to contradictions. In effect his methods have remained merely processes of rapid evaluation which have required verification from external sources before the solutions could be deemed acceptable. ,. Because of its remarkable simplicity, Heaviside's treatment would be of great value if it could be placed on a sound mathematical basis, despite the fact that contour integral methods can now be used to evaluate operational equations with complete mathematical rigor. It would be well to correlate the contour integral methods with Heaviside's methods in theory as well as in application. - 2 - Since with only a few exceptions Heaviside's treatment leads to correct solutions, the situation indicates that a fundamentally correct procedure underlies the work, requiring only a more careful analysis of its basic elements before rigorous operational manipulations may be discovered. Though several attempts 3 have been made to render the treatment rigorous, none of the suggested methods has succeeded in retaining the original simplicity,and our present mode of attack will be different from the above. By but slight modification of the fundamentals of Heaviside's treatment, it will be shown possible to place the work upon a sound basis, retaining the algebraic manipulations and their accompanying simplicity. In the course of the paper the established methods will be applied to pertinent topics covered by operational calculus, permitting the treatment to be extended directly to more involved problems. The ambiguity and uncertainty of earlier operational procedures will be eliminated and the difficulties encountered by Heaviside clarified.* *For the sake of expedience, we shall adopt the present tendency of referring to Heaviside's treatment simply as "operational calculus" and to Carson's infinite integral equation methods and Bromwhich's and Wagner's contour integral methods as "Symbolic calculus". - 2. 3 - THE UNIT FUNCTION To represent a force which is suddenly applied at the time t=o , Heaviside formulated the so-called "unit function", designated by the symbol " j(fJ ", which he defined as a function such that it is zero for time before f=o and unity after of unit function at t=O tinuous at that instant. t:o • He did not specify the value , intimating it to be disconSuch a definition of iffJ is used satisfactorily in symbolic calculus because the latter, being based upon Fourier transforms, is not disturbed by a function with a single discontinuity. However, in Heaviside's calculus, where, for example, infinite derivative series operating on unit function are encountered, a discontinuous unit function presents serious mathematical difficulties, which have caused the methods to be non-rigorous. In the mode of approach which we shall adopt, the discontinuous unit function will be replaced by a continuous one, all of whose derivatives are continuous, though it will be unnecessary to designate the latter function explicitly by a specific expression. Let us consider the electrical phenomenon which is most frequently represented by the unit function, namely that of a constant electromotive force suddenly impressed at time t=o upon a circuit. In such an instance the potential - 4 - established across the terminals of the circuit, though the potential appears to rise immediately, of course requires an appreciable time to build up to its full value. In actuality the exact nature of the potential rise at the terminals of the circuit cannot be determined. The problem of indeterminateness, however, is generally of no consequence, and representation of the phenomenon is considered to be approximated with sufficient accuracy by a stepped function. It is our intention now to examine whether the phenomenon can be represented also by a function which is continuous and all of whose derivatives are continuous. The investigation will be complete if it is shown that the difference in the effects of the forces which are represented by the two functions can be made as small as desired. The response in the neighborhood of t=O is excluded from consideration because at that value for t the continuous and the discontinuous unit functions are not made even approximately equal, though the neighborhood can be made as small as desired. Since only linear differential equations are considered in operational calculus, we may apply freely the superposition theorem (or the Boltzman;Hopkinson theorem). Using electrical nomenclature after the manner of Carson let f1 (tJ be the indicial admittance* of a circuit. 4 , Then *The "indicial admittance" of a circuit is its current response to the application of a discontinuous unit function potential. 5 the current * potential response zdJ due to the application of to the circuit is given by e dJ (I) /dJ = f_~rt-u) de<«) • When is identically zero for negative values of e(fJ t , the current response becomes z'rtJ = Jt f t c-1rt- "') e<u.Jclu • We shall proceed to compare two responses: is the current response when where inctJ second, is given by ~<tJ , jdJ /~ct>, is the discontinuous unit function; and the ~ <ti Jr!; idJ e(t; the first, , is the response when , where .lrt1 ect1 is given by is the continuous unit function which is under consideration. Then the difference in the For physical problems, the difference is a maximum when the indicial admittance is exponentially cumulative, fqrtJ = c, E a.t , and the force function is exponentially dissipative, /tt1= ~ c_ 6 t positive. , where both a and .,6 are real and These conditions then represent the most exacting test which we need apply to the unit function. As an example of a function which satisfies the necessary conditions for .frt1 in that it and all its derivatives *we shall designate current by the letter II z " and the ..,-;-, -, , II J II . • ' imaginary uni' t , 6 are continuous and can be made to approximate a.a :l<tl closely as desired, we may cite Maxwell's "discontinuous" function 5 : ;YI (t) ,.. which may be written {:J) 6 The function becomes discontinuous in the limit as so that h--+ oo ' must be kept finite in our treatment. n It remains to determine whether Maxwell's function is such that, when it is used for ,i(t), small as desired by choosing n sufficiently large. .::1 i(tJ can be made as For simplicity we shall evaluate 4idJ for finite negative values t of not in the neighborhood of negative value of acts as c' '?rt , t , if where n tto • Then for a given is sufficiently large, i rfJ is identified by /Vltf J Jct> Intro- • ducing the above most exacting conditions and the Maxwell function into Eq. (2), we obtain on evaluation that t (0 Here ,:) ict J can be made as small as desired by choosing sufficiently large. when h n The additional difference in response t>o obviously can be made as small as desired by choosing sufficiently large. We observe then that even for the most severe case which we may have occasion to encounter in physical problems, the difference of response to .f ttJ trt1 where 1 rt J is Maxwell' s fun ct ion, and to / <t J f_p <f-J , ;t.1>rt1 is identically zero for made -t<o , where may be.,,.vanishingly , - 7 - small by the proper choice of the parameter n . The above consideration has been limited to application to physical problems, though it seems probable that the procedure may be extended directly to a rigorous mathematical treatment. We observe that Maxwell's function would serve admirably for our proposed unit function since: vatives are continuous; it and all its deri- it can be made to approach the discontinuous force function as closely as desired; and the difference of response to the two functions can be made as small as desired. Of course other functions can be made to serve equally favorably as the unit function. Nevertheless, because the exact nature of the unit function need never be ascertained, it is preferred to leave the function undefined explicitly. There is another important reason for not defining the unit function by any particular expression such as the Maxwell function. a resistance 72 Consider as an illustration the problem of suddenly inserted at time t=o across the terminals of a battery of constant electromotive force Then the potential across .R is £ .ZtfJ and the charge which has passed through the resistance is given by j ,. : If i t .f<u.)du, (5) /cf J is defined by Maxwell's function, we obtain on integration that • - 8 - The factor -j-o .. ,,1.101 cosi,;t) and all its derivatives are continuous, and also, like lfu.xwell's function, the factor can be made to approach the discontinuous unit function as closely as desired by choosing n sufficiently large. Not- withstanding this, if Jc0is defined by Maxwell's function, we cannot write (7) Though in the limit as n- = , the above factor and Maxwell's function are equivalent, we are prohibited from going to that limit when continuous functions are demanded. That is, since in operational equations we permit only continuous functions for operands, when j reappears in an operational equation, /7 must remain finite. It is only in the final solution that the limit can be taken. These considerations lead us to define the unit function which we shall employ in operational calculus in the following general manner. by " .fch ", # The unit function, denoted is a function such that it and all its deriva- tives are continuous and one which can be made to approach the discontin·u ous stepped function of Heaviside as closely as desired (except in the neighborhood of the step, which neighborhood can be made as narrow as desired). - 9 - 3. MULTIPLE ORDER IMPULSES The unit function not being specified explicitly, the derivatives of the function also remain unspecified, though just as in the case of the unit function, this indeterminateness generally introduces no difficulty. The first order impulse, namely zero for all values of t j .Z<fJ , approximates except in the neighborhood of where a sharp peak occurs. t ~o , For our purposes the property of the impulse which is of importance is that the area bounded by the curve ,Id; d~ t and the axis must be unity and that the area must be concentrated as closely to the instant as desired. t=o Both of these conditions are obviously satisfied by the definition of Jrt;. In regard to multiple order impulses an analogous situation exists. It is noted that zero for all values of provided that n t ft-,, f<fJ approximates except in the neighborhood of is flnite. (The infinite order impulse does not exist for any value of t ). The other property of the multiple order impulse which must be considered is the value of the integral 1;(u1 dn-:/ruy,,_ , /_o:, du" up to and includd.ng the of f=O. by parts. where Jcti and all its derivatives , exist in the neighborhood (h-rJ th The expression is readily evaluated on integration We begin with the first order impulse: L t /tt<.) _ 00 d .L(U) du cl{,(. -,= J ((}J 1 d) ., - 10 - a result which is a consequence of the fact that the area bounded by the curve cft. fct; and the t axis is unity and concentrated in the neighborhood of integrating by parts, we obtain I a t d"J(l(J du f {tc) du :t 2 = fruJ ,1,l(ui It dtt. DO Similarly on • Lt/r"J -oO -Oo d.ltu.J c/a. du. ' which becomes The method may be extended to the general case: n jim'fe , t ,;4 O. This identity gives the important property of the (8) nth order impulse. An interesting application of a derivative series is d the expansion of the operator c a.;n , which is defined by the infinite series corresponding to an exponential. the operand is If we obtain frt1 (9) This series is recognized as an expansion about t,o by Taylor's theorem, i. e. ,I (it" C ti /dJ =' ..1 ft .. ct) , (/0) where the unit function has been transferred an interval of time, a. The correlation with a Taylor's series has been only made possible ,. by defining continuous. " lcfJ and all its derivatives as It is observed that, though individual multiple - 11 - order impulses including those of infinite order are not evaluated, the infinite series yields a result. ALGEBRAIC PROPERTIES OF TEE II p II OPERATOR 4. The customary notation wherein "p II denotes the derivative operator throughout. and "? 11 11 will be used denotes Consistent with Heaviside's methods, the 11 operator will be considered simply as abbreviatory for and is not to be treated as a symbolic eqivalent for II p 11 ol II II ;;IT' d II di in the manner of Carson in his infinite integral equation or Bromwich, Wagner, etc. in the case of the contour integral solution. Thee:e three necessary and sufficient conditions -1t- which between permit algebraic treatment -e4' the " ,P II opera tor.s can be established if the inverse operator II I 11 is appropriately p defined and the operand is of appropriate form. It is evident that combinations of the derivative operators and f' h , where a first ,P a. and h are positive real integers satisfy the algebraic laws. We shall define the inverse operator, namely ' f' or 17 - I * The three laws are as follows: {1~ the co:rmnutative law-------------- u~= v~ q b """'b {2 the index law ------------------- u ~ = u (3) the distributive law------------- atu•ll)•a~-a~ where c:c. and b are constants, and u and 11 belong to the class of terms under examination. Of_ cou,,-se the. "p" ope.ra.to,,-5 are not c.on,mvt,d,~e w,·t1,, recpec.t to fvnct-,onr •J t , 12 - by the equation The definition is restricted to the condition that j_t/fu,1,J/ d" .... be finite, a condition which requires that the integraid vanish at the lower limit of Eq. (11). It is the latter result which is sought and is of utmost importance for it permits the commutation of the operators /J and ; is, if t lJ/11•1/clu ; that is finite, we obtain the identities )?Jct; :t)Jd; 1: JrfJ, and as found by ready extension of this relation, a. b j f b "; f f'/> () ='f';° {)~,? ( tt., bJJ f {), where a and h are any real integers, either positive or negative, and the integrands to the?I the specified condition. operation comply with Since the distributive law is also satisfied by the above operators, the algebraic property r· for such operate~ is established, and we may proceed with complete validity to use Heaviside's algebraic manipulations. The condition for algebraic treatment, which we shall call the "algebrization condition", namely that the integrand Jth is such that / t JjtJ<;/dl,f, is finite, generally offers no - «> difficulty in operational calculus, because all operands which are encountered in that calculus include the unit function as a factor. 13 5. INVERSE POLY1WMIAL OPERATORS With the algebraic interpretation of combinations of /' and ) operators established, other operator forms will now be examined. Consider the operational equation (/2) where idJ may be the current response of a series circuit, including unit inductance and resistance a, when unit function potential is impressed on the circuit. The character of tt is limi tecl only in that it must be a finite constant. An operational solution of the equation rests upon the conception of inverse operators. The inverse opera tor J -----i- , f>'a. we define formally as such that (/3) satisfies the algebrization condition. Jd; 1 rh where Let the inverse operator be expanded in negative powers of p I _p+t:t according to the binomial theorem: I ,i{c) := (;0 - aa:i ~ +-/' 3 17 ,, , .. n +(-I);,,~•") _f (/J. (Fl} If the series as a function oft is uniformly convergent, it is obviously an inverse operator, since operation on both sides of Eq. (14) with identity. (f',,.a) reduces the equation to an For a discontinuous unit function the series is convergent and equals -; aI (I- B -at) -Lp rt J as can be readily verified by performing the indicated operations. It is evident 14 that when is a continuous function, the series /tt1 becomes uniformly convergent and. approximates the above value as closely as desired by choosing - ;tdJ sufficiently close to :tJ) ctJ • The resulting solution is in terms of a continuous function, and using the generality of the 1 d-J , we obtain definition of (/5) _!__ fdJ:: _!_ (1-e-at) 1dJ a /J+tN This becomes the solution of Eq. (12). * In a manner similar to the above, any operational expression may be expanded algebraically into an infinite series, provided that the resultant series as a function oft is uniformly :I cfJ In Eq. ( 14), for example, the operand convergent. may be substituted by any function of t provided that the resultant series is uniformly convergent and that the algebrization condition is satisfied. Since the operator is expressible in terms of simple integration, the algebraic property for combination of with established. p , where m is a real integer, is The property can be extended further to include opera tors of the form I --------(f'•f', )(17-p,,_) • • • (p-;o,,) ' where I o, ' I are constants, and, in general, to opera to's such as where Yr~J and Zf_?J Y.J -t ' YtpJ .clp) ' are polynomials in positive integral powers of ? . * ,u~e have b y d e f 1n1 • • t 10n · tha t (f'.,.a.) , :a.. jttJ11t1 : jrtJ ftt; 0 which one may readily show that _;_ (o,. a.) f1t1£1tJ:: /tfJ ..ft!J. /J~"'- r , f rom 15 I The inverse operator might have been expanded p+a. by the binomial theorem in positive powers of p were it not for the fact that the resultant series, (/6) is not uniformly convergent. On the basis of a discontinuous unit function, one may be tempted to consider that all but the first term of the series vanish, except at t=o He will then • obtain the contradiction that the value of the series in positive powers of ? is ;1 .f2;rt-J in contrast to the result already obtained from the series in negative powers of I namely a-(1-c: -c,t ..; ,P , This contradiction has occasionally }-L 7/ f l . been suggested as a distinct flaw in operational calculus. However, evaluation on the basis of a discontinuous unit function when applied to series (16) is entirely without justification because, when the resultant expression is operated upon with values of (f1+-a.J it does not yield , t , notably at t =- 0 • for all .lZ> <f J that is, the series operator ' of (16) does not represent an inverse operator. The situation may be summed up by observing that series (16) as a function of is not uniformly convergent. t The fallacy of the above method is revealed when the inverse operator is expanded into a finite series. Having established the algebraic law for operators, we can ·write the following identity: I --, - - ~ rt J .:: p+a._ [ I L a, a" - - ~ ,, + ~ ~ • .. + <- I),, -12-:- + ,:;l,a an+/ I >,+ I r-' J ,, .... --f!-- ct"+I ' pro. ]:let)' (/7) 16 where n is finite • . The very significant advantage of this expansion over that of Eq. (16) is that converg ence need not be investigated in Eq. (17) since the number of terms therein is finite; Eq. (17) is a true identity. that the value of the expression It can now be observed 1 -- -frh resides almost p+a. entirely in the first and last, or the remainder, term of the series, all other terms approximating zero.~ 1 - -- ~~a 1rfJ The value of has been shown to be ...!.. {1- £-o.-t) 1d J , and subs ti tut ion a of this expression into Eq. (17) demonstrates the importance of the remainder term. It is interesting that, even though a may be negative, corresponding to an exponentially cummula- tive response, all terms of this series approximate zero except the first and the remainder terms, the latter retaining the positive order exponential. The absurdity of ignoring the remainder is evident. It must be recognized that the insistence upon uniformity of convergence of an infinite series is required for valid algebraic manipulations of series operators, since an infinite series can be integrated term by term only if that series is uniformly convergent within the range of integration. Generally expansion of an inverse polynomial in positive powers of p leads to a divergent series, which therefore requires evaluation of the remainder term, while *These terms approximate zero since the unit function is a continuous one. For a finite number of such terms their aggregate value can be made as small as desired. 1? expansion in negative powers of leads to a convergent p series. A very useful operational tool is the so-called Heaviside shifting theorem, given by the equation (18) where ~ Ff,PJ is expressible as the ratio of polynomials in /7 , is a constant, and the algebrization condition is satisfied by the operands. Derivation of the theorem by the Heaviside method may be readily modified to become consistent with the treatment which we have been using, and need not be repeated here. Applying the shifting theorem to the operator , we obtain Consequently the operator operation indicated by the latter at any time. is identical with the p+ a, €-q:Jtr a.~-d ... ... and may be substituted by A similar procedure may be applied to a great many other operational forms. Consider now the differential equation , where / dJ is a known force function, 6. 7-'J is a polynomial in p , and the algebrization condition is satisfied. Then the operational solution is are roots of G71 =0 • Substituting the 18 integral identical to the inverse operator, we obtain {ZZ) The equation is the familiar solution of an nth order linear differential equation with constant coefficients. The Heaviside expansion theorem solution of the equation z' {f) = £. where Yr;n and Y(?J .r{f'J ;/dJ (,?3) ' Zcp1 are polynomials in ,P , requires little elucidation, the proof following directly from the algebraic property of the p operator. The expansion of into partial fractions is permitted, of course, only when the degree of Z~J is higher than that of Y/j-JJ , a condition which is demanded by a similar expansion when ,P is considered purely algebraic. A failure to abide by this limitation results at worse in the loss of impulsive terms. 6. LINEAR DIFFERENTIAL E~UATIONS WITH VARIABLE COEFFICIENTS It has been suggested by Bush r that Heaviside's operational methods may be profitably applied to the case of linear differential equations with variable coefficients, the justification for such treatment being placed by Bush upon the fact that the results in general agree with those obtained by the classical methods. However, we are now in position to establish directly the rigor of operational calculus. 19 Let us consider the simple case, Fig. 1, of a unit function potential applied to a series circuit including an impedance unit, whose impedance operator * is :z {fJ , and a resistance the latter being the only circuit 'R(tJ , element which varies with time. Applying Kirchhoff's potential law to the circuit we obtain where 2·rtJ as usual is the current response. inverse operator I Though the is a function of both 17 and t , it is defined in the customary manner, i. e. by the identity [.Zq:,J + 'l<!IJ] 1 .Z(_pJ+ R {f) :frtJ = ErfJ • asJ Using this definition we may write I z" rt} = - - - - - - f r / J ' 7-tpJ + 'k!tt) or = I if'J {1 -1- - 1 Ll_?J fdJ . (Z6) Rd;] To expand the inverse operator according to the binomial theorem would lead to error since the resulting series would not satisfy the property of an inverse operator defined by Eq. (25). However expansion following the reversal of the *An impedance operator of a circuit element is the ratio of potential a.cross the element and current flowing through it, the operator being expressed as a function of p. 20 position of I L {_?J and RtfJ in the denominator does satisfy the inverse property. 1 ·1 z(J=y;If J { The solution becomes: • I [ J?rtJ-]1 < · ·· +-{-rJ'"' [ 1<d1_j_]+,,, " /-RttJ-+ ,t!/j'J Z (j'J 2 (/' J $JrfJ, where we define, e.g. ' - ] 3 == ,J2rtJ .:/ J}rtJ ; 1! rf 1 ; [ .,;:;nctJ L f/'J .z {f'.J .Zf?J .z f,P.J • The expansion of Eq. (27) is valid only if the resultant . series is uniformly convergent as a function of time. To take a simple example, let where J.., and Ro are constants. is given by Eq. (27); Z<j'J = I' J.. and RdJ = ~ t frf J , Then the current response on performing the indicated operations, we obtain (28) The method can be readily extended to more involved circuits.* One must be reminded, however, that, in general, expansion of the inverse operator must be in terms of negative powers of to obtain a convergent series solution. *By analogy with the successful treatment of linear differential equations with variable coefficients by Heaviside's operational methods, it has been attempted to extend the same method to non-linear equations. Unfortunately this cannot be done. Since the functional operator is not distributive with respect to its operand, the functional operator cannot be expanded into a series in a manner which is requisite to the usual Heaviside treatment. For a solution of the problem see J. R. Carson, Phys. Rev. 17 p 129 (1921). 21 7. FRACTIONAL ORDER DERIVATIVES The function of Heaviside's operational calculus is to simplify operational expressions into forms which can be directly evaluated. The simplification may proceed either to a point where ready integration or differentiation alone remains to be performed, or else it may stop with expressions whose values are kno\m. Operational calculus in itself, however, has no provision for the evaluation of its basic operational forms, such as the derivative or the integral, so that, when half order derivatives are encountered, the basic form, namely or , must be evaluated by other than operational methods. In the study of physical problems, the only fractional order derivatives which have been investigated to any great extent are those of half-integer order. The half order differentiation is defined as an operation such that when it is repeated twice, the resultant operation is a complete The higher order derivatives, /1 differentiation. n is a positive integer, can be treated as , f'"',P z , where • In this way the higher half-integer order derivatives may be obtained , /1 x readily from , as also can half-integer order integrations. Similarly to the extent of physical problems, the general I expression ;:;·;;jrfJ:/rfJ may be derived by operational manipulations 22 I from We observe then that it is required to evaluate 17 x /d;. I only the expression p~.ldJ by other than operational methods before Heaviside's treatment may be applied to half-integer order differentiation and integration operations. I ,P ~ / There are several ways of evaluating application of symbolic calculus , I it is found that = p:E/2(/J l 7Jit the perhaps being the most Using the discontinuous unit function satisfactory. where z {t J .1:z, cf) I ,'1; (f} • To evaluate ' ~ 2 _Jrf J is the continuous function, we use the superpo- ;/{I) sition theorem equations, i. e. ? i _f (f J = i I:- U:- d /(v. J { J! (f·k) ' oo where, consistent with previous notation, Here /rlJ has been considered to be made up of infinitesimal steps whose tread lengths approach zero in the limit as the summation of effects of each incremental rise is expressed as an integral. The resulting function J _p~.fd1 and all its derivatives are continuous, and the function approaches * {;r' , /2rtJ as closely as desired by choosing close to /JJ (f J • Jee J sufficiently To write the integral of Eq. ( 29) every I time /1 :z .frt; is evaluated would be awkward, and therefore a new notation must be used to designate such a function. It is evident that the function cannot strictly be written simply * We have that t jJ-(uJciL(t<J-= .f<o> :,;ctJ ]irn. :/(tJ-4/t} Therefore 7 ~,m Jrh ➔ :lpct_) k•-= It .t<- ~.J 4> «=- ... {JJff-t<..) = /,.D(f J 0 (JTt 23 -~ lr!J because the latter is both discontinuous at t= 0 TJTt° and imaginary for negative values of t . We could denote by the integral of Eq. (29) a symbol such as L~ J a-1] , 'Jr t :I> /I C. "c" destgnates that the function and all its deriva- where tives are continuous and that the function approaches as closely as desired. , f.z,d) ,r-7 T.lft However, it is unnecessary to . d complicate notation by this symbol and we shall 1nfaed employ the customary expression - ' frf1 w with the understanding that the term expresses a continuous function. 4 The expression of Eq. (29). p r,/(!J is obtained by differentiation Since direct differentiation of the right hand side as it stands is not possible, it is first integrated by parts, i. e. Now differentiating the equation, we obtain f 1frf} = - / f -0.. I {Jr(f-t<) Recalling the property of the second order impulse and employing the above established notation, we could write .J p;,./[/J= [_zf;t;z Jc+ {Jmpu/s1ve term cd t=o] , 3 since ,~ldJ, as well as all its derivatives, is continuous and approaches as closely as desired when not in the neighborhood of f:o. t is Similarly we obtain for higher order derivatives (30) 24 where n is a positive integer and II r denotes the Gamma II if function. Again to simply notation, we shall return to the A more customary notation, omitting the brackets. I For integrations, ,fJ -;:I .I(fJ i. e. ,P-:i/rt.J =/?i ,/rt;, = ,;:11, / ' z It; . . ' d / u,_) , d :/[A,.J • we observe that a= - o0 for which we obtain ;° -:1,/rl> = i~ f[j!' < Therefore, we write as before and in general ' {3/) brackets to be omitted in the usual notation. The definition of the half order derivative permits it to be incorporated into the realm of operatori which obey algebraic :properties, so that any operator which is a function of integral and half-integral order differentiations and integrations may be treated algebraically. The Heaviside shifting theorem also applies to such operators. 8. JNFINITE NON-DIDUCTIVE CAELE The familiar partial differential equation for the potential, ec~t) , along a non-inductive cable is given by 25 where R is the distributed series resistance of the cable and C is its distributed capacita nce to ground, while the leakage conductance and series inductance are considered neg ligible. All circuit parameters are assumed constant. It is easily verified by substitution into the partial differential equation that a solution of the equation is where in the customary manner the exponential operator is defined by the infinite series corresponding to the exponential, e.g. ] j,d J. The solution follows from consideration of the algebraic property .of the ? operator raised to half-integer powers. For the specific example of a unit function electromotive force impressed at 'J!-= 0 upon a non-inductive cable which extends to infinity in the positive i: direction, the potential equation becomes e { ~ t J = c - ✓PR. c ' 1t I rt J , which can be readily shovm to be uniformly convergent for all values of ~ and t. the cable, we have that we obtain Now if if~f) d)e; t J =- f?irz, t) , is the current along and sub st i tu ting for e[z, tJ 26 At the head of the cable, i. e. ~=a , the current is simply t(OJJ:/ij[f</; so that the impedance operator of the 172'. infinite cable is ✓jc Consider now the problem of determining the response of the infinite cable when the latter is terminated at its source by a resistance 1? 0 • The impedance operator {R,C) ;i: (/'J of the complete circuit becomes and the current response e<O= /(fJ l idJ to unit function electromotive force is Fig. ,2 (S'Z) We shall treat an operator of this form, namely rt J = .r- L. ( where a re-- J(IJ (3'3) '? + ra is a positive real constant. The solution of this equation will be o.btained in two forms: namely, as an infinite convergent series and as an asymptotic series. Our treatment will differ from others in that the complete remainder terms for both solutions will be obtained, terms which Heaviside failed to consider • * Though remainder terms have previously been evaluated by putting the original operational expression into an integral which is integrated by parts, such a method lacks the flexibility of the complete operational solution since the integral form is preferably introduced only for the final remain de r term in the manner now to be described. 27 The convergent series is obtained by expansion of the operator in negative powers of ? , according to the binomial theorem: The expansion is exact since the remainder term is included. To evaluate the remainder, first the half-powers of its denominator are eliminated: and, using the shifting theorem for the denominator factor Since £ -af is greatest in the range dominant to the absolute value of R t)o at t=o , a is obtained by removing that exponential from the equation, i. e. or ~fat_t_) lrf J , r(; +z) where e is a real constant such that o~ e ~ I .(:?SJ • The complete solution is now obtained by evaluating Eq. (34) term by term using Eq. (31): 28 The presence of the positive order exponential in the remainder term renders the remainder rapidly more uncertain with increasing values of t so that an asymptotic form of solution is required and is next obtained. Expanding the original operator, Eq. (33), in positive powers of p according to the birfmial theorem, we have The expansion again is exact~ On the elimination of half-order derivatives from the denominator of the remainder, the latter becomes ,,.,, :2 :c-,J Let n+' = 171 .2 .,,,,., (-ff J-r /·rie ..lrfJ. ,I' - c., , an integer, and apply the shifting theorem: .R =(,f)~-f;_ /rt) -(t:-/" l.'z:-tJU t911(uJlf/,(, The first term R, may be reduced directly, since 0 _f!_ $(CJ=£ /.,-a, to 7 (J6) where the impulsive terms are ignored because the asymptotic solution is not applied to the neighborhood of For the second term t=O • R.e of the remainder, the integral 1.irlJ 29 is separated according to Then But so that Therefore the complete remainder is (5'.9) To evaluate this expression we recall that integration and differentiation operations are commutative provided that the operand vanishes at the definite limit of the integral. In Eq. (39) the operand under consideration is ~-arfa?f{tJ t , which indeed vanishes in the limit as approaches infinity. Then using the commutative property, we may"shift" the multiple derivative operator (a?Jh, past the first exponential, bring it into the integrand, "shift" it past the second exponential, and complete its journey by operation on (ap .Id J • The important aspect of the integral is that the impulsive terms are not present in the range of t over which the integral extends. We find for t >o 30 A dominant to the absolute value of R is obtained if the exponential is removed from the integra nd past the integral sign; then ., o, e, 1 J t >0 , The complete solution obtained on term by term evaluation of Eq. (37) becomes The impulsive terms are dropped since • It will be noted that the error incurred by stopping the series with the m th term is less than the magnitude of the rn th term. Usually the diverg ent series obtained by expansion of an operator in positive powers of p is asymptotic and Heaviside generally evaluated such a series by stopping it with its smallest term, presuming that the uncertainty of the remainder is less than that smallest term. example satisfies this cond ition. occasionally fails. The above However the procedure Consider the problem of a sinusoidal 31 electromotive force impressed suddenly on an infinite non-inductive cable. 1 - z:rt) = - .z {?J The current response may be written: me e (f) = £ f lR c- Jwt -; cc J C /4 ' where the real part of the solution is the response to a potential f cos wt .2'rfJ • Converting the exponential term to its operational equivalent, .s i rtJ = J; ff /' -J,e_:z u.J Expanding in positive powers of~ according to the binomial theorem, we obtain The remainder term is fr) . (jJ ,-, .c t (j 6''£,1"'t,-j"'ufj' J{4.J du Separating the upper limit and evaluating as in the preceding example, we find where where n The complete solution is is made even and 0, , {)-? Ci> o) are real quantities such 32 that c,<:Je21<1 evidently is The steady state term , which, we note, was obtained from the remainder term of the expansion. Therefore a failure to take the remainder into account would lead to an absurd result. The example illustrates the necessity for the development of an asymptotic series in a manner which permits of the evaluation of the uncertainty consequent to stopping short the series. Sometimes the error resulting from the ommission of the remainder term is not as obvious as above,for example when the remainder contains a negative order exponential. Since an asymptotic series cannot reveal the presence of a negative order exponential, it is to be expected that if the solution contains such exponentials, the latter will persist in the remainder term when the result is expressed in asymptotic form. Heaviside's usual methods would fail in these instances and he has used intuitive methods to handle the situation, methods which are not rigorous and not explicitly defined. A conspicuous example of such a problem is that of an infinite non-inductive cable terminated by inductance. :By the application of the methods herein estab- lished, the problem is treated in detail in the appendix of this paper. The solution is made cumbersome at one point due to the fact that the remainder of the asymptotic expans ion of 1 f t- (ol.+j(3) I 1 2 /cf J does not lend its elf to ready - evaluation when ex 33 - is real and positive and (3 is real. We have examined the method of treating inverse polynomials in integral and half-integral powers of p • An analogous method, though more cumbersome, may be applied to the more advanced operational expressions. For example, in the case of operators pertaining to the inductive transmission line, the expansion may be made to proceed not as a power series in ~ but in terms of Bessel functions, as used by Heaviside, permitting the evaluation of a dominant to the remainder of the series. Space does not permit a detailed exposition, and the latter procedure can be derived by utilizing Heaviside's own methods, with the slight modifications which have been discussed in the course of this paper. 9. SYSTEM INITIALLY NOT IN EQ,UILIBRIUM Thus far we have been dealing with solutions of differential equations under the simplified boundary conditions pertaining to a circuit undisturbed prior to application of the electromotive force which is under consideration. The methods already developed are readily extended to a treatment of general boundary conditions. 34 We h a ve obse rv ed that operational equations for lumped parameter circuits state explicitly the entire history of the dependent variable, but in the case of systems which are initially (f-= 0 ) not in equilibrium, the values of the variable are both unknown and of no consequence prior to t=O The obvious procedure in • setting up the operational equation is to assume that the system is in equilibrium before t=o and that at this instant an impulsive force is introduced such as to create a state identical with the boundary conditions. Consider an nth order differential equation: We are to determine the dissipation of the v.-ariabie following an initial excitation, specified by z"<tJ z·raJ, The operational equation assumes the following form: where the right-hand side represents the impulsive force which is necessary to produce the initial conditions. There remains then only to identify the with these initial conditions. left-hand polynomial in ? b coefficients Divide the equation by the , (-'17) 35 where 1<,..., is the coefficient of p -"" ,· etc., and c c k I -= ' { 6>,-1 - q; , - , "o• J Cl,., .2, ,, , i. e. are . . . c ,., constants. As we approach values of t t= o from the side of positive , we find that, since the integration terms approach zero, :But this approach is exactly what we mean by the first boundary condition, namely i · roJ , so that K. ~ z· roJ • Now differentiate both sides of Eq. (47) with respect to t .• _,., ,.:{ C, jJ r C,z + , • • ., C,, p C<,,p"+"• -+o,;:,~Qo Approaching t = t> as before, we have ; The impulsive term on the right hand side vanishes except in the neighborhood of and the neighborhood is as small as desired, so that we obtain in .k',~rfJ exactly what we mean by the second boundary condition, or The method is extended by successive differentiations, giving for the general term: The b coefficients can now be evaluated in terms of the specified initial conditions: .-, 36 6,, = a,, z'(o J The operational equation becomes: + ••. (,t/8) -1-p" a,, /Co) }lrfJ. Immediately after the impulsive force is applied, the force vanishes and the proper initial conditions are assumed; • therefore the validity of the equation is established. -If The solution of the above equation pro- ceeds in the usual operational manner. If an additional force is impressed on the system, a function representing the force is added to the righthand side of the equation according to the principle of superposition. In electrical circuit problems, it is a series of simultaneous differential equations rather than an order differential equation which is involved. nth For a *H. Jeffreys introduces the above general boundary conditions, though for a more restricted treatment of the fundamental operator ) , "Operational Methods in Mathematical Physics" Cambridge Tract, 1927. 36 A passive network of n branches under initial excitation, the first current equation of a series of simultaneous equations is where e. g. branches Jz l.1<s indicates the mutual inductance betiveen the ands. That the boundary con d itions specified by are satisfied by this equation is evident by comparison with Eq. ( 48). In setting up the current equation it is more convenient to express the terms in dimensions of potential, to which fo·rnn the above equation may be turned by operatin g on the equation with ; . Furthermore, to consider a more general problem, the presence of sources of potential within the branches will be taken into account. Since inital conditions have already been introduced into the equations, potential functions are considered to vanish for negative values o f t and consequently include the unit function , as a factor. The first of the system of equations becomes: Again the solution proceeds in the customary operational manner. 37 In the field of partial differential equations, an analogous method may be employed in the treatment of boundary conditions. As an example, we shall consider the wave equation: ) \: (-r:, -t) I ")-i:-<- c~ d 'e ( .,,., f) = () at~ governed by the initial conditions: ('19) In the manner of the previous problem, the equation is written operationally {St>) where the right-hand side corresponds to the force function necessary to produce the required initial conditions. ascertain the function j,rr J and ./4 r·o on both sides of the equation with , we first operate : includes ~d; as a factor, as we approach Since tc~ 1 ,P ~ To , the first and last terms of the equation tend to zero,leaving c2 ecx, t) - j (7tJ ;/{fJ Therefore, comparing with the first bounda!"J condition Eq. (49), we obtain 38 Similarly, operation on Eq. ( 50) with .!. l" leads to the second boundariJ condition The operational equation becomes In the customary manner the equation is integrated with respect to z, treating ? as a constant, the justi- fication for such a procedure resting upon the algebraic property of the/' operator. The solution may be expressed in terms of the particular integral, J fr., t J , and the solution of the complimentary equation: or The familiar method of determining the particular integral by operational methods, let ting 2> = j"Jt , gives The operator is expressed in partial fractions, --'- J>"- C ( .,e; = -<.f' 1 I .J) + I!- C C - 7> - J=. C , ) and each fraction may be substituted by its equivalent, e. g. I £ --?JCJt ./.Z C -oo Z< .. c;/1,( , 39 where the lower limit is chosen at - 00 in order to simplify the derivation. Evaluating the first term z; cx_,tJ of the particula r integral: or But the function ~ ::/(t+;) a peak concentrated at of magnitude c , 21=-c-t with the as a function of u , represents which encompasses an area axis. u Therefore the ex- pression becomes or In a similar manner,we find Since the remaining terms .,;_; rx., tJ of the particular inte- gral are of the same form as those already derived except for the presence of a ? Proceeding to the limit as factor, we have ./rt; is made to approach the discontinuous unit function, the particular integral becomes: t Ir~.,tJ-=-f {f,(x,cf)+£,{x-ct}j+jj/,(,(x•C'1)r£;(x-(1AJ}ct<A.. 0 J t>fJ (.5SJ 40 By change of variable the integral may be written in a more familiar form: L{,t,f)= ~ {£,(xt-ct}+E,(y-ct)+ 2 :j x,.cf £'~(uJd'1,, t>o. CS(.) x-ct The complete solution of the wave equation becomes: . c=(X_,t) =::-F, (f- x•ct f) +~ (/-+ ~) + f {£,fr+ct)-1-£,{)(·CtJj...-Jc/£-l(UJdl,f. 7 {.5 ) y-ct The two functions, F, and ~ , are determined by the particular conditions specified at the termina.ls of the line, the functions representing reflected waves. If an electromotive force is impressed at the terminals, the corresponding force function is also involved in the two boundary functions. As in the field of lumped parameter circuits, the operational equations pertaining to distributed parameter circuits describe an entire past history of the equation; or at least a history which simulates the actual in regard to the production of the initial conditions. In contrast to the differential equation, which is true for only particular intervals of its independent variables, the operational equation is true for all values of the time variable. It is this completeness which renders operational calculus so useful, for the initial conditions are inherently stated in the equations and do not require further attention. 41 10 • SlJJl1TIVrARY A mathematically sound treatment of operational calculus has been shov.rn to be possible following several modifications of Heaviside's methods. The basic diffi- culty in Heaviside's presentation, namely that which accompanies differentiations of a discontinuous function, has been eliminated by the restriction that operands e-f of operational equations, as well as all successive derivatives of the operands, must always be continuous. consequence, the unit function As a :ictJ has been defined as a function such that it and all its derivatives are continuous and. which function can be made to approach the Heaviside's discontinuous stepped function as closely as desired. To render the operators p I I commutative, ;o and P t- is defined by the operation /;, .. dt-t. , where the integrand f JctJ is required to satisfy the condition that j-= ljru;/ d« be finite. Linear differential equations with variable coefficients are shown to be amenable to solution by operational calculus. I :i The half order differentiation, r' , is defined as an operation, such that when it is repeated twice, it performs the same operation as e, complete differentiation. 42 Its basic evaluation is found from other than operational I ? :;;~{I;: Gt .7'.o<t J calculus methods to be is the discontinuous unit function. , where ;/.I)rfJ For our purposes, using the continuous unit function, we have by the superposition theorem that t I 17.:£ /(fJ ,:1./'{l<..J = /u=- oo {.,,-{f- fA) which, as well as all its derivatives, is continuous and which approaches in the limit as The expression is abbreviated by • • The alge- braic law can be readily shovm to extend to operators which are functions of integral and half-integral powers of r . In the treatment of as:rmptotic series, the remainder terms must be determined before the series can be conI sidered evaluated. When inverse polynomials in ,? and ? are expanded into either the convergent or .z asymptotic form of series, the remainder terms may be obtained if the finite series expansions are used, where the finite operational series are true identities with respect to the original operators. Example:, of the procedure are presented. The operational methods herein established are applied to the problems involving systems vtl1ich are initially not in equilibrium, where both ordinary a~d partial differential equations are considered. 43 A d eri va. tion of a n a symptotic expansion, •i:h ic h inelud es the r ern.a, in cl.e r term, of v1here oc and (! are r eal an d o< is positive, i s g iven in the appendix in the course of treatment of the non-inductive ca bl e t er minated by in ductance. The deriva tion supplies the only deficiency _which has be en l) resent t hus far to the d etermination of an asymptotic solution with rema inder term for any operational equa tion involving polynomials in half-intege r powers of p . The writer takes pleasure in acknowledg ing his indebtedness to Dr. 1Torg a n Ward and to Dr. D. H. Weinstein of California In s titute of Technology who h.ave kindly assisted in the preparation of this thesis. - 44 - APPENDIX 1'TO:N-INDUCTIVE CABLE TERMINATED BY INDUCTANCE A problem which has been the center of extensive investigation is that of determining the asymptotic solution for the response to unit function electromotive force of a non-inductive cable terminated at its source by inductance. It has been mentioned, p 32, that the difficulty lies in the presence of negatjve order exponential terms which appear in the remainder of the asymptotic expansion and which are not readily evaluated /0 • Consider the circuit, Fig. 3; the impedance operator , so that of the cable has been found to be the impedance operator (R,C) of the complete circuit becomes .:z. <;n =,P ;...o + f-:?;., ~ .z:. f"C ;t -----edJ• ftt; .Z<;11 1 T response ith • The current to unit function electromotive force is { /) Then the potential (ctJ at the head of the line, Fig. is (2/ We shall treat an operator of this form, namely (-3) (3), 45 Let us expand the operator of Eq.(3) in positive powers of according to the binomial theorem. , V f o ( J == I .J/ ~f ,d J = [ 3 5 .l z /- fa17J., + (ctpJ - (a_pJ + ••• ,r-,J ,,_, (C?j7)cz..,., 3 ;;:(»-I 0_p) + r <- 1) (cf_?) n :i;, ~ [t:;pJ ] :ltfJ. +/ ,!.h We shall evalue,te first the remainder term Rc/J = 1-1)" C4 fJ::z. (aJJJ'/• ,-/ ' Clearing the fractional order derivative from the denominator by opera ting on }<dJ with -?.... (t:1JI) .,_ - I (c/j'//,__ I , we obtain J (qp)i - I ' (q/' J 3 - :irf J • (.:5) I Now we expand the operator into partial fractions, and let h be even so that By R, dJ , where 1-n we denote the terms of the remainder which involve only integral powers of 17 • Using the relation --./2, Itf> = cM/tfJ 1- b already evaluated, we have tbat The multiple differentiation is now performed and the terms combined: f The neighborhood of t~o ) 0 , (B) is excluded from consideration since the asymptotic series is not applicable to this region. The other terms of the remainder, to be grouped under are 46 R {fJ "" r<?,PJ,,,[-'-l+i/3 -1-iri ~ Ja_ I + v •,1 -t. ,o- er 2{o- -1+1 .J) 2(/.J - -1-;u'j I .2a. For the first of these fractions, 1~r :ZrtJ • .Za. I , we transform R-2,, rt; the inverse operator to its equivalent, i.e. t i ·I ~/ , Then separating the integral according to -oo and evaluating -r; ;td J , - oo 00 we have The value of the definite integral approximates fdJ t- 0,, as a is made to approach the discontinuous unit function. To evaluate the second term, we shift the derivative operator (a/J~ within the integral in the manner already discussed on p 29. We find for t >6 R.1, rfJ A dominant to the absolute value of is obtained if the exponential is removed from the integrand past the integral sign; then i. ~-<,Id} = ~a+ .Jr/: ... f) (i-J n,~ -J , a<.'9,(/ ~ t>o. {IZ) The positive order exponential cancels that of Eq. (8) when the complete remainder RclJ is derived. The second and third terms, ~, <rtJ and ~,3 ctJ respectively, of .R dJ Eq. (9), present certain difficulties in their 2 47 evaluation. 122,zrf; Let us consider a general expression of which and R 2 , 3 dJ f D ,c:I( J where o<. are examples, namely - (~ - . ) e>!+Jf3 ,., € -(<> •j(' ) f f t E (ol•,1°(3) "- -oo j n ;/ooctu r is real and positive, and (3 is real. is the remainder after the mth (/ '3) This expression term of the asymptotic expansion of To the writer's knowledge no rigorous method of evaluating a useful dominant of the remainder has yet appeared in literature :If- • For the remainder of the particular operator series in our problem Heaviside obtained an exponential f, with the uncertainty less than the smallest term of the series when the series is stopped with that term, but no proof was furnished by Heaviside though his conclusions are correct,as we shall observe presently. The following treatment of ldJ, Eq. (14), obtaining a complete and useful asymptotic expansion, was outlined to the writer by Dr. 1viorgan Ward. · The expression is first expanded in convergent form, either by integration by parts or by familiar operational procedure: *For the specific case of ~~ 0 , T.J.Stieltjes, Acta Math.IX p 167 (1886), has shown that the remainder is less tr,..a,n the smallest term of the asymptotic series, when the series is stopped with that term. 1 Treating the problem by contour integration T.J.I'A Bromwich Pree.Phys.Soc. 41 p 420 (1929) (discussion of an article written by W. E. Sumpner), derives the negative order exponential but does not evaluate the uncertainty of the remainder. 48 ~'=, , and we let where The series is a hypergeometric one, namely Ia; = 2 ff ,F, r 1; } ; - nt J id; An asymptotic expansion is obtained by employing the Kummer function where ; the contour is designated by Fig.4, and 1a1r9 .:z- J <f JT Wplane requires that the singularity at w=-.r be outside the contour, as shown. The 0 Kummer function is identified with the N~- 4 - hypergeometric function by: For the problem under consideration, we shall let o<•::£, r~ i .i'= - l'ft Then But Therefore, we obtain on substitution ·(It IctJ = f -~ - 0 v-r1 ;r.. r.1,. ..!. -cit :2,-'' ...!_ !i" J] ..lrf1 . (15) To evaluate the contour integral we first expand by the binomial theorem ~ O- elf) - I w-- )1/""' r .z. ~,.,.cir"'(~ J .. ,. • .,. 6r-J hT .,. c,f;) (/- Jf) ,.,.-,.J - I ? and substitute into the integrand: ; I _f I I 9(f, j, -?t) = -L-'- ~-~{(-w-f:i_ (-,.,.;; r-,.,.-J,' _ ... ,. r-,J'"1-xr),.,-:; c-tl,._.,f-w)"''i(/- .-r)-'jdw-.J.1f {-,-1t j"' c1t ut; rr1tJ,., r~tJ"'" r1t • But it is kno~~ that 49 where the contour is identical with that of Fig. 4. Then To evaluate the remainder, we shall deform the contour as shown in Fig. 5, where the radius of the circle about the origin r, is made vanishingly small and the lines approach the real axis. W pl&1ne ~ and are made to A dominant to the w= .-,to toJ•J~Jt l---<1t rad. r absolute value of the remainder may be ob- r: y{' tained by permitting the factor r, C (1-c:TT) - I to assume its greatest absolute value in the range of inte gration, i.e. from Fig. 5 I (1-;5f'j ~ (3 , since kl I-=: J The re- • mainder term of Eq. (16) therefore becomes: e , rr- .... -i:- J~ cM)'"•:. 101 <. I * On substituting into Eq. (15), we obtain the final solution: •The remainder term, it will be noted, is multiplied by the term of the asymptotic series following the last considered. remainder The Rrcc i, Eq. (13), is -("'-•J·f) t- ;l?_ cf) ., = [ E v-<o1•Jr) ~ ~ (3 I rr-,,.,-i)(o";Jr:)'" .. 't- ,... _!. ] ~ .1d) (/8) • * The uncertaint;'}r factor e is different in the various equations where it appears, subject only to the cond.i tion that 1e1..:.. 1 • 50 The expansion is extremely useful as it now permits us to obtain complete asymptotic solutions for any operational equation involving polynomials in half integer powers of p. We now apply the above results to the problem on hand, namely the evaluation of P.,, 2 lf J and Eq.(9) with the remainder, J?rr!J, J.?-l 3 <f J ,Eq. ( 9). Comparing Eq.(13), already evaluated in Eq.(18), we have that And substituting~ ; _(f)t I ff"l r<.,,:i_(f) £ = -j- C-:;-+J~ .. .2fl + .J(J" ' n,.: ~, 7, /(), , .. ; 181 <.I 3 q,..+r . /(-;,.-..:...it> ' t> , 0, (1'3) In a similar manner, we obtain the conjugate remainder term: I 'J? .,,3 I ( (C)" __ E: Jj) •"5"-J;r ±. ~ ] ? + ~ ..3(3 ..3 1 rr- ... --i-J (..!) ..... :i"; t Jf)l<I . ~ t>o (20) The complete remainder is evaluated by the combination of the four constituents: 'R=-k, .. 'R.,_, ... 'l<.,... +k.,,a Eq's (8,12,19,20); The final potential equation therefore becomes: 'f & + ~( 31, 0. .,, {) I t1 -l -+ - - (~) 2 rr-... ~]:) t ,.__i j ,• I 81 < I , t >O , f1< = >I , 7, IO, ' · • The exponential term which we have obtained agrees with that given by Heaviside. It is unnecessary, however, despite Heaviside's statement, to stop the series with its smallest *In the evaluation of ~-(01 '"J(3) , it must be recalled that 51 term in order to ascertain the uncertainty of the remainder. A treatment by convergent series of the non-inductive cable terminated by inductance permits ready evalua tion of the uncertainty of the remainder after the nth Expanding the operator in negative po-wers of I V,: ct J = (c,,f') /2"- + I 3 p term. , we obtain ..lei J The remainder term, 'k?dJ , is expanded into partial fr a ctions, following the clearing of fractional order derivatives from the denominator, as for t h e asymptotic series: :J (_ap)-,, - I 2,-, Lc!J. Ccy) ~ Applyin g the shifting theorem to each fr a ction and co mbining t er ms, we obta in 52 - It is evident that a dominant to ,frtJ may be obtained by proper removal of exponentials and trigonometric functions from the integrand, and one form of the remainder is certainly given by On integrating and taking the limit as the continuous unit function is made to approach the discontinuous one, RrfJ" 1 t 3 I 3 I r.!.../h"'z I • (_t/"'']; r<-zi.~-:zl ~ UJ» .. ~l a. {ea: .. ,z) [ ./ • o~/0/~J, t":;o. Finally the potential equation is obtained by performing the indicated operations of Eq. (21), i.e. 3 - - ' (,.!:) z- - - I f-) i ~ rt J - rc~J a + • .. .,. ~ 3 2 ..J _I CL ,,_, (-t I t J - - (;;) 3 S h r ti n,I) The expansion is useful for small values of ;;;:t .' as-.;-t" increases, greatly more terms of the series are required to maintain accuracy of computations. (.24) 53 REFERENCES 1. O. Heaviside Proc.Roy.Soc. A, LII p 504 (1893) LIV p 105 (1894) "Electromagnetic Theor-J 11 Benn Bros. Ltd., London "Electrical Papers" Copley Pub., Boston 2. II (1898), III (1912). II (1925). The treatments consist of the integra l equation methods attributed to .J. R. Carson and the contour integral methods attributed to T. J. I'A :Bromwich and K. w. Wa gner. Both treatments are discussed earlier and ind ependently by G. Giorgi. J. R. Carson Bell Sys. Tech. J. I p 43 (1922l. T . .J.I'A.Bromwich Pree.London Math.Soc. XV p 401(1916. K. VI . Wagner Archiv fuer Electrot. IV p 159(1916 . G. Giorgi Atti della Ass .Ellettro. Ital. IX p 651 ( 1905). .r . Smith .J.Franklin Inst. Phil. Mag . Math. Ann. cc p 519(1925l. XII :p 201(1931. vc p 557(1926. .r. R• Carson J. c. Maxwell Bell Sys. Tech .J. IV p 698(1925). 6. V. Bush "Operational Circuit Analysis" Wiley & Sons, n.Y. p 258(1929). 7. V. :Bush l oo. cit~ 8. o. Heaviside "Electromagnetic Theory" Benn :Bros. Ltd., London .J. w. E. Sumpner N. Weiner 5. "Electricity and J\l[agnetism" Clarendon Press, Oxford Ip 7 (1881). p 353(1929). II p 37(1898). T. .J.I'A Bromwich "Infinite Series", 10. J. R. Carson Macmillan & Co. London (1926), 2nd Ed. aevised p.333. IV pp 739, 756 (1925). Bell Sys. Tech . .J. o. Heaviside loo. cit. E • .J. Berg, p. 42; p 489. s. J. Haefner, J.J.Smith, .J.Franklin Inst. CCV p 229 (1928). -:-