On Integrable Delta Functions on the Levi-Civita Field 1

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1 ISSN , p-adic Numbers, Ultrametric Analysis and Applications, 018, Vol. 10, No. 1, pp c Pleiades Publishing, Ltd., 018. RESEARCH ARTICLES On Integrable Delta Functions on the Levi-Civita Field 1 Darren Flynn ** and Khodr Shamseddine *** Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T N, Canada Received November 7, 017 Abstract In this paper, we develop a theory of integrable delta functions on the Levi-Civita field R as well as on R and R 3 with similar properties to the one-dimensional, two-dimensional and threedimensional Dirac Delta functions and which reduce to them when restricted to points in R, R and R 3, respectively. First we review the recently developed Lebesgue-like measure and integration theory over R, R and R 3. Then we introduce delta functions on R, R and R 3 that are integrable in the context of the aforementioned integration theory; and we study their properties and some applications. DOI: /S X Key words: non-archimedean analysis, Levi-Civita field, power series, measure theory and integration, double and triple integrals, Dirac delta function, Green s function, solutions of ODE s and PDE s. 1. INTRODUCTION In various branches of physics, one encounters sources which are nearly instantaneous if time is the independent variable) or almost localized if the independent variable is a space coordinate). To avoid the cumbersome studies of the detailed functional dependencies of such sources, one would like to replace them with idealized sources that are truly instantaneous or localized. Typical examples of such sources are the concentrated forces and moments in solid mechanics, the point masses in the theory of the gravitational potential, and the point charges in electrostatics. The field of real numbers R does not permit a direct representation of the improper) delta functions used for the description of impulsive instantaneous) or concentrated localized) sources. Of course, within the framework of distributions, these concepts canbe accountedforinarigorousfashion, butat theexpense oftheintuitive interpretation. The existence of infinitely small numbers and infinitely large numbers in the non-archimedean Levi- Civita field R allows us to have well-behaved delta functions. For example, the function δ : R R,given by δx) 3 4 d 3 d x ) if x <dand 0 otherwise, where d is a positive infinitely small number, is a one-dimensional) continuous and piece-wise infinitely differentiable) delta function; it assumes an infinitely large value 3/4d 1 )at0, it vanishes at all other real points and its integral is equal to one. We recall that the elements of the Levi-Civita field R and its complex counterpart C are functions from Q to R and C, respectively, with left-finite support denoted by supp). That is, below every rational number q, thereareonlyfinitely many points where the given function does not vanish. For the further discussion, it is convenient to introduce the following terminology. The text was submitted by the authors in English. 1 This paper was presented by the second author at the Sixth International Conference on p-adic Mathematical Physics and its Applications which was held in Mexico City, October 3-7, 017. ** flynnd3@myumanitoba.ca *** Khodr.Shamseddine@umanitoba.ca 3

2 ON INTEGRABLE DELTA FUNCTIONS 33 Definition 1.1. λ,,, q )Forx 0in R or C, we let λx) minsuppx)), which exists because of the left-finiteness of suppx); and we let λ0) +. Moreover, we denote the value of x at q Q with brackets like x[q]. Given x, y 0in R or C, wesayx y if λx) λy); andwesayx y if λx) λy) and x[λx)] y[λy)]. Finally,foranyq Q,wesayx q y if x[p] y[p] for all p q in Q. At this point, these definitions may feel somewhat arbitrary; but after having introduced an order on R, we will see that λ describes orders of magnitude, the relation corresponds to agreement up to infinitely small relative error, while corresponds to agreement of order of magnitude. The sets R and C are endowed with formal power series multiplication and componentwise addition, which make them into fields [6, 9] in which we can isomorphically embed R and C respectively) as subfields via the map Π:R, C R, C defined by x if q 0 Πx)[q]. 1.1) 0 else Definition 1.. Order in R)Letx, y R be given. Then we say that x>yor y<x)ifx y and x y)[λx y)] > 0; andwesayx y or y x)ifx y or x>y. It follows that the relation or ) defines a total order on R which makes it into an ordered field. Note that, given a<bin R, wedefine the R-interval [a, b] {x R : a x b}, with the obvious adjustments in the definitions of the intervals [a, b), a, b], anda, b). Moreover, the embedding Π in Equation 1.1) of R into R is compatible with the order. The order leads to the definition of an ordinary absolute value on R: x if x 0 x max{x, x} x if x<0; which induces the same topology on R called the order topology or valuation topology) as that induced by the ultrametric absolute value: e λx) if x 0 x u 0 if x 0, as was shown in [13]. Moreover, two corresponding absolute values are defined on C in the natural way: x + iy x + y ; and x + iy u e λx+iy) max{ x u, y u }. Thus, C is topologically isomorphic to R provided with the product topology induced by or u )inr. We note in passing here that u is a non-archimedean valuation on R resp. C); that is, it satisfies the following properties 1. v u 0 for all v R resp. v C)and v u 0if and only if v 0;. vw u v u w u for all v, w R resp. v, w C); and 3. v + w u max{ v u, w u } for all v, w R resp. v, w C): the strong triangle inequality. Thus, R, u ) and C, u ) are non-archimedean valued fields. Besides the usual order relations on R, some other notations are convenient.

3 34 FLYNN, SHAMSEDDINE Definition 1.3., ) Letx, y R be non-negative. We say x is infinitely smaller than y and write x y) ifnx < y for all n N; wesayx is infinitely larger than y and write x y) if y x. Ifx 1, we say x is infinitely small; if x 1, wesayx is infinitely large. Infinitely small numbers are also called infinitesimals or differentials. Infinitely large numbers are also called infinite. Non-negativenumbers that are neither infinitely small nor infinitely large are also called finite. Definition 1.4. The Number d) Letd be the element of R given by d[1] 1 and d[t] 0for t 1. Remark 1.5. Given m Z,thend m is the positive R-number given by dd }{{ d} if m>0 m times d m. 1 if m 0 1 d m if m<0 Moreover, given a rational number q m/n with n N and m Z), then d q is the positive nth root of d m in R that is, d q ) n d m ) and it is given by d q 1 if t q [t]. 0 otherwise It is easy to check that d q 1 if q>0and d q 1 if q<0in Q. Moreover, for all x R resp. C), the elements of suppx) can be arranged in ascending order, say suppx) {q 1,q,...} with q j <q j+1 for all j; andx can be written as x x[q j ]d q j, where the series converges in the valuation topology. Altogether, it follows that R resp. C) is a non-archimedean field extension of R resp. C). For a detailed study of these fields, we refer the reader to the survey paper [9] and the references therein. In particular, it is shown that R and C are complete with respect to the natural valuation) topology. It follows therefore that the fields R and C are just special cases of the class of fields discussed in [5]. For a general overview of the algebraic properties of formal power series fields in general, we refer the reader to the comprehensive overview by Ribenboim [4], and for an overview of the related valuation theory to the books by Krull [], Schikhof [5] and Alling [1]. A thorough and complete treatment of ordered structures can also be found in [3]. Besides being the smallest ordered non-archimedean field extension of the real numbers that is both complete in the order topology and real closed, the Levi-Civita field R is of particular interest because of its practical usefulness. Since the supports of the elements of R are left-finite,it is possible to represent these numbers on a computer; and having infinitely small numbers in the field allows for many computational applications. One such application is the computation of derivatives of real functions representable on a computer [10], where both the accuracy of formula manipulators and the speed of classical numerical methods are achieved.. MEASURE THEORY AND INTEGRATION ON R, R AND R 3 Using the nice smoothness properties of power series see [7] and the references therein), we developed a Lebesgue-like measure and integration theory on R in [8, 1] that uses the R-analytic functions functions given locally by power series- Definition.4) as the building blocks for measurable functions instead of the step functions used in the real case. This was possible in particular because the family Sa, b) of analytic functions on a given interval Ia, b) R where Ia, b) denotes any one of the intervals [a, b], a, b], [a, b) or a, b)) satisfies the following crucial properties. j1

4 ON INTEGRABLE DELTA FUNCTIONS Sa, b) is an algebra that contains the identity function;. for all f Sa, b), f is Lipschitz on Ia, b) and there exists an anti-derivative F of f in Sa, b), which is unique up to a constant; 3. for all differentiable f Sa, b),iff 0on a, b) then f is constant on Ia, b);moreover,iff 0 on a, b) then f is nondecreasing on Ia, b). Notation.1. Let a<bin R be given. Then by lia, b)) we will denote the length of the interval Ia, b), thatis lia, b)) length of Ia, b) b a. Definition.. Let A R be given. Then we say that A is measurable if for every ɛ>0 in R, there exist a sequence of mutually disjoint intervals I n ) and a sequence of mutually disjoint intervals J n ) such that I n A J n, li n ) and lj n) converge in R, and lj n ) li n ) ɛ. Given a measurable set A, then for every k N, we can select a sequence of mutually disjoint intervals ) ) I k n and a sequence of mutually disjoint intervals J k n such that l In k ) and l Jn) k converge in R for all k, In k I k+1 n A J k+1 n Jn k and ) l Jn k ) l In k d k for all k N. SinceR is Cauchy-complete in the order topology, it follows that lim l I k ) n and k lim l Jn) k both exist and they are equal. We call the common value of the limits the measure of A k and we denote it by ma). Thus, ma) lim k ) l In k lim k ) l Jn k. Contrary to the real case, { sup li n ):I n s are mutually disjoint intervals and and } I n A { inf lj n ):J n s are mutually disjoint intervals and A } J n need not exist for a given set A R. However, as shown in [1], if A is measurable then both the supremum and infimum exist and they are equal to ma). This shows that the definition of measurable sets in Definition. is a natural generalization of that of the Lebesgue measurable sets of real analysis that corrects for the lack of suprema and infima in non-archimedean ordered fields. It follows directly from the definition that ma) 0 for any measurable set A R and that any interval Ia, b) is measurable with measure mia, b)) lia, b)) b a. It also follows that if A is a countable union of mutually disjoint intervals I n a n,b n )) such that b n a n ) converges then A is

5 36 FLYNN, SHAMSEDDINE measurable with ma) b n a n ).Moreover,ifB A R and if A and B are measurable, then mb) ma). In [1] we show that the measure defined on R above has similar properties to those of the Lebesgue measure on R. For example, we show that any subset of a measurable set of measure 0 is itself measurable and has measure 0. We also show that any countable unions of measurable sets whose measures form a null sequence is measurable and the measure of the union is less than or equal to the sum of the measures of the original sets; moreover, the measure of the union is equal to the sum of the measures of the original sets if the latter are mutually disjoint. Furthermore, we show that any finite intersection of measurable sets is also measurable and that the sum of the measures of two measurable sets is equal to the sum of the measures of their union and intersection. It is worth noting that the complement of a measurable set in a measurable set need not be measurable. For example, [0, 1] and [0, 1] Q are both measurable with measures 1 and 0, respectively. However, the complement of [0, 1] Q in [0, 1] is not measurable. On the other hand, if B A R and if A, B and A \ B are all measurable, then ma) mb)+ma \ B). The example of [0, 1] \ [0, 1] Q above shows that the axiom of choice is not needed here to construct a nonmeasurable set, as there are many simple examples of nonmeasurable sets. Indeed, any uncountable real subset of R, like [0, 1] R for example, is not measurable. Then we define in [1] a measurable function on a measurable set A R using Definition. and analytic functions Definition.4 below). Definition.3. A sequence a n ) in R or C) is said to be regular if the union of the supports of all members of the sequence is a left-finite subset of Q. Definition.4. Let a<bin R be given and let f : Ia, b) R. Thenwesaythatf is analytic on Ia, b) if for all x Ia, b) there exists a positive δ b a in R, and there exists a regular sequence a n x)) in R such that, under weak convergence, f y) a n x)y x) n for all y x δ, x + δ) Ia, b). n0 Definition.5. Let a<bin R be given and let f : Ia, b) R be analytic. Then there is a rational number if) called the index of f and defined by if) :min{λfx)) x Ia, b)}. It is shown in [13] that the above minimum must exist and that λfx)) if) for almost every x Ia, b) d λb a) R.Moreover,ifx Ia, b) d λb a) R satisfies λfx)) if), thenλfy)) if) for all y satisfying y x d λb a). Definition.6. Let A R be a measurable subset of R and let f : A R be bounded on A. Then we say that f is measurable on A if for all ɛ>0in R, there exists a sequence of mutually disjoint intervals I n ) such that I n A for all n, l I n ) converges in R, ma) li n ) ɛ and f is analytic on I n for all n. In [1], we derive a simple characterization of measurable functions and we show that they form an algebra. Then we show that a measurable function is differentiable almost everywhere and that a function measurable on two measurable subsets of R is also measurable on their union and intersection. We define the integral of an analytic function over an interval Ia, b) and we use that to define the integral of a measurable function f over a measurable set A. Before we do that, we recall the following result whose proof can be found in [6]. Proposition.7. Let a<bin R and let f : Ia, b) R be analytic on Ia, b). Then f is Lipschitz on Ia, b);

6 lim fx) and lim fx) exist; x a + x b ON INTEGRABLE DELTA FUNCTIONS 37 the function g :[a, b] R,givenby fx) if x Ia, b) gx) lim fξ) if x a ξ a + lim fξ) if x b, ξ b extends f to an analytic function on [a, b] when Ia, b) [a, b]. Definition.8. Let a<bin R, let f : Ia, b) R be analytic on Ia, b), and let F be an analytic anti-derivative of f on Ia, b). Then the integral of f over Ia, b) is the R number f lim F x) lim F x). x b x a + Ia,b) The limits in Definition.8 account for the case when the interval Ia, b) does not include one or both of the end points; and these limits exist by Proposition.7 above. Now let A R be measurable, let f : A R be measurable and let M be a bound for f on A.Then for every k N, there exists a sequence of mutually disjoint intervals In k ) such that In k A, l I k ) n converges, ma) l In) k d k,andf is analytic on In k generality, we may assume that In k I k+1 n for all n N. Without loss of ) 0,and for all n N and for all k N. Since lim n l I k n since I f ) Ml I k n k n proved in [1] for analytic functions), it follows that f 0for all k N. lim n I k n Thus, I f converges in R for all k N [11]. n k ) We show that the sequence I f converges in R; and we define the unique limit as the n k integral of f over A. Definition.9. Let A R be measurable and let f : A R be measurable. Then the integral of f over A, denoted by A f,isgivenby f lim f. A I n li n) ma) I n A I n s are mutually disjoint f is analytic on I n n It turns out that the integral in Definition.9 satisfies similar properties to those of the Lebesgue integral on R [1]. In particular, we prove the linearity property of the integral and that if f M on A then A f MmA),wheremA) is the measure of A. We also show that the sum of the integrals of a measurable function over two measurable sets is equal to the sum of its integrals over the union and the intersection of the two sets.

7 38 FLYNN, SHAMSEDDINE In [8], which is a continuation of the work done in [1] and complements it, we show, among other results, that the uniform limit of a sequence of convergent power series on an interval Ia, b) is again a power series that converges on Ia, b). Then we use that to prove the uniform convergence theorem in R. Theorem.10. Let A R be measurable, let f : A R,foreachk N let f k : A R be measurable on A, and let the sequence f k ) converge uniformly to f on A. Thenf is measurable on A, lim k A f k exists, and lim f k f. k A A In [14] we generalize the results of [8, 1] to two and three dimensions. In particular, we define a Lebesgue-like measure on R resp. R 3 ). Then we define measurable functions on measurable sets using analytic functions in two resp. three) variables and show how to integrate those measurable functions using iterated integration. The resulting double resp. triple) integral satisfies similar properties to those of the single integral in [8, 1] as well as those properties satisfied by the double and triple integrals of real calculus. In order to have basic regions, like disks for example, measurable, it turns out that the socalled simple regions defined below, rather than rectangles, are the best choice for the building blocks for measurable sets. We recall the following definitions from [14] which will be needed later in this paper. Definition.11 Simple Region). Let G R.ThenwesaythatG is a simple region if there exist a b in R and analytic functions h 1,h : Ia, b) R, with h 1 h on Ia, b) such that G {x, y) R : y Ih 1 x),h x)),x Ia, b)} or G {x, y) R : x Ih 1 y),h y)),y Ia, b)}. Definition.1 λ x and λ y ofasimpleregion). Let A R be a simple region. If A {x, y) R : y Ih 1 x),h x)),x Ia, b)} we define λ x A) λb a) and λ y A) ih x) h 1 x)) on Ia, b) where ih x) h 1 x)) is the index of the analytic function h h 1 on Ia, b). On the other hand, if A {x, y) R : x Ih 1 y),h y)),y Ia, b)}, we define λ y A) λb a) and λ x A) ih y) h 1 y)) on Ia, b). If λ x A) λ y A) 0then we say that A is finite. Definition.13 Analytic Functions on R ). Let A R be a simple region. Then we say that f : A R is an analytic function on A if, for every x 0,y 0 ) A, there exist a simple region A 0 containing x 0,y 0 ) that satisfies λ x A 0 )λ x A) and λ y A 0 )λ y A), and a regular sequence a ij ) i,j0 such that for every s, t R,ifx 0 + s, y 0 + t) A A 0 then fx 0 + s, y 0 + t) a ij s i t j fx 0,y 0 )+ a ij s i t j, i,j0 where the power series converges in the weak topology. i,j0 i+j 0 Given a simple region S R andananalyticfunctionf : S R,wedefine the index of f on S by if) min{λfx, y)) x, y) S}, which is shown to exist [14]. We note that λfx, y)) if) for almost every x, y) S d λxs) R d λys) R) and for any such point x, y) S d λxs) R d λys) R), wehavethatλfx,y )) if) for all x,y ) S satisfying x x d λxs) and y y d λys). With the above definitions, we can proceed to define measurable sets, measurable functions, and integration just as we did in R, replacing intervals by simple regions. We can then extend the measure theory and integration to R 3, R 4, etc. in an inductive way and obtain similar properties for the resulting integrals as those for the single integral defined above.

8 ON INTEGRABLE DELTA FUNCTIONS THE DELTA FUNCTION ON THE LEVI-CIVITA FIELD In the following, capitalizing on the existence of infinitely small and infinitely large numbers and the newly developed integration theory on R, R and R 3, we will introduce a measurable function that has similar properties to the Dirac Delta function and reduces to it when restricted to R. Definition 3.1. Let δ : R R be given by { 3 δx) 4 d 3 d x ) if x <d 0 if x d. Proposition 3.. Let I R be an interval. If d, d) I then δx) 1. Moreover, if d, d) I then x I x I δx) 0. Proof. Note that δx) is measurable on I [1]. If d, d) I then δx) δx) x I x d,d) x d,d) 3 4 d 3 d x ) 3 [d 4 d 3 x ] [ ] ) d 1 d d 3 x3 d 3 4 d 3 d 3 ) 3 d3 1. If d, d) I then δx) 0for all x I; and hence δx) x I x I 00. Proposition 3.3. Let I R be an interval containing d, d). Thenδx) has a measurable antiderivative on I that is equal to the Heaviside function on I R. Proof. Let H : I R be given by 0 if x d Hx) 3 4 d 3 d x 1 3 x3 )+ 1 if d<x<d. 1 if x d Then Hx) is measurable and differentiable on I with H x) δx) on I. Moreover, 0 if x<0 Hx) R 1/ if x 0, 1 if x>0 which is the so-called Heaviside function.

9 40 FLYNN, SHAMSEDDINE Proposition 3.4. Let α R \{0} be given, and let I R be any interval satisfying Then δαx) 1 α. Proof. Note that, by definition of the delta function, we have that { 3 δαx) 4 d 3 d αx) ) if αx <d 0 if αx d { 3 4 d 3 d αx) ) if x < d α 0 if x d. α It follows that [ 3 3 δαx) 4 d 3 d αx) ) x 4 d 1 d α x 3 3 x I ) x d α, d α x I )] d α d α ) d α, d α I. 1 α. Lemma 3.5. Let f : I0, 1) R be analytic with if) 0. Then for every x I0, 1) and for every n N,wehavethatλf n) x)) 0. Proof. Let x I0, 1) and let q<0in Q be given. Since f is analytic on the finite interval I0, 1), there exists δ>0in R such that for all y x δ, x + δ) I0, 1), wehavethat f n) x) fy) y x) n. n! n0 n0 Since if) 0, it follows that, for almost every h 0,δ) R,wehaveλfx + h)) 0. In other words, for almost every h 0,δ) R we have that fx + h)[q] 0.But ) f n) x) fx + h)[q] h n f n) x)[q] [q] h n f n) x)[q] [0] h n. n! n! n! So for almost every h 0,δ) R we have n0 n0 f n) x)[q] h n 0. n! Since the above is a real power series, this is possible only if f n) x)[q] 0for all n N. Therefore, for any n N and x I0, 1), wehavethatλf n) x)) 0. Theorem 3.6. Let a<bin R be given and let f : Ia, b) R be analytic on Ia, b) with if) 0. Then for any x Ia, b) and for any n N,wehavethat λf n) x)) nλb a). Proof. Define F : I0, 1) R by F x) fb a)x + a). Then F is analytic on I0, 1) and if )if) 0; hence, by the previous lemma, for all x I0, 1) and n N,wehavethat ) λ F n) x) 0. n0

10 Note that, for all x I0, 1) and n N,wehavethat ON INTEGRABLE DELTA FUNCTIONS 41 It follows that F n) x) b a) n f n) b a)x + a). 0 λf n) x)) λb a) n f n) b a)x + a)) nλb a)+λf n) b a)x + a)); and hence λf n) b a)x + a)) nλb a). Proposition 3.7. Let a<bin R be such that λb a) < 1 and let f : Ia, b) R be analytic on Ia, b) with if) 0.Thenforanyx 0 [a + d, b d], wehavethat fx)δx x 0 ) 0 fx 0 ). x Ia,b) Proof. Fix x 0 [a + d, b d].sincef is a finite analytic function, there exists a η>0in R with λη) λb a) such that, for any x Ia, b) satisfying x x 0 <η,wehavethatfx) f k) x 0 ) x x 0 ) k. k0 Therefore, fx)δx x 0 ) fx)δx x 0 ) x Ia,b) x x 0 d,x 0 +d) x x 0 d,x 0 +d) k0 f k) x 0 ) x x 0 ) k δx x 0 ) fx 0 )δx x 0 ) + x x 0 d,x 0 +d) x x 0 d,x 0 +d) fx 0 )+ x x 0 d,x 0 +d) f k) x 0 ) x x 0 ) k δx x 0 ) Now, for any x x 0 d, x 0 + d),wehavethat x x 0 <d, and hence f k) x 0 ) x x 0 ) k δx x 0 ) x x 0 d,x 0 +d) It follows that Thus λ x x 0 d,x 0 +d) x x 0 d,x 0 +d) f k) x 0 ) f k) x 0 ) x [x 0 d,x 0 +d] x x 0 ) k δx x 0 ) f k) x 0 ) x x 0 ) k δx x 0 ). x x 0 ) k δx x 0 ) λ f k) x 0 ) d k δx x 0 ). f k) x 0 ) d k. ) f k) x 0 ) d k.

11 4 FLYNN, SHAMSEDDINE However, since if) 0we can apply Theorem 3.6 to get that for all k N, λf k) x 0 )) > k and hence ) f λ k) x 0 ) d k > 0. Thus, Therefore, It follows that λ x x 0 d,x 0 +d) x x 0 d,x 0 +d) x Ia,b) f k) x 0 ) x x 0 ) k δx x 0 ) > 0. f k) x 0 ) x x 0 ) k δx x 0 ) 0 0. fx)δx x 0 ) 0 fx 0 ). Proposition 3.8. Let a<b<cin R be such λb a) < 1 and λc b) < 1; let g :[a, b] R and h :[b, c] R be analytic functions satisfying gb) hb) and ih) ig) 0; and let f :[a, c] R be given by { gx) if x [a, b) fx) hx) if x [b, c]. Then for any x 0 [a + d, c d], wehavethat fx)δx x 0 ) 0 fx 0 ). x [a,c] Proof. Without loss of generality, we may assume that b 0.Fixx 0 [a + d, c d].if x 0 d then by Proposition 3.7 we are done; so without loss of generality we may assume that x 0 <d. Thus, we have that fx)δx x 0 ) gx)δx x 0 )+ hx)δx x 0 ). x [a,c] x [x 0 d,0] x [0,x 0 +d] Both g and h are analytic functions defined on [a, 0] and [0,c], respectively; and hence they both can be expanded as power series centered at 0. Thus, gx) α k x k and where α k gk) 0) hx) k0 β k x k, k0 and β k hk) 0) for k 0, 1,,...

12 ON INTEGRABLE DELTA FUNCTIONS 43 Since λb a) λ a) λa) < 1 and λc b) λc) < 1 both power series will have radii of convergence infinitely larger than d, and hence they will converge everywhere on [x 0 d, 0] and [0,x 0 + d], respectively. Thus, gx)δx x 0 ) α k x k δx x 0 ) x [x 0 d,0] x [x 0 d,0] k0 and x [0,x 0 +d] hx)δx x 0 ) x [0,x 0 +d] β k x k δx x 0 ). k0 Therefore, x [a,c] fx)δx x 0 ) x [x 0 d,0] α 0 α k x k δx x 0 )+ β k x k δx x 0 ) k0 x [0,x 0 +d] k0 δx x 0 )+β 0 δx x 0 ) + + x [x 0 d,0] x [x 0 d,0] x [0,x 0 +d] α k x k δx x 0 ) β k x k δx x 0 ). x [0,x 0 +d] However, α 0 g0) f0) h0) β 0, and hence α 0 δx x 0 )+β 0 δx x 0 )f0) δx x 0 )f0). x [x 0 d,0] x [0,x 0 +d] x [x 0 d,x 0 +d] Thus, x [a,c] fx)δx x 0 )f0) + x [x 0 d,0] α k x k δx x 0 )+ x [0,x 0 +d] β k x k δx x 0 ). But λ α k x k δx x 0 )+ β k x k δx x 0 ) x [x 0 d,0] λ α k d) k x [0,x 0 +d] δx x 0 )+ β k d) k x [x 0 d,0] x [0,x 0 +d] δx x 0 ) > 0 as in the proof of Proposition 3.7. Thus, α k x k δx x 0 )+ x [x 0 d,0] x [0,x 0 +d] β k x k δx x 0 ) 0 0,

13 44 FLYNN, SHAMSEDDINE and hence x [a,c] fx)δx x 0 ) 0 f0). However, f0) 0 fx 0 ) [6]; therefore fx)δx x 0 ) 0 fx 0 ). x [a,c] Uniform differentiability on an interval of R is defined in the same way as in the real case. Definition 3.9. Let a<bin R be given and let f : Ia, b) R be differentiable with derivative f on Ia, b). Thenwesaythatf is uniformly differentiable on Ia, b) if for every ɛ>0 in R there exists δ>0in R such that for all x, y Ia, b) 0 < y x <δ fy) fx) y x f x) <ɛ. Lemma Let f : I0, 1) R be analytic with if) 0.Thenf is uniformly differentiable on I0, 1). Proof. Firstwenotethat,byTheorem3.6,wehavethatλf n) x)) 0 for all n N and for all x I0, 1). Nowletɛ>0 in R be given and let δ min { d 4 ɛ, d }. Then for x, y I0, 1) satisfying 0 < y x <δ,wehavethat fy) fx)+f x)y x)+ n f n) x) y x) n, n! where the power series converges in the order topology since λf n) x)) 0 for all n N and since 0 < y x <δ 1 so that It follows that and hence lim n fy) fx) f x)y x) fy) fx) y x f n) x) y x) n 0. n! n f x) < <ɛ. n0 f n) x) y x) n < n! n0 d n 3 n! d n n! y x < ) dɛ n n0 d 1 d ɛ d 1 n! dn y x) ; ) d n 3 d 4 ɛ n! Theorem Let a<bin R be given and let f : Ia, b) R be analytic on Ia, b). Thenf is uniformly differentiable on Ia, b).

14 ON INTEGRABLE DELTA FUNCTIONS 45 Proof. Let F : I0, 1) R be given by F x) d if) fa +b a)x). ThenF is analytic on I0, 1) with if )0; and hence, by Lemma 3.10 above, F is uniformly differentiable on I0, 1). Nowfix ɛ>0 in R. SinceF is uniformly differentiable on I0, 1), there is a δ>0in R such that if x, y I0, 1) and y x < δ b a then F y) F x) F x)y x) <d if) ɛ. However, F y) F x) F x)y x) d if) fa +b a)y) d if) fa b a)x) d if) f a +b a)x)b a)y x) d if) fa +b a)y) fa +b a)x) f a +b a)x)b a)y x). Thus, if x, y I0, 1) and y x < δ b a then we have that fa +b a)y) fa +b a)x) f a +b a)x)b a)y x) <ɛ. Now let u, v Ia, b) be such that v u <δ;andlet x u a b a and y v a b a. Then u a +b a)x, v a +b a)y, x, y I0, 1) and y x < δ b a. It follows that fv) fu) f u)v u) fa +b a)y) fa +b a)x) f a +b a)x)b a)y x) <ɛ. Thus, f is uniformly differentiable on Ia, b). Notation 3.1. In the following, and to avoid confusion with the number d, we will use D x to denote the differential operator d dx, moreover we will use Dn dn x to denote dx. n Proposition Let x 0, a<b,andɛ>0in R be given; and let f :[x 0 ɛ, x 0 + ɛ] [a, b] R be a double) power series. Then D x fx, y) fx, y). x Proof. Let N N be such that d N <ɛ.bytheorem3.11abovewehavethatf is uniformly differentiable with respect to x and hence fx + d N+k,y) fx, y) lim k d N+k fx, y) uniformly). x Moreover, by definition D x fx, y) lim k However, by Theorem 3.9 in [8] we have that fx + d N+k,y) fx, y) lim k d N+k This completes the proof of the proposition. fx + d N+k,y) fx, y) d N+k. fx + d N+k,y) fx, y) lim k d N+k fx, y). x

15 46 FLYNN, SHAMSEDDINE Corollary Let x 0, a<b,andɛ>0 in R be given; and let f :[x 0 ɛ, x 0 + ɛ] [a, b] R be analytic on :[x 0 ɛ, x 0 + ɛ] [a, b]. Then D x fx, y) fx, y). x Proof. This follows immediately from the fact that analytic functions are given locally by power series. Proposition 3.15 Leibniz s Rule). Fix x 0 R and let ɛ>0 in R be given. Let α, β :[x 0 ɛ, x 0 + ɛ] R be analytic functions with αx) βx) for all x [x 0 ɛ, x 0 + ɛ]. LetS bethesimpleregion given by S {x, y) R : y [αx),βx)],x [x 0 ɛ, x 0 + ɛ]} and let f : S R be analytic. Then fx, y) fx, βx))β x) fx, αx))α x)+ D x y [αx),βx)] Proof. The proof is identical to that of the real case. y [αx),βx)] fx, y). x Proposition Let x 0, a<b,andɛ>0 in R be given and let μ :[x 0 ɛ, x 0 + ɛ] [a, b] be a non-constant analytic function. Let g :[x 0 ɛ, x 0 + ɛ] [a, μx)] R and h :[x 0 ɛ, x 0 + ɛ] [μx),b] R be analytic and let f :[x 0 ɛ, x 0 + ɛ] [a, b] be given by { gx, y) if y μx) fx, y) hx, y) if y>μx). Then D x fx, y) fx, y) x if and only if fx, y) is continuous. Proof. Observe that D x fx, y) D x gx, y)+d x hx, y). y [a,μx)] But by proposition 3.15) we have that gx, y) gx, μx))μ x)+ and So D x y [a,μx)] D x D x y [μx),b] hx, y) y [μx),b] fx, y) gx, μx))μ x)+ y [μx),b] y [a,μx)] gx, y) x x hx, y) hx, μx))μ x). y [a,μx)] gx, y) x

16 ON INTEGRABLE DELTA FUNCTIONS 47 + x hx, y) hx, μx))μ x) y [μx),b] x fx, y)+[gx, μx)) hx, μx))] μ x). Since μ is a non-constant analytic function we know that μ 0; and it follows that the above expression equals x fx, y) if and only if gx, μx)) hx, μx)) for all x [x 0 ɛ, x 0 + ɛ], that is if and only if f is continuous at y μx) and hence everywhere since g and h are analytic). 4. EXAMPLES IN ONE DIMENSION In this section, we present two simple examples in which we illustrate the applications of the delta function defined on R above. Example 4.1. [Solving Poisson s Equation in One Dimension] Suppose that we wish to find the solution to the real differential equation ẍt) ft) on the interval [0, + ) and subject to the initial conditions x0) 0, ẋ0) 0. To begin, we observe that the piecewise analytic solution to Gt, t )δt t ) is t A 1 t t )+B 1 t t d Gt, t ) A t t )+B d 3 d t t ) 1 6 t t ) 4 ) t d<t <t+ d, A 3 t t )+B 3 t t + d where A 1, A, A 3, B 1, B and B 3 are constants to be determined. To ensure that our solution satisfies the given initial conditions we must have that the real parts of G0,t ) and G t 0,t ) equal zero; and to accomplish that, it is enough to set Gr, t )0 and G t r, t )0where r R is any number that is infinitely small in absolute value we will use r d since that turns out to be a convenient choice). In order to apply Proposition 3.16 we require that G be continuous so that D t t Gt, t ) t t Gt, t ))andthat G t t, t ) be continuous so that D t t t Gt, t ) t t Gt, t )). Using the initial conditions and continuity of Gt, t ) and its derivative at t t ± d,wecansolvefora 1, B 1, A, B, A 3,andB 3,toget t t d t t d Gt, t ) 1 t t ) d d 3 d t t ) 1 6 t t ) 4 ) t d<t <t+ d. 0 t t + d Note that when restricted to real points, the real part of Gt, t ) reduces to the classical Green s function for Dt. Applying Proposition 3.16, we obtain that Dt Gt, t )ft ) t Gt, t )ft ) It follows that t [0,d 1 ] t [0,d 1 ] t [0,d 1 ] t [0,d 1 ] 0 ft). δt t )ft ) Gt, t )ft ) [0] is a real) solution to the equation üt) ft)

17 48 FLYNN, SHAMSEDDINE with the initial conditions G0,t )ft ) [0] t [0,d 1 ] and hence we must have that xt) t [0,d 1 ] Now, if we set ft) t then we see that Gt, t )ft ) Gt, t )ft ) t [0,d 1 ] G t 0,t )ft ) [0] 0; Gt, t )ft ) [0]. t [0,d 1 ] t t d)t + t [0,t+d] t [0,t d] t [t d,t+d] But 1 t t ) d + 3 d 8 d 3 t t ) 16 )) t t ) 4 t. t t d )t 1 t d)3 6 and Thus, 1 6 t3. t [t d,t+d] t [0,d 1 ] t [0,t d] 1 t t ) d + 3 d 8 d 3 t t ) 16 )) t t ) 4 t 7 5 td d3. Gt, t )ft ) 1 6 t d) td d t3 and hence the real solution is xt) Example 4. Damped Driven Harmonic Oscillator). Consider now an underdamped, driven harmonic oscillator with mass m, viscous damping constant c, spring constant k, and driving force ft).letxt) be the position of the oscillator at time t with x0) 0 and ẋ0) 0. The oscillator s equation of motion is With the following change of variables equation 4.1) takes the form ẍt)+ c k mẋt)+ xt) ft) m m. 4.1) γ c k mk and ω 0 m, ẍt)+γω 0 ẋt)+ω0 xt) ft) m. Since the oscillator is underdamped we have that γ ω0 ω 0 < 0 which is equivalent to γ<1. To solve the equation of motion we first find the Green s function for the differential operator Dt +γω 0 D t + ω0 );thatis,wefind a solution for the differential equation ) t +γω 0 t + ω 0 Gt, t )δt t ).

18 ON INTEGRABLE DELTA FUNCTIONS 49 First we observe that the analytic solution to the homogeneous partial differential equation ) t +γω 0 t + ω 0 G hom t, t )0 is G hom t, t )e γω 0t t ) A sinωt t )) + B cosωt t )) ), where ω 1 γ ω 0 and where A and B are arbitrary constants. One particular solution to the inhomogeneous partial differential equation ) t +γω 0 t + ω 0 G inhom t, t ) 3 4 d 3 d t ) is given by G inhom t, t ) 3 d ω0 d 3 t t ) + γt t ) + 1 ) 4γ 4 ω 0 ω0. Since 0 if t d δt) 3 4 d 3 d t ) if d<t<d, 0 if d t we must have e γω 0t t ) A 1 sinωt t )) + B 1 cosωt t ))) Gt, t e γω 0t t ) A sinωt t )) + B cosωt t ))) ) + 3 d t t ) ω0 4 + γt t ) ω γ ) ω0 e γω 0t t ) A 3 sinωt t )) + B 3 cosωt t ))) if t t d if t d<t <t+ d if t t + d where A 1,A,A 3,B 1,B,B 3 are constants to be determined by the initial conditions. As in the previous example the real part of our Green s function must satisfy the same initial conditions as the desired solution. To this end we require G d, t )0and G t d, t )0. Solving for the relevant constants yields A 3 0and B 3 0. Again, as in the previous example, requiring continuity of G and G/t givesustheremaining4 constants: A 1 3 ω0 d 3 e γω 0d γ 3 ω ω0 d 3 e γω 0d 3γ B 1 3 ω0 d 3 e γω 0d 3 ω0 d 3 e γω 0d A 3 ω0 d 3 e γω 0d ω 0 3γ + γ 1 ) ) cosωd) d ω 0 ω ) cosωd) γ3 + ω 0 ω 0 γ 3 3γ + ω 0 3γ γ3 + ω 0 ω 0 γ 3 3γ + ω 0 ω 0 γ 1 ) d γ 1 ) d γ 1 ) d γ 1 ) d γ d 1 4γ ω 0 ω0 γ + d 1 4γ ω ω 0 ω0 ) sinωd) γ + d 1 4γ ω ω 0 ω0 ) sinωd) γ d 1 4γ ω ω 0 ω0 ) cosωd) γ d 1 4γ ω ω 0 ω0 ) ) sinωd) ) ) sinωd) ) ) cosωd) ) ) cosωd) ) ) sinωd)

19 50 FLYNN, SHAMSEDDINE B 3 ω0 d 3 e γω 0d γ 3 3γ + γ 1 ) ) sinωd) γ d + d 1 ) ) 4γ ω 0 ω 0 ω ω 0 ω0 cosωd). While at first glance these constants seem too cumbersome, we have that 1 A 1 0 ω and B 1 0 0; and hence { 1 Gt, t ) R ω e γω 0t t ) sinωt t )) if t>t 0 0 if t t which is the classical Green s function for this problem. Now, suppose that the driving force is given by ft) me γω0t. Then the equation of motion becomes ẍt)+γω 0 ẋt)+ω0 xt) e γω 0t. Thus, as in the previous example, we can obtain the real solution as the real part of Gt, t ) ft ) m. Therefore, t [0,d 1 ] xt) 0 t [0,d 1 ] But Gt, t )0for t >t+ d, and hence Gt, t ) ft ) m Thus, xt) 0 t [0,d 1 ] Gt, t )e γω 0t Gt, t ) ft ) m. t [0,t+d] Gt, t )e γω 0t. t [0,t+d] e γω 0t A sinωt t )) + B cosωt t ))) + e γω 0t + e γω 0t t [t d,t+d] t [t d,t+d] t [0,t d] 3 ω0 d t t ) + γt t ) + 1 ) 4γ 4 ω 0 ω0 A 1 sinωt t )) + B 1 cosωt t ))) [ e γω 0t A sinωd) sinωt) sinωd) cosωt) cosωd) + A 1 + B 1 ω ω ω 3 ) ω0 d 3 4γ 3 ω γ γω0 3 e γω0d e γω0d )+ 3 d ω0 γ ω0 + d ) ] ω0 e γω0d + e γω0d ) 0 e γω 0t cosωt) 1 ω, which agrees with the classical solution.

20 ON INTEGRABLE DELTA FUNCTIONS THE DELTA FUNCTION ON R AND R 3 Using the integration theory developed in [14] it is possible to define integrable delta functions on R and R 3. Definition 5.1. Let δ : R R be given by δ x, y) δx)δy), where δ is the delta function defined above on R. Proposition 5.. Let S R be measurable. If d, d) d, d) S then δ x, y) 1. If d, d) d, d) S then Proof. If d, d) d, d) S then δ x, y) S S S δ x, y) 0. x,y) d,d) d,d) x d,d) x d,d) δx) δx) 1. δx)δy) y d,d) δy) If d, d) d, d) S, thenδ x, y) 0everywhere on S; and hence δ x, y) 00. S S Proposition 5.3. Let S R be a simple region with λ x S) < 1 and λ y S) < 1, let f : S R be an analytic function with index if) 0on S. Then, for any x 0,y 0 ) S that satisfies x 0 a, x 0 + a) y 0 a, y 0 + a) S for some positive a d in R,wehavethat fx, y)δ x x 0,y y 0 ) 0 fx 0,y 0 ). x,y) S Proof. First we note that δ x x 0,y y 0 )0everywhere except on the simple region x 0 d, x 0 + d) y 0 d, y 0 + d). Thus, fx, y)δ x x 0,y y 0 ) fx, y)δ x x 0,y y 0 ). x,y) S x x 0 d,x 0 +d) y y 0 d,y 0 +d) Now, for a fixed x x 0 d, x 0 + d), hy) :fx, y) is an analytic function on y 0 a, y 0 + a) which contains y 0 d, y 0 + d) ; and hence, by Proposition 3.7, we have that hy)δy y 0 ) 0 hy 0 )fx, y 0 ). y y 0 d,y 0 +d)

21 5 FLYNN, SHAMSEDDINE Furthermore, gx) :fx, y 0 ) is analytic on x 0 a, x 0 + a) which contains x 0 d, x 0 + d); and hence gx)δx x 0 ) 0 gx 0 )fx 0,y 0 ). x x 0 d,x 0 +d) Thus, x,y) S fx, y)δ x x 0,y y 0 ) x x 0 d,x 0 +d) 0 δx x 0 ) y y 0 d,y 0 +d) δx x 0 )fx, y 0 ) δy y 0 )fx, y) x x 0 d,x 0 +d) 0 fx 0,y 0 ). Definition 5.4. Let δ 3 : R 3 R be given by δ 3 x, y, z) δx)δy)δz). It follows immediately from Definitions 5.1 and 5.4 that δ 3 x, y, z) δ x, y)δz) δ x, z)δy) δx)δ y,z). Proposition 5.5. Let S R 3 be measurable. If d, d) d, d) d, d) S then δ 3 x, y, z) 1. If d, d) d, d) d, d) S then S S δ 3 x, y, z) 0. Proof. If d, d) d, d) d, d) S then δ 3 x, y, z) S x,y,z) d,d) d,d) d,d) x,y) d,d) d,d) x,y) d,d) d,d) δ x, y) δ x, y) 1. δ 3 x, y, z) z d,d) δz) If d, d) d, d) d, d) S, thenδ 3 x, y, z) 0everywhere on S; and hence δ 3 x, y, z) 00. S S

22 ON INTEGRABLE DELTA FUNCTIONS 53 Proposition 5.6. Let S R 3 be a simple region with λ x S) < 1, λ y S) < 1, λ z S) < 1, and let f : S R be an analytic function on S with if) 0 on S. Then, for any x 0,y 0,z 0 ) S that satisfies x 0 a, x 0 + a) y 0 a, y 0 + a) z 0 a, z 0 + a) S for some positive a d in R,wehavethat fx, y, z)δ 3 x x 0,y y 0,z z 0 ) 0 fx 0,y 0,z 0 ). x,y,z) S Proof. First we note that δ 3 x x 0,y y 0,z z 0 )0everywhere except on the simple region x 0 d, x 0 + d) y 0 d, y 0 + d) z 0 d, z 0 + d). Thus, fx, y, z)δ 3 x x 0,y y 0,z z 0 ) x,y,z) S fx, y, z)δ 3 x x 0,y y 0,z z 0 ) x,y,z) x 0 d,x 0 +d) y 0 d,y 0 +d) z 0 d,z 0 +d) x,y) x 0 d,x 0 +d) y 0 d,y 0 +d) δ x x 0,y y 0 ) z z 0 d,z 0 +d) fx, y, z)δz z 0 ). Now, for a fixed x, y) x 0 d, x 0 + d) y 0 d, y 0 + d), hz) :fx, y, z) is an analytic function on the interval z 0 a, z 0 + a) which contains z 0 d, z 0 + d); and hence, by Proposition 3.7, we have that hz)δz z 0 ) 0 hz 0 )fx, y, z 0 ). z z 0 d,z 0 +d) Furthermore, gx, y) :fx, y, z 0 ) is analytic on the simple region S xy : x 0 a, x 0 + a) y 0 a, y 0 + a) containing x 0 d, x 0 + d) y 0 d, y 0 + d); and hence, by Proposition 5.3, we have that gx, y)δ x x 0,y y 0 ) 0 gx 0,y 0 )fx 0,y 0,z 0 ). Thus, x,y) x 0 d,x 0 +d) y 0 d,y 0 +d) fx, y, z)δ 3 x x 0,y y 0,z z 0 ) x,y,z) S δ x x 0,y y 0 ) x,y) x 0 d,x 0 +d) y 0 d,y 0 +d) z z 0 d,z 0 +d) 0 δ x x 0,y y 0 )fx, y, z 0 ) x,y) x 0 d,x 0 +d) y 0 d,y 0 +d) 0 fx 0,y 0,z 0 ). fx, y, z)δz z 0 ) ) As in the classical case it is also possible to define the delta function in spherical coordinates, in particular we have that δ sph r r )F r, φ, θ)δ 3 r r,φ φ,θ θ ),

23 54 FLYNN, SHAMSEDDINE where of course r is the point r, φ, θ), r is the point r,φ,θ ),andf is some as yet unknown function. Naturally we must have that δ sph r r )1, from which it follows that r R 3 F r,φ,θ )δ 3 r r,φ φ,θ θ )r sin θ 1. r R + φ [0,π] θ [0,π] However, we already know that δ 3 r r,φ φ,θ θ )1 r R + φ [0,π] θ [0,π] since δ 3 r r,φ φ,θ θ ) is normalized by definition, so we obtain δ 3 r r,φ φ,θ θ ) r R + φ [0,π] θ [0,π] F r,φ,θ )δ 3 r r,φ φ,θ θ )r sin θ and hence r R + φ [0,π] θ [0,π] F r,φ,θ ) 1 r sin θ. Definition 5.7. We define δ sph : R 3 R by δ sph r r ) δr r,φ φ,θ θ ) r. sin θ Note that as in the classical case, if a problem has spherical symmetry then the delta function takes the form δ sph r r )Fr )δr r ) and since r sinθ )4πr it follows that and hence φ [0,π] θ [0,π] F r ) 1 4πr δ sph r r ) δr r ) 4πr. Example 5.8 Electric Field of a thick spherical shell). Suppose we wish to find the electric field of a thick spherical shell centred at the origin with inner radius R 1, outer radius R and a uniform charge density. One way to accomplish this is to solve the differential equation implied by Gauss s law: Er) ρr)

24 ON INTEGRABLE DELTA FUNCTIONS 55 where Er) is the electric fieldatthe pointr andρr), the chargedensitymultiplied byaconstant that depends on the system of units used, is given by 0 if r<r 1 ρr) ρ 0 if R 1 r R. 0 if r>r As in previous examples we can solve this differential equation by finding the Green s function Gr, r )G r e r + G φ e φ + G θ e θ corresponding to the operator, in particular Gr, r ) must satisfy Gr, r )δ sph r r ). 5.1) However we have spherical symmetry about the origin; so we may infer that G φ G θ 0 and δ sph r r ) δr r ) 4πr. Thus, equation 5.1) reduces to 1 r r G r r, r ) ) δr r ) r 4πr. Solving this differential equation yields c 1 if r r d G r r, r r ) 1 3 4πr 4 d 3 d r r ) 1 3 r r ) 3) + c r if r d<r <r+ d if r r + d c 3 r where c 1, c,andc 3 are constants of integration. We know that Er) 0for r<r 1 since there is no charge inside the shell. To ensure our solution satisfies this initial condition we must have G r 0,r ) 0 0 and as in previous examples we accomplish this by setting G r d, r )0.Using this initial condition as well as the continuity of G r,weareabletosolvefortheconstantsin G r r, r ),infactwefind that 1 G r r, r 4πr if r r d ) 1 3 4πr 4 d 3 d r r ) 1 3 r r ) 3) + 1 if r d<r <r+ d. 8πr 0 if r r + d Now that we know the Green s function of the operator and have made it satisfy the relevant boundary conditions we can solve for the real) electric field of the spherical shell by recalling that Er) E r r)e r 0 G r r, r )ρr )r sin θ e r 4π G r r, r )ρr )r e r. If r<r 1 then we have that r R + φ [0,π) θ [0,π] If R 1 r R then we have that ρ 0 r E r r) 0 4π 4πr + r [R 1,r d] r [r d,r+d] E r r) 0 0. r R d 4πr 4 d 3 r r ) 13 ) r r ) ) 8πr ρ 0 r

25 56 FLYNN, SHAMSEDDINE ρ 0 0 r 3 3r R1 3 ). Finally if r>r then we get E r r) 0 4π ρ 0 r 4πr ρ 0 R 3 3r R1 3 ). r [R 1,R ] Hence the electric field of a uniformly charged thick spherical shell is given by 0 if r<r 1 E r r) ρ 0 3r r 3 R 3 ) 1 if R 1 r R. ρ 0 3r R 3 R1 3 ) if r>r While the examples given in this section are admittedly simple they serve to illustrate how to use the newly defined delta functions. In the future we plan to engage in a more detailed study of the delta functions in two and three dimensions as well as in constructing more complex and challenging examples. REFERENCES 1. N. L. Alling, FoundationsofAnalysisoverSurrealNumberFieldsNorth Holland, 1987).. W. Krull, Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167, S. Priess-Crampe, Angeordnete Strukturen: Gruppen, Korper, projektive Ebenen Springer, Berlin, 1983). 4. P. Ribenboim, Fields: algebraically closed and others, Manuscripta Math. 75, ). 5. W. H. Schikhof, Ultrametric Calculus: An Introduction to p-adic Analysis Cambridge Univ. Press, 1985). 6. K. Shamseddine, New Elements of Analysis on the Levi-Civita Field, PhD thesis, Michigan State University, East Lansing, Michigan, USA 1999). Also Michigan State University report MSUCL K. Shamseddine, A brief survey of the study of power series and analytic functions on the Levi-Civita fields, Contemp. Math. 596, ). 8. K. Shamseddine, New results on integration on the Levi-Civita field, Indag. Math. N.S.) 4 1), ). 9. K. Shamseddine, Analysison the Levi-Civita field and computational applications, J. Appl. Math. Comput. 55, ). 10. K. Shamseddine and M. Berz, Exception handling in derivative computation with non-archimedean calculus, in Computational Differentiation: Techniques, Applications, and Tools, pp SIAM, Philadelphia, 1996). 11. K. Shamseddine and M. Berz, Convergence on the Levi-Civita field and study of power series, in Proc. Sixth International Conference on p-adic Functional Analysis, pp Marcel Dekker, New York, NY, 000). 1. K. Shamseddine and M. Berz, Measure theory and integration on the Levi-Civita field, Contemp. Math. 319, ). 13. K. Shamseddine and M. Berz, Analyticalpropertiesofpowerserieson Levi-Civitafields, Ann. Math. Blaise Pascal 1 ), ). 14. K. Shamseddine and D. Flynn, Measure theory and Lebesgue-like integration in two and three dimensions over the Levi-Civita field, Contemp. Math. 665, ).