Appendix F Asymptotic Expansion of Single Integrals
Asymptotic analysis is a powerful analytical approach to obtaining elegantly simple analytic approximations to problems that contain either a parameter or a variable whose magnitude becomes either large or small in comparison to some value that is characteristic of the problem. Its elegance lies in the fact that the results may usually be expressed in a single dominant term that contains all of the essential physics of the problem instead of through the subtle interaction of a large (perhaps infinite) number of terms in a summation. The basic idea behind this general approximating method of analysis may be illustrated by the evaluation of the real exponential integral [1] Z 1 1 t E1.x/ e dt; (F.1) x t where x>0is real-valued. This function possesses the convergent series expan- sion [2] X1 xn E .x/ D C ln.x/ C (F.2) 1 nnŠ nD1 P 1 1 for x>0, where limn!n kD1 k ln.n C 1/ D 0:57721 : : : is Euler’s constant. Although this series converges for all positive values of x, it becomes computationally useless for x 1. To obtain a useful expression for the value of this function for large values of its argument, repeated integration by parts results in [1] ˇ Z t ˇ1 1 e ˇ 1 t E1.x/ D ˇ e dt t t 2 x ˇ x Z x t ˇ1 1 e e ˇ 1 t D C ˇ C 2 e dt x t 2 t 3 x x e x 1Š 2Š 3Š nŠ D D 1 C C C. 1/n C R .x/; x x x2 x3 xn nC1 (F.3)
777 778 F Asymptotic Expansion of Single Integrals where the remainder after n terms is given by Z 1 e t R .x/ D . 1/nC1.n C 1/Š dt: nC1 nC2 (F.4) x t
Since the integral appearing in this remainder term is bounded by e x=xnC2 for x>0, then the magnitude of this remainder term is bounded as
.n C 1/Š jR .x/j < e x: (F.5) nC1 xnC2 The remainder after n terms is then seen to be bounded in magnitude by the magni- tude of the first term neglected in the series summation e x 1Š 2Š 3Š nŠ S .x/ D 1 C C C. 1/n : (F.6) n x x x2 x3 xn
However, if one considers the expansion given in (F.6) as an infinite series, the result is divergent. Nevertheless, for sufficiently large values of x>0, the series summation given in (F.6) is rapidly convergent for a finite number of terms n.An estimate of the optimum number of terms to be used in this expansion for a given value of x may be obtained from the ratio of successive terms as jun.x/=unC1.x/j D n=x 1, so that the optimum number of terms to be used in the summation Sn.x/ for an estimate of E1.x/ for a given large value of x is approximately given by the greatest integer in x, as illustrated in Fig. F.1 when x D 5:7. In that case the 5 optimum number of terms is given by n D 5 where 5e Sn.5/ D 0:8704, which is in 5 good agreement with the actual value of 5e E1.5/ D 0:8663. Inclusion of additional terms in the summation only results in a decrease in accuracy. Most importantly, since the remainder after the first (or dominant) term becomes exponentially small as x !1, the approximation E1.x/ S1.x/ becomes increasingly accurate as x increases. This example then leads to the following distinction between an asymptotic expansion and a power series expansion of some function: For the power series expansion XN f.x/Š un.x/ nD0 of a given function f.x/, the approximation to the value of f at some fixed value of x improves in some well-defined sense as N !1, while for an asymptotic expansion
f.x/D Sn.x/ C RnC1.x/ of that function, the approximation to f.x/by the series summation Sn.x/ improves in some (as yet undefined) sense for fixed n as x !1. The first (or dominant) term F Asymptotic Expansion of Single Integrals 779
1
0.95
0.9 (x) n
S 0.85 x xe
0.8
0.75
0.7 2 3 4 5 6 7 8 9 10 n
Fig. F.1 Dependence of the series summation Sn.x/ approximation of the exponential integral E1.x/ for x D 5:7 as a function of the number n of terms in the summation. The open circles x connected by the solid line segments describe the values of the quantity xe Sn.x/,whilethedashed curves describe upper and lower envelopes to these values. Notice that the approximate values 5 oscillate about the actual value of 5e E1.5/ D 0:8663
Fig. F.2 Statue of the Norwegian mathematician Niels Henrik Abel (1802–1829) wrestling with a sea serpent on the royal palace grounds in Oslo, Norway. (Photograph by K. E. Oughstun) in the asymptotic expansion of of f.x/ represents the asymptotic approximation of that function as x !1. However, care must always be taken to ensure that the given asymptotic expansion is not only well defined mathematically but is also properly applied and interpreted. If not, critical errors may result. Such was the motivation for Abel (see Fig. F.2) to lament in 1828 that 780 F Asymptotic Expansion of Single Integrals
Divergent series are the invention of the devil, and it is a shame to base on them any demon- stration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes. . . A brief outline of the essential theory is presented here. A more detailed de- velopment may be found in the texts by E. T. Copson [3], N. Bleistein and R. Handelsman [4], and J. D. Murray [5].
F.1 Foundations
Definition 1. Spherical Neighborhood. A spherical neighborhood of a point z0 is given by the set of points in the open disc
jz z0j <ı (F.7) if z0 is at a finite distance from the origin (i.e., if jz0j < 1), while it is given by the open region jzj >ı (F.8) if z0 is the point at infinity. Definition 2. O-order. Let f.z/ and g.z/ be two functions of the complex variable z that possess definite limits as z ! z0 in some domain D. Then
f.z/ D O.g.z// as z ! z0 (F.9) if there exist positive constants K and ı such that jf.z/j Kjg.z/j whenever 0< jz z0j <ı.Ifjf.z/j Kjg.z/j for all z 2 D, then f.z/ D O.g.z// in D. Definition 3. o-order. Let f.z/ and g.z/ be two functions of the complex variable z that possess definite limits as z ! z0 in some domain D. Then
f.z/ D o.g.z// as z ! z0 (F.10) if there exists a positive constant ı such that jf.z/j jg.z/j for any >0when- ever 0 Thus, as long as g.z/ is not zero in a neighborhood of z0, except possibly at the point z0 itself, then f.z/ f.z/ D o.g.z// H) ! 0 as z ! z0; (F.11) ˇg.z/ ˇ ˇ ˇ ˇf.z/ˇ f.z/ D O.g.z// H) ˇ ˇ K as z ! z0; (F.12) g.z/ where K is some positive constant. The O-order is clearly more important than the o-order in asymptotic analysis since it provides more specific information about the F.1 Foundations 781 behavior of the function at the point under consideration. For example, if f.z/ ! 0 as z ! z0, then the O-order states how rapidly f.z/ approaches zero at that point whereas the o-order merely confirms that f.z/ approaches zero at that point. In particular f.z/ D o.g.z// as z ! z0 H) f.z/ D O.g.z// as z ! z0; (F.13) while the opposite is not necessarily true. As an example, consider the exponential function f.z/ D e z which is an entire function of complex z D x C iy. With z 2 D1 W 0 x iy n f.z/ D e e D o.x / 8 n>0as jzj!1in D1: Any function of the complex variable z that is o.z n/ for all n>0as jzj!1 demonstrates the exponential character of the function. However, for the domain z D2 W 0 Theorem 7. Let fn.z/ D Ofgn.z/g in some domain D for n D 1;2;:::;N. Then ( ) XN XN anfn.z/ D O janjjgn.z/j (F.14) nD1 nD1 for all z 2 D, where the coefficients an, n D 1;2;:::;N are complex constants. Proof. Since fn D Ofgng in D, then there exists a set of real positive constants Kn such that jfnj Knjgnj in D.LetK D maxfKng, so that ˇ ˇ ˇ ˇ ˇXN ˇ XN XN ˇ ˇ ˇ anfnˇ janjjfnj K janjjgnj; nD1 nD1 nD1 which is precisely the statement of the above order relation. ut Theorem 8. Let f.z/ D Ofg.z/g for all z 2 D. Then for any ˛>0, jf.z/j˛ D O fjg.z/j˛g (F.15) for all z 2 D. Proof. Since f D Ofgg in D, then jf j Kjgj8z 2 D for some positive constant K. For any ˛>0, jf j˛ jKgj˛ K˛jgj˛ D Kjgj˛; which is precisely the statement of the above order relation. ut 782 F Asymptotic Expansion of Single Integrals Theorem 9. Let fi .z/ D Ofgi .z/g in some domain D for i D 1;2;:::;n, and let jgi .z/j jg.z/j for all z 2 D for each i D 1;2;:::;n. Then Xn ai fi .z/ D Ofg.z/g (F.16) iD1 for all z 2 D, where the coefficients ai , i D 1;2;:::;n, are all constants. Proof. Since fi D Ofgi g for i D 1;2;:::;n, then there exist positive constants Ki , i D 1;2;:::;n, such that jfi j Ki jgi j for all z 2 D.LetK D maxfKi g and let ai , i D 1;2;:::;nbe arbitrary constants. Then ˇ ˇ ˇ ˇ ˇXn ˇ Xn Xn Xn ˇ ˇ ˇ ai fi ˇ jai jjfi j K jai jjgi j Kjgj jai jDAKjgj; iD1 iD1 iD1 iD1 which is precisely the statement of the above order relation. ut Theorem 10. Let fi .z/ D Ofgi .z/g in some domain D for i D 1;2;:::;n. Then ( ) Yn Yn fi .z/ D O gi .z/ (F.17) iD1 iD1 for all z 2 D. Proof. Since fi D Ofgi g for i D 1;2;:::;n, then there exists a set of positive constants Ki , i D 1;2;:::;n, such that jfi j Ki jgi j for all z 2 D.LetK D maxfKi g, i D 1;2;:::;n. Then ˇ ˇ ˇ ˇ ˇYn ˇ Yn Yn Yn ˇ ˇ n ˇ fi ˇ jfi j Ki jgi j K jgi j; iD1 iD1 iD1 iD1 which is precisely the statement of the above order relation. ut Notice that these four theorems remain valid if the order symbol O is changed to the (weaker) order symbol o. Order relations can be integrated (a smoothing operation), but they cannot, in general, be differentiated with respect to some other variable. If a function f.z; / is a function of the two variables z and and if f.z; /D Ofg.z; /g as z ! z0 for all 2 D , then for two values 1; 2 2 D with 2 > 1, ˇZ ˇ Z Z ˇ 2 ˇ 2 2 ˇ ˇ ˇ f.z; /d ˇ jf.z; /jd K jg.z; /jd ; 1 1 1 F.2 Asymptotic Sequences, Series and Expansions 783 so that Z Z 2 2 f.z; /d D O g.z; /d (F.18) 1 1 ‹ as z ! z0. However, it is not always true that @f . z; /=@ D [email protected]; /=@ g as z ! z0; order relations under differentiation must then be considered on a case by case basis. F.2 Asymptotic Sequences, Series and Expansions Definition 4. Asymptotic Sequence. A finite or infinite sequence of functions f n.z/g is an asymptotic sequence as z ! z0 if there exists a spherical neighbor- hood of z0 within which none of the functions n.z/ vanish (except possibly at the point z0) and nC1.z/ D of n.z/g as z ! z0 (F.19) for all n. That is .z/ lim nC1 D 0 z!z0 n.z/ for all n. n As an example, the sequence of functions f.z z0/ g, n D 0; 1; 2; : : : is an asymptotic sequence as z ! z0 with finite jz0j since .z/ .z z /nC1 nC1 D 0 D .z z / ! 0 z ! z : n 0 as 0 n.z/ .z z0/ The sequence of functions fz ng, n D 0; 1; 2; : : : is an asymptotic sequence as z !1since .z/ z .nC1/ nC1 D D z 1 ! 0 z !1: n as n.z/ z z an Finally, the sequence of functions fe z g with real-valued an satisying anC1 >an is an asymptotic sequence as z !1since z a C nC1.z/ e z n 1 D D z .anC1 an/ ! 0 z !1: z a as n.z/ e z n Because of the order relation operations given in (F.14)–(F.18), new asymptotic sequences may be formed by the appropriate combination of existing asymptotic sequences. In particular, integration of an asymptotic sequence over some vari- able other than the asymptotic variable [cf. (F.18)] results in another asymptotic sequence, while differentiation does not, in general. 784 F Asymptotic Expansion of Single Integrals Definition 5. Asymptotic Expansion in the Sense of Poincare´P (1886). If f n.z/g 1 is an asymptotic sequence of functions as z !1, then the series nD1 an n.z/, not necessarily convergent, where the coefficients an are independent of the variable z, is said to be an asymptotic expansion of a function f.z/ in the sense of Poincare´ with respect to the asymptotic sequence f n.z/g if for every value of N , XN f.z/ D an n.z/ C of N .z/g (F.20) nD1 as z ! z0. If the asymptotic expansion given in (F.20) of a function f.z/ exists for a given asymptotic sequence f n.z/g, it is unique and the expansion coefficients are uniquely determined as f.z/ a1 D lim ; (F.21) z!z0 1.z/ f.z/ a1 1.z/ a2 D lim ; (F.22) z!z0 2.z/ : : P N 1 f.z/ nD1 an n.z/ aN D lim : (F.23) z!z0 N .z/ If a function f.z/ possesses an asymptotic expansion in this sense, one then writes X1 f.z/ an n.z/ as z ! z0: (F.24) nD1 A partial sum of this series is called an asymptotic approximation to the function f.z/. Since NX 1 f.z/ an n.z/ D aN N .z/ C of N .z/g nD1 as z ! z0, then NX 1 f.z/ D an n.z/ C Of N .z/g (F.25) nD1 is an asymptotic approximation to f.z/ as z ! z0 with an error of order Of N .z/g, which is of the same order of magnitude as the first term omitted in the series. The first nonzero term in the asymptotic expansion is called the leading or dominant term in the expansion. If a1 ¤ 0, then f.z/ a1 1.z/ as z ! z0 in the appropriate domain, meaning that f.z/= 1.z/ ! a1 as z ! z0. In many practical cases, the F.2 Asymptotic Sequences, Series and Expansions 785 asymptotic sequence may not be known and the leading term in the expansion is all that is or can be determined, and frequently is all that is required. The asymptotic expansion of a given function f.z/ depends upon the specific sequence of functions f n.z/g that is chosen, so that a function may possess several asymptotic expansions. This interesting property may be quite useful when expan- sions are required in different domains. However, care must always be taken not to mistakenly use an asymptotic expansion outside of its domain of validity, regard- less of how tempting the result may appear, as the consequences may lead to Abel’s lament. n The most common asymptotic sequence is the power sequence f.z z0/ g as z ! z0 in some domain D containing the point z0. Without any loss in generality, the point z0 may either be taken as the point at infinity through the change of variable D 1=.z z0/, or else as the origin with the change of variable D z z0.Ifz0 is the point at infinity, then a typical power series expansion of the function f.z/ about that point is of the form X1 a f.z/ g.z/ n as z !1; (F.26) zn nD1 where g.z/ is some function, valid in some domain of the variable z. Asymptotic ex- pansions that are based on asymptotic power sequences are called asymptotic power series. For example, the asymptotic expansion of the exponential integral given in (F.3) is an asymptotic power series in terms of the sequence f. 1/nC1.n 1/Šz ng as z !1with g.z/ D e z in the domain jarg.z/j < =2. As another example, consider the function f.z/ D 1=.z 1/ for jzj >1which possesses the power series expansion 1 X1 1 as z !1; z 1 zn nD1 in terms of the power sequence fz ng. Since 1 X1 1 as z !1; z2 1 z2n nD1 then one also has the power series expansion 1 X1 1 .z C 1/ as z !1; z 1 z2n nD1 in terms of the power sequence fz 2ng. Not only can a function possess more than one asymptotic expansion in a given domain, a given asymptotic expansion may also be the expansion for more than one function. As an illustration, consider the asymptotic expansion of the exponential 786 F Asymptotic Expansion of Single Integrals function e z in terms of the power sequence fz ng as z !1in the domain jarg.z/j < =2, given by X1 ˛ e z n as z !1; jarg.z/j < : zn 2 nD0 whose coefficients are determined from the limiting procedure given in (F.21)– (F.23) as e z ˛n D lim D 0; z!1 z n for all n D 0; 1; 2; : : : . Thus, if a given function f.z/ possesses the asymptotic power series expansion X1 a f.z/ g.z/ n as z !1; jarg.z/j < ; zn 2 nD1 then also X1 a f.z/ C e z g.z/ n as z !1; jarg.z/j < : zn 2 nD1 Theorem 11. Linear Superposition of Asymptotic Expansions. If f.z/ and g.z/ possess the asymptotic expansions X1 f.z/ an n.z/; nD1 X1 g.z/ bn n.z/; nD1 with respect to the asymptotic sequence f n.z/g as z !1in a common domain D, then X1 ˛f .z/ C ˇg.z/ .˛an C ˇbn/ n.z/; (F.27) nD1 as z !1in D, where ˛ and ˇ are complex constants. Proof. It follows from from (F.25) that ˇ ˇ ˇ ˇ NX 1 ˇ NX 1 ˇ O ˇ ˇ f.z/ an n.z/ C f N .z/gH)ˇf.z/ an n.z/ˇ K1j N .z/j; nD1 ˇ nD1 ˇ ˇ ˇ NX 1 ˇ NX 1 ˇ O ˇ ˇ g.z/ bn n.z/ C f N .z/gH)ˇg.z/ bn n.z/ˇ K2j N .z/j; nD1 nD1 F.2 Asymptotic Sequences, Series and Expansions 787 so that ˇ ˇ ˇ ˇ ˇ NX 1 ˇ ˇ ˇ ˇ˛f .z/ C ˇg.z/ .˛an C ˇbn/ n.z/ˇ ˇ " nD1 # " # ˇ ˇ ˇ ˇ NX 1 NX 1 ˇ ˇ ˇ D ˇ˛ f.z/ an n.z/ C ˇ g.z/ bn n.z/ ˇ ˇ nD1 ˇ ˇ nD1 ˇ ˇ ˇ ˇ ˇ ˇ NX 1 ˇ ˇ NX 1 ˇ ˇ ˇ ˇ ˇ j˛j ˇf.z/ an n.z/ˇ Cjˇj ˇg.z/ bn n.z/ˇ nD1 nD1 j˛jK1j N .z/jCjˇjK2j N .z/jDKj N .z/j; where K Dj˛jK1 CjˇjK2. ut Theorem 12. Product of Asymptotic Power Series. If f.z/ and g.z/ possess the asymptotic power series expansions XN a ˚ f.z/ D n C O z .N C1/ ; zn nD1 XN b ˚ g.z/ D n C O z .N C1/ ; zn nD1 as z !1in a common domain D, then XN c ˚ f.z/g.z/ D n C O z .N C1/ (F.28) zn nD1 as z !1in D, where cn D a0bn C a1bn 1 C Can 1b1 C anb0. The proof of this theorem follows directly from the product of the two asymptotic power series. It then follows from the result given in (F.28) that Œf .z/ m, where m is a positive integer, possesses an asymptotic power series as z !1in D, and hence that any polynomial of f.z/ and any rational function of f.z/, where the denominator has no zeros in the domain D, also possess asymptotic power series as z !1in D. In particular,P if the coefficient a0 ¤ 0 in the asymptotic power series expansion 1 n D f.z/ nD1 an=z as z !1in , then 1 1 X1 d C n (F.29) f.z/ a zn 0 nD1 as z !1in D. To establish this result and determine explicit expressions for the coefficients dn, notice first that 1=f.z/ tends to the finite limit 1=a0 d0 as z !1. 788 F Asymptotic Expansion of Single Integrals One then has that 1=f.z/ 1=a 1 1 0 D z 1=z f.z/ a0 1 1 D z 2 a0 C a1=z C O.1=z / a0 a .a C a =z C O.1=z2// D z 0 0 1 2 a0Œa0 C a1=z C O.1=z / a C O.1=z/ a D 1 ! 1 d ; O 2 2 1 a0Œa0 C a1=z C .1=z / a0 2 2 1=f.z/ 1=.a0 a1=z/ 1=f.z/ 1=a0 C a1=a0z a1 a0a2 D ! 3 d2; 1=z 1=z a0 and so on for higher order coefficients. Theorem 13. Integration of Asymptotic Power Series. If f.z/ has the asymptotic power series expansion XN a ˚ f.z/ D n C O z .N C1/ zn nD1 as z !1in a domain D, then Z 1 a XN a f. / a 1 d nC1 (F.30) 0 nzn z nD1 as z !1in D for all simply connected paths of integration in D. P ˚ N n O .N C1/ Proof. Since f.z/ a0 a1=z D nD2 an=z C r.z/, where r.z/ D z as z !1in D, then for any simply connected path of integration in D, Z Z Z 1 a XN 1 1 1 f. / a 1 d D a d C r. /d 0 n n z nD2 z z Z XN a 1 D n C r. /d : .n 1/zn 1 nD2 z ˇ ˇ Since jr.z/j K ˇz .N C1/ˇ, if the path of integration is taken along a path in D with fixed argument of the complex variable z, then ˇZ ˇ Z ˇ 1 ˇ 1 ˇ ˇ .N C1/ K N ˇ r. /d ˇ K j j dj jD jzj ; z jzj N F.2 Asymptotic Sequences, Series and Expansions 789 so that Z 1 a XN a f. / a 1 d D n C o z N C1 :ut 0 .n 1/zn 1 z nD2 Term by term differentiation of an asymptotic expansion, on the other hand, does not necessarily result in an asymptotic expansion except in the case of a function whose derivative possesses an asymptotic power series, as stated in the following: Theorem 14. Differentiation of Asymptotic Power Series. If the analytic function f.z/ and its first derivative f 0.z/ both possess asymptotic power series expansions as z !1in some domain D, then X1 a X1 na f.z/ n H) f 0.z/ n (F.31) zn znC1 nD0 nD1 as z !1in D. Proof. Since f 0.z/ possesses an asymptotic power series expansion as z !1in D, then X1 b f 0.z/ n : zn nD0 Since f 0.z/ is continuous in D, then Z z 0 f.z/ f.z0/ D f . /d z0 Z z z 0 b1 D b0.z z0/ C b1 ln C f . / b0 d : z0 z0 D 0 O 2 Since f.z/ !R an0 as z !1in o, and since f .z/ b0 b1=z D f1=z g so that the integral z f 0. / b b1 d is convergent as z !1, then b D b D 0, z0 0 0 1 so that the above expression becomes Z 1 X1 b a f.z/ D f 0. /d nC1 as z !1: 0 nzn z nD1 However, X1 a a f.z/ n as z !1: 0 zn nD1 Since an asymptotic power series expansion is unique in a given domain, then bnC1 D nan. ut 790 F Asymptotic Expansion of Single Integrals If the function f.z/ is both single-valued and analytic for all jzj >R, then it possesses a convergent power series expansion that is valid for all arg.z/ in the domain jzj >R. The uniqueness property of asymptotic power series then states that this convergent power series must be the asymptotic power series for f.z/ as z !1based on the asymptotic sequence fz ng. The preceding two theorems then state that this asymptotic series expansion is both differentiable and integrable for all jzj >R. If the function f.z/ is not analytic everywhere in the region jzj >R, then it cannot have a single asymptotic power series expansion in the sequence fz ng that is valid for all arg .z/. Different expansions will then be necessary for different ar- gument domains and caution must be exercised not to use the wrong expansion in a given subdomain. F.3 Integration by Parts In those cases where it is possible, a straightforward method of obtaining an asymp- totic expansion of the integral representation of a given function is to use the method of integration by parts, as illustrated in the following example: Consider the integral representation of the normalized incomplete gamma function [2] Z x .a;x/ D e t t a 1dt; (F.32) 0 where .aC 1/ D a .a/: (F.34) R 1 t Since .1/D 0 e dt D 1, one then has that .nC 1/ D nŠ (F.35) for n D 0; 1; 2; : : : , so that 0Š D .1/D 1. A power series expansion of the normalized incomplete gamma function may be obtained by expanding the integrand in (F.32) in a power series in the variable t and integrating term by term as Z ! Z x X1 . 1/n X1 . 1/n x .a;x/ D t n t a 1dt D t nCa 1dt nŠ nŠ 0 nD0 nD0 0 X1 . 1/n D xa xn; (F.36) .a C n/nŠ nD0 F.3IntegrationbyParts 791 which converges for all x. However, this power series representation is only useful for small values of x. For large values of x, an asymptotic power series will clearly be of more use. This divergent series representation is obtained from the integral representation (F.32), which may be rewritten as Z Z 1 1 .a;x/ D e t t a 1dt e t t a 1dt 0 x D .a/ .a;x/ (F.37) where Z 1 .a;x/D e t t a 1dt; (F.38) x is the incomplete gamma function [2]. Successive integration by parts of this integral representation of the incomplete Gamma function then gives Z 1 .a;x/ D e t t a 1dt let u D t a 1;dv D e t dt x Z 1 D e xxa 1 C .a 1/ e t t a 2dt let u D t a 2;dv D e t dt x : : ˚ D e x xa 1 C .a 1/xa 2 C C.a 1/ .a N C 1/xa N Z 1 C.a 1/.a 2/ .a N/ e t t a N 1dt: (F.39) x For any fixed value of N>a 1, ˇZ ˇ Z ˇ 1 ˇ 1 ˚ ˇ t a N 1 ˇ a N 1 t a N x ˇ e t dtˇ F.4 The Method of Stationary Phase The method of stationary phase was originally developed by G. G. Stokes [6] in 1857 and Lord Kelvin [7] in 1887 for the asymptotic approximation of Fourier trans- form type integrals of the form Z b f. /D g.t/ei h.t/dt (F.42) a as !1, where a, b, g.t/, h.t/, and t are all real-valued. The functions h. / and g. / are assumed to be analytic functions of the complex variable D t C i in some domain containing the closed interval Œa; b along the real axis [3]. The exponential kernel ei h.t/ appearing in the integrand is then purely oscillatory so that as becomes large, its oscillations become very dense and destructive interference occurs almost everywhere. The exceptions occur at any stationary phase point tj defined by 0 h .tj / 0; (F.43) since the phase term h.t/ is nearly constant in a neighborhood of each such point, as well as from the lower and upper end points t D a and t D b of the integration domain since the effects of destructive interference will be incomplete there. Assume that there is a single interior stationary phase point at t D t0 where 00 a 1 h.t/ D h.t / C h00.t /.t t /2 C (F.44) 0 2 0 0 ˚ 2 so that h.t/ h.t0/ D O .t t0/ , whereas in a neighborhood of any other point 2 Œa; b , h.t/ h. / D O f.t /g. The expansion given in (F.44) then suggests the change of variable 2 h.t/ h.t0/ ˙s ; (F.45) 2 00 2 00 where Cs is used when h .t0/>0while s is used when h .t0/<0. For values of t in a neighborhood of t0, (F.45) may be inverted with use of the Taylor series expansion given in (F.44) as F.4 The Method of Stationary Phase 793 2 1=2 ˚ O 2 t t0 D 00 s C s ; (F.46) jh .t0/j 00 00 valid when either h .t0/>0or h .t0/<0. With the Taylor series expansion 0 g.t/ D g.t0/ C g .t0/.t t0/ C ; (F.47) the integral in (F.42) becomes Z b f. / D g.t/ei h.t/dt a Z 1=2 s2 2 2 i h.t0/ ˙i s O 00 g.t0/e e Œ1 C fsg ds jh .t0/j s Z 1 Z 1=2 1 1 2 2 2 i h.t0/ ˙i s O 2 ˙i s 00 g.t0/e e ds C s e ds jh .t0/j 1 1 as !1, so that Z b 2 1=2 ˚ i h.t/ i h.t0/ ˙i =4 O 1 g.t/e dt 00 g.t0/e e C (F.48) a jh .t0/j 00 as !1, where C =4 corresponds to the case when h .t0/>0while =4 00 corresponds to h .t0/<0. The end point contributions to the asymptotic behavior of this integral are ob- tained through a straightforward integration by parts as Z ˇ Z b ˇb b i h.t/ g.t/ ˇ d g.t/ i h.t/ g.t/e dt D ˇ e dt i h0.t/ dt i h0.t/ a a a 1 g.b/ g.a/ ˚ ei h.b/ ei h.a/ C O 2 ; (F.49) i h0.b/ h0.a/ as !1. Combination of (F.48) and (F.49) then results in the asymptotic approx- imation Z b 2 1=2 i h.t/ i h.t0/ ˙i =4 g.t/e dt 00 g.t0/e e a jh .t0/j 1 g.b/ g.a/ ˚ C ei h.b/ ei h.a/ C O 2 i h0.b/ h0.a/ (F.50) as !1. As an example, consider the asymptotic behavior of the Fourier integral of a con- tinuous function f.t/ that identically vanishes outside of the finite domain Œ˛; ˇ ; 794 F Asymptotic Expansion of Single Integrals i.e., has compact support. Since h.t/ D t does not possess any stationary phase points, the asymptotic behavior is due solely to the endpoints. Repeated integration by parts N times then results in the expression Z ˇ NX 1 i nC1 ˚ f.t/e i!tdt D f .n/.˛/e i!˛ f .n/.ˇ/e i!ˇ C R .!/; ! N ˛ nD0 where Z N ˇ i .N / i!t RN .!/ f .t/e dt: ! ˛ By the Riemann–Lebesgue lemma [8], if f .N /.t/ is continuous in the interval Œ˛; ˇ , then the integral appearing in the remainder term RN .!/ tends to zero in the limit as N ! !1, so that RN .!/ D Of! g. The above result is then the asymptotic series as ! !1of the Fourier transform of the N th-order continuous function f.t/with compact support Œ˛; ˇ . F.5 Watson’s Lemma An important class of integrals that is amenable to asymptotic analysis is the class of Laplace integrals Z 1 f.x/D .t/e xtdt; (F.51) 0 where f .t/g is integrable over every finite interval Œ0; T . Watson’s lemma provides an asymptotic expansion for such Laplace transform integrals for the fairly wide class of functions of the form .t/ D t g.t/; (F.52) where g.t/ possesses a Taylor series expansion about t D 0 with g.0/ ¤ 0, and where is real-valued with > 1 in order to ensure convergence of the integral appearing in (F.51) at t D 0.Ifg.t/ possesses a zero of order r at t D 0, then the quantity t r is combined with the factor t to create a new function g.t/ that does not vanish at t D 0. With the method of analysis given by Murray [5] as a guide, consider the asymp- totic expansion of the function f.x/given by the Laplace integral representation Z T f.x/D t g.t/e xtdt (F.53) 0 F.5 Watson’s Lemma 795 for real-valued x>0as x !1and for any finite or infinite value of T>0, where g.t/ possesses a Taylor series expansion about t D 0 with g.0/ ¤ 0, and where > 1. In order that the integral appearing in (F.53) converges as T !1,itis required that the inequality jg.t/j X1 g.n/.0/ XN g.t/ D t n D a t n C r .t/; (F.54) nŠ n n nD0 nD0 .n/ where an D g .0/=nŠ, n D 0; 1; 2; : : : , and where N C1 jrn.t/j Consider the final change of variable D xT.1 C u/ in the last integral above, which then becomes Z Z 1 1 x . CN C2/ CN C1e d D T CN C2e xT e xT u.1 C u/ CN C1du: xT 0 Because .1 C u/a XN ˚ . CnC1/ . CN C2/ f.x/D an. C n C 1/x C O x nD0 F.5 Watson’s Lemma 797 so that Z T X1 . C n C 1/g.n/.0/ t g.t/e xtdt (F.62) .nC 1/x CnC1 0 nD0 as x !1. This result is known as Watson’s lemma (see Ex. 2 of [1]). Notice that all of the contributions to the above asymptotic expansion as x !1 arise from the region about the point t D 0 irrespective of the order of the zero of t g.t/. Also notice that the upper limit T does not appear in the asymptotic expansion given in (F.62). When T>R, separate the integral into the sum of two integrals, the first from 0 to T1 Since jg.t/j Under the change of variable t D T1.1Cu/, the integral appearing on the right-hand side of the above inequality becomes Z Z 1 1 .x c/t C1 .x c/T1 .x c/T1u t e dt D T1 e .1 C u/ e du T1 Z0 1 C1 .x c/T1 Œ.x c/T1 u X1 m 1 .t/ D amt r ; jtj Z 1 1 X m .t/e zt dt a z m=r (F.65) m r 0 mD1 as jzj!1in the sector jarg.z/j 2 < 2 . Proof. For any fixed positive integer M , there exists a constant C such that ˇ ˇ ˇ ˇ ˇ MX 1 ˇ ˇ X1 ˇ ˇ m ˇ ˇ m ˇ ˇ r 1ˇ ˇ r 1ˇ ˇ .t/ amt ˇ D ˇ amt ˇ mD1 mDM ˇ ˇ ˇX1 ˇ M ˇ m 1 ˇ r 1 ˇ r ˇ D t ˇ aM Cm 1t ˇ mD1 M 1 0 t r K j .t/j M 1 bt Ct r e ; for real t 0. Then Z Z 1 MX 1 1 zt m 1 zt .t/e dt D am t r e dt C RM 0 mD1 0 M 1 X m D a z m=r C R ; m r M mD1 where the remainder after M 1 terms is given by Z ( ) 1 X1 m 1 zt RM D amt r e dt 0 mDM Z ( ) 1 MX 1 m 1 zt D .t/ amt r e dt: 0 mD1 With x when x>b. Since jarg.z/j 2 , then x jzj sin , and so x>bwhen jzj >bcsc . Consequently, if jarg.z/ 2 and jzj >bcsc , then F.5 Watson’s Lemma 799 ˇ ˇ C jzjM=r M ˇzM=r R ˇ ; (F.66) M .jzj sin b/M=r r ˚ M=r which is bounded as jzj!1. Hence, RM D O z . ut As an extension of Watson’s lemma as expressed in (F.62), consider the asymp- totic behavior of the class of real-valued integrals of the form Z ˇ 2 f.x/D .t/e xt dt; (F.67) ˛ as x !1, where ˛ and ˇ are positive constants and where .t/ possesses the Taylor series expansion X1 n .t/ D ant ; jtj t D 1=2 I 0 t ˇ; t D 1=2 I ˛ t 0; so that Z 2 Z 2 1 ˇ 1 ˛ f.x/D 1=2 . 1=2/e x d C 1=2 . 1=2/e x d : (F.69) 2 0 2 0 Assuming for the moment that R>max.˛;ˇ/, application of Watson’s lemma to each of the integrals appearing in (F.68) gives ( ) Z 2 Z 2 1 X1 ˇ ˛ f.x/ D a .n 1/=2e x d C . 1/n .n 1/=2e x d 2 n nD0 0 0 Z X1 T n 1=2 x a2n e d ; (F.70) nD0 0 where T is any positive number. The integral appearing in this expression is given by Z Z Z T 1 1 .2n 1/=2e x d D .2n 1/=2e x d .2n 1/=2e x d 0 0 Z T 1 1 e xT D .2n 1/=2e d C O x.2nC1/=2 x 0 ..2n C 1/=2/ e xT D C O x.2nC1/=2 x 800 F Asymptotic Expansion of Single Integrals ˚ as x !1, where the O e xT =x term results from the second integral after re- peated integration by parts. Substitution of this result in (F.70) then gives Z ˇ X1 xt2 .2nC1/=2 .t/e dt a2n ..2n C 1/=2/x (F.71) ˛ nD0 p as x !1. Since .1=2/ D and ..2nC1/=2/ D ..2n 1/=2/ ..2n 1/=2/, then the asymptotic series in (F.71) may be expressed as Z ˇ 2 1=2 a 3a .t/e xt dt a C 2 C 4 C 0 2 (F.72) ˛ x 2x 4x as x !1. As before, all of the contributions to the asymptotic expansion arise from the neighborhood about the point t D 0. F.6 Laplace’s Method Laplace’s method [9] considers the asymptotic behavior of integrals of the type Z ˇ f.x/D g.t/exh.t/dt (F.73) ˛ as x !1, where x is real and positive, g.t/ is a real-valued, continuous function on the interval ˛ t ˇ, and where h.t/, together with its first two derivatives h0.t/ and h00.t/, are real-valued and continuous on the interval ˛ t ˇ with ˛ and ˇ both real. The essence of Laplace’s method is that the dominant contributions to the asymptotic behavior of the integral appearing in (F.73) as x !1arise from each neighborhood of the points in the interval ˛ t ˇ where h.t/ attains relative maxima. In general, the function h.t/ will possess several relative maxima in the interval Œ˛; ˇ , including possibly the points at either endpoint, as depicted in Fig. F.3. To accomodate this possibility, the integral in (F.73) is then separated into the sum of several integrals over each of the subintervals where h.t/ attains a single maximum value in each subinterval. For the example illustrated in Fig. F.3, this separation is given by Z Z Z t1 t2 ˇ f.x/D g.t/exh.t/dt C g.t/exh.t/dt C g.t/exh.t/dt; ˛ t1 t2 where the dominant term in the first integral arises from the right neighborhood of the endpoint at t D ˛, the dominant term in the second integral from the maximum at t D p1, and the dominant term in the third integral from the maximum at t D p2. F.6 Laplace’s Method 801 h(t) abt1 p1 t2 p2 t Fig. F.3 Example of a continuous function h.t/ with several relative maxima in the interval ˛ t ˇ. The left endpoint at t D ˛ is a relative maximum while the right endpoint at t D ˇ is not Consider the case when h.t/ has a single maximium at the interior point t D a, where ˛ g1. / g.a /; h1. / h.a /; g2. / g.a C /; h2. / h.a C /: Each integral appearing in the expression (F.74) is now of the form where the maxi- mum value of the exponential function in the integrand occurs at the lower endpoint D 0. Without any loss of generality, attention is now focused on the asymptotic be- havior of integrals of the form Z T f.x/D g.t/exh.t/dt; (F.75) 0 802 F Asymptotic Expansion of Single Integrals where h.0/ is the maximum value of h.t/ in the interval 0 t T with T>0. The function h.t/ may then either possess a genuine maximum at the origin with h0.0/ D 0 and h00.0/ < 0, or it may not, in which case h0.0/ < 0. These two cases must then be treated separately. Case 1: Let h0.0/ D 0, h00.0/ < 0, and h.0/ > h.t/ for all t 2 .0; T /. Assume that g.t/ and h00.t/ are both real continuous functions on 0 t T . Since h00.t/ is con- tinuous and h.0/ is the maximum value of h.t/ on this interval, then there exists a ı-neighborhood of the point t D 0 such that h00.t/ < 0 for 0 t ı 1 h.t/ h.0/ D h00. /t 2; 2 when 0 t ı, where h00. / < 0. Define the variable s by the relation h.t/ h.0/ s2; (F.76) so that 2 exh.t/ D exh.0/e xs ; (F.77) and the extension of Watson’s lemma given in (F.72) applies. Assume that g.t/ possesses the Taylor series expansion 1 g.t/ D g.0/ C g0.0/t C g00.0/t 2 C (F.78) 2 in a neighborhood of the origin that is valid for some finite radius of convergence. Furthermore, substitution of the Taylor series expansion of h.t/ about t D 0 in (F.76) results in the expression 1 1 h00.0/t 2 C h000.0/t 3 C D s2; 2 3Š so that 2 1=2 ˚ t D s C O s2 : (F.79) h00.0/ Substitution of this expression in (F.78) then gives 2 1=2 ˚ g.t/ D g.0/ C g0.0/ s C O s2 (F.80) h00.0/ F.6 Laplace’s Method 803 The change of variable from t to s is now made in (F.75) with the above substitu- tions as h i Z 00 1=2 1=2 h .0/ T 2 2 2 f.x/ exh.0/ e xs Œg.0/ C Ofsg ds h00.0/ 0 Z Z 1=2 1 1 2 2 2 g.0/ exh.0/ e xs ds C eh.0/xO se xs ds ; h00.0/ 0 0 1=2 ˚ g.0/ eh.0/x C eh.0/xO x 1 ; 2h00.0/x which then gives the result Z T 1=2 ˚ xh.t/ h.0/x h.0/xO 1 g.t/e dt g.0/ 00 e C e x ; (F.81) 0 2h .0/x as x !1. With the above result in hand, attention is now turned to the asymptotic behavior of the integral Z T2 f.x/D g.t/exh.t/dt; T1 where T1 and T2 are both positive numbers and where h.0/ is the maximum value 0 00 of h.t/ in the interval T1 t T2 with h .0/ D 0 and hR .0/ < 0. The preceding 1 xs2 1=2 derivationR then applies, leading to the definite integrals 1 e ds D . =x/ 1 xs2 and 1 se ds D 0. Because of the˚ vanishing of this second integral, the sec- ond term in˚ this result is˚ not eh.0/xO x 1 as˚ it is in (F.81). As a consequence, O 2 O 3 O 2 terms of s and s in (F.79) and s in (F.80) are required.R The first 1 2 xs2 nonzero contribution to this second term arises from the integral 1 s e ds D .1=2/. =x3/1=2, so that Z T2 2 1=2 ˚ xh.t/ h.0/x h.0/xO 3=2 g.t/e dt g.0/ 00 e C e x ; (F.82) T1 h .0/x as x !1. For the general integral given in (F.73), where the maximum value of the function h.t/ in the interval Œ˛; ˇ occurs at t D a, with ˛ Case 2: Let h.0/ > h.t/ for all t 2 .0; T / with h0.0/ < 0. Then as x !1, the dominant contribution to the integral Z T f.x/D g.t/exh.t/dt 0 comes from a ı-neighborhood of the point at t D 0. The mean value theorem then states that there exists a value with 0< <ı h.t/ h.0/ D h0. /t; when 0 t ı, where h0. / < 0. Consider then the change of variable h.t/ h.0/ D s (F.84) so that, from the Taylor series expansion of h.t/ about t D 0, one obtains 1 ˚ t D s C O s2 : (F.85) h0.0/ Under this change of variable, the above integral becomes Z 0 h .0/T 1 O xh.0/ xs f.x/ g.0/ C fsg e e 0 ds 0 Z Zh .0/ g.0/ 1 1 xh.0/ xs xh.0/O xs 0 e e ds C e se ds ; h .0/ 0 0 so that Z T g.0/ ˚ xh.t/ xh.0/ xh.0/O 2 g.t/e dt 0 e C e x (F.86) 0 h .0/x as x !1. For the general case where the maximum value of h.t/ in the interval ˛ t ˇ occurs at the lower endpoint t D ˛ with h0.˛/ < 0, the above result becomes Z ˇ g.˛/ ˚ xh.t/ xh.˛/ xh.˛/O 2 g.t/e dt 0 e C e x (F.87) ˛ h .˛/x as x !1. On the other hand, when the maximum value of h.t/ in the interval ˛ t ˇ occurs at the upper endpoint t D ˇ with h0.ˇ/ > 0, one obtains Z ˇ g.ˇ/ ˚ xh.t/ xh.ˇ/ xh.ˇ/O 2 g.t/e dt 0 e C e x (F.88) ˛ h .ˇ/x as x !1. F.7 The Method of Steepest Descents 805 As an illustration of Laplace’s method, consider obtaining an asymptotic approx- imation of the gamma function Z Z 1 1 .xC 1/ D t xe t dt D e tCx ln t dt 0 0 for real values of x as x !1. With the change of variable t D x , this integral becomes Z 1 .xC 1/ D xxC1 ex.ln /d ; 0 which is the same form as the integral in (F.74) with g. / D 1 and h. / D ln with h0.1/ D 0 and h00.1/ < 0. Direct application of the result given in (F.83) then yields the asymptotic approximation .xC 1/ xx.2 x/1=2e x (F.89) as x !1.Ifx D n is an integer, then .nC 1/ D nŠ and the above result yields Stirling’s formula p nŠ nn 2 ne n (F.90) as n !1. F.7 The Method of Steepest Descents Originated by B. Riemann [10] in 1876 and then fully developed by P. Debye [11] in 1909, the method of steepest descents provides a much needed generalization of Laplace’s method to integrals in the complex plane. The method is applicable to the specific class of integrals of the form Z f. /D g.z/e h.z/dz; (F.91) C where C is some piecewise continuous contour in the complex z-plane, g.z/ and h.z/ are both analytic functions of the complex variable z in some domain D which contains the contour C , both independent of with a real positive number. Since h.z/ D .x;y/ C i .x;y/ is analytic in some domain D, it then fol- lows from the Cauchy–Riemann conditions that h.z/ cannot have either maxima or minima in that domain, only saddle points. Assume that the point z0 D x0 C iy0 is a relative maximum of .x;y/ @ .x; y/ @ .x; y/ @ .x; y/ @ .x; y/ D ; D (F.92) @x @y @y @x 806 F Asymptotic Expansion of Single Integrals are satisfied at z0, then r D 0 at that point also. Consequently @ .x; y/ @ .x; y/ @ .x; y/ @ .x; y/ h0.z/ D C i D C i D 0 (F.93) @x @x @y @y at z D z0. Since .x;y/ and .x; y/ are then both potential functions, each sat- isfying Laplace’s equation r2 D 0, r2 D 0, then by the maximum modulus theorem, both .x;y/ and .x; y/ cannot have either a maximum or a minimum in the domain of analyticity D of h.z/. The point z0 is then a saddle point of both .x;y/ and .x; y/, and hence of h.z/. A first-order saddle point (a saddle point of order one) of h.z/ at the point z D z0 satisfies the relations 0 00 h .z0/ D 0; h .z0/ ¤ 0; (F.94) while an nth-order saddle point (a saddle point of order n) at the point z D z0 satisfies the relations .1/ .2/ .n/ .nC1/ h .z0/ D h .z0/ D Dh .z0/ D 0; h .z0/ ¤ 0: (F.95) Since @ @ @ @ r r D C D 0 (F.96) @x @x @y @y by virtue of the Cauchy–Riemann conditions, then the family of isotimic1 contours .x;y/ D constant are everywhere orthogonal to the family of isotimic contours .x; y/ D constant in the domain of analyticity D of h.z/. The contour lines along the direction of r , where .x;y/changes most rapidly, are then the contours D constant. A path of steepest descent through the saddle point z0 D x0 C iy0 is then defined by the contour .x; y/ D .x0;y0/. Two comments regarding the path of steepest descent are in order here. First, .x; y/ =fh.z/g generally produces an oscillatory contribution ei .x;y/ in the integrand of (F.91) with oscillation frequency that increases with the asymptotic parameter ; however, this oscillation identically vanishes along any path of steepest descent through the saddle point, reinforcing the central importance of this specific path of integration in the asymptotic expansion procedure for contour integrals of the form given in (F.91). Second, because of Cauchy’s theorem, the original contour of integration C appearing in (F.91) can always be deformed into the path of steepest descent through a given saddle point provided that the saddle point is in the domain of analyticity D of the function h.z/, and provided that appropriate care is given to any singularities of the function g.z/ that may be crossed in that deformation. In a neighborhood of an isolated first-order saddle point at z D z0, z0 2 D,the function h.z/ possesses the Taylor series expansion 1 From the Greek: iso (of equal) timos (worth). F.7 The Method of Steepest Descents 807 1 ˚ h.z/ D h.z / C h00.z /.z z /2 C O .z z /3 : (F.97) 0 2 0 0 0 With the identifications 00 i˛ 00 h .z0/ ae ;aDjh .z0/j >0; (F.98) i z z0 re ;rDjz z0j 0; (F.99) the above expansion becomes 1 ˚ .x;y/ C i .x;y/ D C i C ar2ei.2 C˛/ C O r3 ; (F.100) 0 0 2 where 0 .x0;y0/ and 0 .x0;y0/. Upon equating real and imaginary parts, one obtains the pair of expressions 1 ˚ .x;y/ D C ar2 cos .2 C ˛/ C O r3 ; (F.101) 0 2 1 ˚ .x; y/ D C ar2 sin .2 C ˛/ C O r3 : (F.102) 0 2 There are then two isotimic contours .x;y/ D 0 which, for sufficiently small radial distances r from the saddle point z0, are tangent to the two orthogonal lines that are given by the solutions of cos .2 C ˛/ D 0, so that 1 1 D ˛ and its continuation D C ˛ ; (F.103) 2 2 2 2 1 1 D C ˛ and its continuation D C ˛ : (F.104) 2 2 2 2 The local valley regions below the saddle point where .x;y/ < 0 are then given by ˛ 3 ˛ 5 ˛ 7 ˛ < < ; & < < : (F.105) 4 2 4 2 4 2 4 2 The two isotimic contours .x; y/ D 0 along which .x;y/ changes most rapidly as one moves away from the saddle point z0, i.e., the paths of steepest descent and ascent from z0, are tangent at z D z0 to the two orthogonal lines that are given by the solutions of sin .2 C ˛/ D 0, so that ˛ ˛ D and its continuation D ; (F.106) 2 2 ˛ 3 ˛ D and its continuation D : (F.107) 2 2 2 2 808 F Asymptotic Expansion of Single Integrals Fig. F.4 Local geometric (x,y) structure in the complex 0 z-plane of .x;y/ (x,y) 0 (x,y) 0 (x,y) 0 The geometric structure of this local behavior about an isolated first-order saddle point is illustrated in Fig. F.4. With the behavior about the saddle point determined, the next step in the method is to deform the contour of integration C , assuming that each of the original end- points lie on opposite sides of the saddle point in the region satisfying .x;y/ < 0, so that it lies along the path of steepest descent through the saddle point, as depicted by the contour .x; y/ D 0 in the shaded region of Fig. F.4. Along this steepest descent path .z/ 0 D h.z/ h.z0/ 1 D h00.z /.z z /2; with h00.z /<0; (F.108) 2 0 0 0 which is real-valued since the imaginary parts of h.z/ and h.z0/ cancel along the path of steepest descent. From Laplace’s method, define the real variable by the relation 2 h.z/ h.z0/ (F.109) when z is along the steepest descent path through the saddle point, which then deter- mines z D z. / as a function of . Under this change of variable the contour integral appearing in (F.91) becomes Z f. / D g.z/e h.z/dz C Z b 2 dz D e h.z/ g.z. //e d ; (F.110) a d F.7 The Method of Steepest Descents 809 where a >0and b >0correspond to the endpoints of the deformed contour through the saddle point under the coordinate transformation given in (F.109). Con- sequently, Z 1 2 dz f. / e h.z/ g.z. //e d (F.111) 1 d as !1. Upon expanding the left-hand side of (F.109) in a Taylor series about the saddle point at z D z0, one obtains 1 ˚ h00.z /.z z /2 C O .z z /3 D 2; 2 0 0 0 so that 2 1=2 z z0 D 00 : (F.112) h .z0/ 00 i˛ Since h .z0/ D ae is complex-valued, one must choose the appropriate branch of 00 1=2 the quantity Œ 1=h .z0/ when z lies along the path of steepest descent through that saddle point. This branch choice is determined by the sense of direction that the path makes through the branch point at z D z0. Suppose that when the original contour C is deformed to lie along the steepest descent path through z0, it progresses from the region where .x;y/ < 0 with 3 ˛ ˛ arg.z z0/ D 2 2 to the region with arg.z z0/ D 2 2 , as depicted in 00 Fig. F.4, where ˛ argfh .z0/g.Theappropriate choice in (F.112) must then be the 00 one such that argf 1=h .z0/g gives >0when z is in the final region arg.z z0/ D ˛ 2 2 , so that ( ) 1=2 ˇ ˇ 1 ˛ 00 ˇ 00 ˇ i˛ arg 00 D H) h .z0/ D h .z0/ e ; h .z0/ 2 2 and 1=2 ˚ 2 i ˛ 2 z z D e . 2 2 / C O 0 jh00.z /j 0 2 1=2 ˚ O 2 D i 00 C : (F.113) h .z0/ If the direction of integration were reversed, then the above result would be replaced by the expression 2 1=2 ˚ O 2 z z0 D i 00 C I h .z0/ however, it is advised that each case be treated individually. 810 F Asymptotic Expansion of Single Integrals The asymptotic expansion of the integral in (F.111) is now completed with sub- stitution of the Taylor series expansion 0 1 00 2 g.z. // D g.z0/ C g .z0/.z z0/ C g .z0/.z z0/ C 2 2 1=2 ˚ 0 O 2 D g.z0/ C g .z0/ 00 C ; (F.114) h .z0/ so that Z 1=2 1 2 2 h.z0/ f. / g.z0/ 00 e e d C ; h .z0/ 1 R p 1 2 as !1. Since 1 e d D = , one finally obtains the general result 2 1=2 1 h.z0/ h.z0/O f. / g.z0/ 00 e C e ; (F.115) h .z0/ 00 1=2 as !1, where the specific branch of the quantity Œ 1=h .z0/ must be chosen so as to be consistent with the direction of the deformed contour of integration through the saddle point at z D z0. For the cases given in (F.113), the above result becomes r 2 g.z0/ f. / p e h.z0/ei. ˛/=2; 00 jh .z0/j as !1. References 1. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. New York: MacMillan, 1943. Chap. VIII. 2. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions,vol.55ofAp- plied Mathematics Series. Washington, D.C.: National Bureau of Standards, 1964. 3. E. T. Copson, Asymptotic Expansions. London: Cambridge University Press, 1965. 4. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Dover, 1975. 5. J. D. Murray, Asymptotic Analysis. New York: Springer-Verlag, 1984. 6. G. G. Stokes, “On the discontinuity of arbitrary constants which appear in divergent develop- ments,” Trans. Camb. Phil. Soc., vol. X, pp. 106–128, 1857. 7. L. Kelvin, “On the waves produced by a single impulse in water of any depth, or in a dispersive medium,” Proc. Roy. Soc., vol. XLII, p. 80, 1887. 8. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. New York: MacMillan, 1943. Sect. 9.41. 9. P. S. de Laplace, Theorie´ Analytique des Probabilities´ . Paris: V. Courcier, 1820. 10. B. Riemann, Gesammelte Mathematische Werke. Leipzig: Teubner, 1876. 11. P. Debye, “Naherungsformeln¨ fur¨ die zylinderfunktionen fur¨ grosse werte des arguments und unbeschrankt¨ verander liche werte des index,” Math. Ann., vol. 67, pp. 535–558, 1909. Appendix G Proof of Theorem 1 A proof of Theorem 1 in Sect. 9.3.1, due to Sherman, Stamnes, and Lalor [1], which states that Q if UQ .p;q;!/2 TN for some positive even integer N , then for z >0, the spectral wave field UQ .r;!/D UQ .x; y; z;!/given in (9.194)–(9.195) with k a positive real constant satisfies UQ .x; y; z;!/D UQ 0.x; y; z/ C R.x; y; z/; (G.1) where Z Q ik.pxCqyCmz/ UQ 0.x; y; z/ D .p;q/UQ .p;q;!/e dpd q; (G.2) D ˚ H R O N and where .x; y; z/ D q.kR/ as kR !1uniformly with respect to 1 and 2 for 2 2 all real 1; 2 such that ı< 1 1 2 1 for any positive constant ı<1, is given here based upon a modification of the proof of Theorem 1 by Chako [2] or Theorem 3 by Focke [3]. Proof. Construct three neutralizer functions j .p; q/, j D 1; 2; 3, each of which is a real continuous function of p and q with continuous partial derivatives of all orders for all p; q and which satisfy 0 j .p; q/ 1 with 1.p; q/ C 2.p; q/ C 3.p; q/ D 1: Choose positive constants C1;C2;C3;C4 such that q 2 2 1 C 2 811 812 G Proof of Theorem 1 q D D 2 1 D3 p C1 C2 1 C3 C4 Fig. G.1 Graphical illustrationp of the notation used in the proof of Theorem 1. The shaded area indicates the region wherep m D 1 p2 q2 is real-valued, and the unshaded area indicates the region where m D i p2 C q2 1 is pure imaginary With these four constants specified, the neutralizer functions j .p; q/, j D 1; 2; 3 are now required to satisfy the conditions p 1; for p2 C q2 C .p; q/ D p 1 ; 1 0; for p2 C q2 C p2 1; for C p2 C q2 C .p; q/ D p 2 p 3 ; 2 0; for p2 C q2 C & p2 C q2 C p 1 4 1; for p2 C q2 C .p; q/ D p 4 : 3 2 2 0; for p C q C3 G Proof of Theorem 1 813 Finally, three overlapping regions D1;D2;D3 in the p; q-plane are defined as n ˇ p o ˇ 2 2 D1 .p; q/ p C q C2 C " ; n ˇ p o ˇ 2 2 D2 .p; q/ C1 " p C q C4 C " ; n ˇ p o ˇ 2 2 D3 .p; q/ C3 " p C q ; where ">0is a positive constant that satisfies the three inequalities "<1 C2; " Each of these three regions is illustrated in Fig. G.1. Features of note that are impor- tant for this proof are the following: Each of the neutralizer functions j .p; q/, j D 1; 2; 3, vanishes both for points .p; q/ … Dj and for points .p; q/ in some neighborhood of the boundary @Dj of Dj The stationary phase point .ps;qs/ lies within the interior of D1 and is exterior to both regions D2 and D3 p The direction cosine m DC1 p2 q2 is real-valued for all points .p; q/ 2 D1 p 2 2 And m DCi p C q 1 is pure imaginary for all points .p; q/ 2 D3. Separate the angular spectrum integral for UQ .r;!/ that is given in (9.194) and (9.195) into three parts as UQ .r;!/D UQ 1.r;!/C UQ 2.r;!/C UQ 3.r;!/; where Z Q ik.pxCqyCmz/ UQ j .r;!/ j .p; q/UQ .p;q;!/e dpd q Dj for j D 1; 2; 3. Each of these integrals is now separately treated. Consider first the asymptotic approximation of the integral representation of UQ 1.r;!/which is in a form that is appropriate for the method of stationary phase. Since the stationary phase point .ps;qs/ is the only critical point of significance in the domain D1, and since the critical points of the phase function k r D k.px C qy C mz/ on the boundary @D1 of D1 do not contribute to the integral because of the neutralizer function 1.p; q/, and since the region ˝2 which contains 814 G Proof of Theorem 1 the neighborhood ˝1 of the stationary phase point .ps;qs/ lies entirely within D1, then it follows from the proof of Theorem 1 in Chako [2] that UQ 1.r;!/D UQ 0.x; y; z/ C R.x; y; z/; ˚ R O N where .x; y; z/ D .kR/ as kR !1with fixedq k>0, uniformly with 2 2 respect to 1; 2 for all real 1 and 2 such that ı< 1 1 2 1 for any positive constant ı with 0<ı<1. Consider next the asymptotic approximationp of the integral representation of Q 2 2 U3.r;!/p. In this case, the quantity m DCi p C q 1 is pure imaginary with 2 im p .C3 "/ 1<0for all .p; q/ 2 D3. For a sufficiently large value of 2 2 2 R D .x x0/ C .y y0/ C .z z0/ , a positive constant a then exists such that z >a. In that case Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ Q ˇ ikmz UQ 3.r;!/ ˇUQ .p;q;!/ˇ e dpd q D3 p Z ˇ ˇ 2 ˇ ˇ e k.z a/ .C3 "/ 1 ˇUQ .p;q;!/ˇ eikmadpd q: D3 Q Since UQ .p;q;!/ 2 TN , it then follows from the first condition [see (9.204)] in the definition of TN that the final integral appearing in the above inequality converges to a finite, nonnegative constant M3 that is independent of the direction cosines 1; 2; 3. Since z D z0 C 3R [see (9.201)] and since 3 >0, then the above in- equality becomes ˇ ˇ p 2 ˇ ˇ kŒ.z0 a/CRı .C3 "/ 1 UQ 3.r;!/ M3e ; ˇ ˇ ˚ ˇ ˇ N independent of 1; 2. Hence, UQ 3.r;!/ D O .kR/ uniformly with respect to 1; 2 as kR !1for all positive even integer values N . Consider finally the asymptotic approximation of the integral representation of UQ 2.r;!/, which turns out to be much more involved than the previous two cases. The analysis begins with the change of variables p D sin ˛ cos ˇ; q D sin ˛ sin ˇ; with Jacobian J .p; q=˛; ˇ/ D sin ˛ cos ˇ. In addition, 1=2 m D 1 p2 q2 D cos ˛; where ˛ is, in general, complex as the region D2 extends from inside the inner region where m is real-valued into the outer region where m is pure imaginary, as illustrated in Fig. G.1. In addition, let the angles #; ' describe the intersection point G Proof of Theorem 1 815 on the unit sphere with center at the origin r D 0 in coordinate space with the line through the origin that is parallel to the direction of observation r D .x; y; z/ as it recedes to infinity through the fixed point r0 D .x0;y0; z0/ with z >0, where 1 D sin # cos '; 2 D sin # sin '; 3 D cos #; with 0 #< =2and 0 '<2 . Under these two changes of variables, the integral for UQ 2.r;!/becomes Z Z 2 ikRŒsin # sin ˛ cos .ˇ '/Ccos # cos ˛ UQ 2.r;!/D A.˛; ˇ/e d˛dˇ; 0 C where ik.px0Cqy0Cmz0/ A.˛; ˇ/ 2.p; q/V .p; q; m/e sin ˛; with V.p;q;m/ defined in (9.204). The contour of integration C in the complex p 0 ˛-plane, which extends from ˛1 D arcsin C1 " along the real ˛ -axis to =2 1 p and then to the endpoint at ˛4 D =2 i cosh C4 C " , depicted in Fig. G.2, then results in a complete, single covering of the original integration domain D2. Notice that, in contrast with the original form of the integral, the phase ˚ sin # sin ˛ cos .ˇ '/ C cos # cos ˛ '' ' C Fig. G.2 Contour of integra- tion C for the ˛-integral in UQ 2.r;!/ 816 G Proof of Theorem 1 appearing in the above transformed integral for UQ 2.r;!/is analytic and the ampli- tude function A.˛; ˇ/ is continuous with continuous partial derivatives with respect to ˛ and ˇ up to order N over the entire integration domain D2. However, the con- tour of integration C is now complex, as illustrated in Fig. G.2, and this results in the 0 00 phase ˚ being complex-valued when ˛ D ˛ C i˛ varies from =2 to ˛4 along C . As ˛ varies over the contour C with the angle ˇ held fixed, the phase function ˚ varies over a simple curve C.ˇ/ of finite length in the complex ˛-plane. Since the partial derivative @˚ D sin # cos ˛ cos .ˇ '/ cos # sin ˛ @˛ ˇ is an entire function of complex ˛ for all ˇ, then " # 1 @˛ @˚ D @˚ ˇ @˛ ˇ is an analytic function of complex ˛ for all ˛; ˇ provided that @˚ ¤ 0. Since @˛ ˇ cos # D >ı, then @˚ ¤ 0 when cos ˛ D 0. It then follows that the zeros of 3 @˛ ˇ @˚ @˛ ˇ occur for those values of ˛ that satisfy the relation tan ˛ D tan # cos .ˇ '/: On the portion of the contour C over which ˛ is real, ˛1 ˛ =2, and since cos # D 3 C1 " (see Fig. G.1), then # ˛ =2 and one obtains the inequality tan ˛>tan # cos .ˇ '/. On the portion of the contour C over which ˛ is complex, As a consequence of the prescribed properties for both the function V.p;q;m/D Q mUQ .p;q;!/[cf. (9.204)] and the neutralizer function 2.p; q/, the function A.˛; ˇ/ possesses N continuous partial derivatives with respect to the variable ˛ 2 C for all G Proof of Theorem 1 817 fixed ˇ 2 Œ0; 2 /.1 Because the quantity @˛ is an analytic function of ˛ 2 C ,the @˚ ˇ @˛ product A.˛; ˇ/ @˚ ˇ also possesses N continuous partial derivatives taken with respect to ˛ along the contour C with fixed ˇ 2 Œ0; 2 /. Notice that differentiation with respect to ˚ along the contour C.ˇ/with fixed ˇ is equivalent to differentiation with respect to ˛ along C with fixed ˇ followed by multiplication by the quantity @˛ @˛ @˛ @˚ ˇ. Because @˚ ˇ is analytic with respect to ˛, the quantity A.˛; ˇ/ @˚ ˇ has N continuous partial derivatives with respect to ˚ 2 C.ˇ/ for all fixed ˇ 2 Œ0; 2 /. The ˚-integral Z @˛ I.kR;ˇ/ D A ˛.˚/;ˇ eikR˚d˚ C.ˇ/ @˚ ˇ appearing in the integral expression for UQ 2.r;!/is now integrated by parts N times by integrating the exponential factor eikR˚ each time and differentiating the remain- ing factor, with the result [4] I.kR;ˇ/ D LN .˚4/ LN .˚1/ C RN .kR; ˇ/; where ( " #) ˇ NX 1 ˇ @n @˛ ˇ eikR˚j L .˚ / D i n 1 A ˛.˚/;ˇ ˇ ; N j n ˇ nC1 @˚ @˚ ˇ .kR/ nD0 ˇ ˚D˚j with ˚j D sin # sin ˛j cos .ˇ '/ C cos # cos ˛j for j D 1; 4, where ˛1 and ˛4 denote the endpoints of the contour C (see Fig. G.2), and where Z ( " #) @N @˛ R .kR; ˇ/ D . ikR/ N A ˛.˚/;ˇ eikR˚d˚: N N C.ˇ/ @˚ @˚ ˇ ˇ As a consequence of the prescribed properties for the neutralizer function 2.p; q/, @nA.˛;ˇ/ the function A.˛; ˇ/ and all N of its partial derivatives @˛n , n D 0; 1; : : : ; N 1, taken along the contour C , vanish at the endpoints ˚1 and ˚4 of the contour C . Each @nA.˛;ˇ/ of the N partial derivatives @˚n taken along the contour C.ˇ/ then also vanish at the endpoints ˚1 and ˚4, so that LN .˚1/ D LN .˚4/ D 0: Furthermore, because the integrand in the above expression for the remainder term RN .kR; ˇ/ is continuous along the contour C.ˇ/ and as this contour is of finite 1 Notice that the derivatives with respect to ˛ must be taken along the contour C since both V.p;q;m/ and 2.p; q/ are defined only for real-valued p and q. That is, variation of ˛ along the contour C corresponds to p; q varying along the real axis. 818 G Proof of Theorem 1 length for all #, ', and ˇ, then that integral is bounded by some positive constant M2 independent of #, ', and ˇ, so that N jRN .kR; ˇ/j M2.kR/ : Consequently, ˇ ˇ ˇ ˇ N UQ 2.r;!/ 2 M2.kR/ ; ˚ N so that UQ 2.r;!/ D O .kR/ uniformly with respect to 1; 2 as kR !1with fixed k>0. In summary, following the method of proof given by Sherman, Stamnes, and Lalor [1], it has been established that the three terms UQ j .r;!/, j D 1; 2; 3, whose sum gives UQ .r;!/, satisfy the order relations ˚ UQ .r;!/ D UQ .x; y; z/ C O .kR/ N ; 1 0˚ UQ .r;!/ D O .kR/ N ; 2 ˚ N UQ 3.r;!/ D O .kR/ ; uniformly with respect to 1; 2 as kR !1with fixed k>0. This then completes the proof of the theorem. References 1. G. C. Sherman, J. J. Stamnes, and E.´ Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys., vol. 17, no. 5, pp. 760–776, 1976. 2. N. Chako, “Asymptotic expansions of double and multiple integrals occurring in diffraction theory,” J. Inst. Math. Appl., vol. 1, no. 4, pp. 372–422, 1965. 3. J. Focke, “Asymptotische Entwicklungen mittels der Methode der stationaren¨ phase,” Ber. Verh. Saechs. Akad. Wiss. Leipzig, vol. 101, no. 3, pp. 1–48, 1954. 4. A. Erdelyi,´ “Asymptotic representations of Fourier integrals and the method of stationary phase,” SIAM J. Appl. Math., vol. 3, pp. 17–27, 1955. Appendix H The Radon Transform The Radon transform, named after the Czech mathematician Johann Radon (1887– 1956) has its origin in his 1919 paper [1] (which then led to the Radon-Nikodym´ theorem). For the purpose of this brief development, it is best introduced in connec- tion with the projection-slice theorem in Fourier analysis in the following manner. Consider a two-dimensional scalar (object) function O.r/ D O.x;y/, which may be expressed as Z O.R/ D O.r/ı.R r/d 2r; (H.1) D where D is any region containing the point R, and where ı.r/ is the two-dimensional Dirac delta function which has the Fourier integral representation Z 1 1 ı.R r/ D eik .R r/d 2k: 2 (H.2) 4 1 Let nO be the unit vector in the dirction of the vector k so that k D knO. With this substitution, (H.2) becomes Z 1 1 ı.R r/ D d 2keik Re ikr On; 2 4 1 and since Z 1 e ikr On D dx ı.x r nO/e ikx; 1 one then obtains the expression Z Z 1 1 1 ı.R r/ D d 2keik R dx ı.x r nO/e ikx: 2 (H.3) 4 1 1 Substitution of this result into (H.1) then gives Z Z 1 1 1 O.R/ D d 2keik R dx P.nOI x/e ikx; 2 (H.4) 4 1 1 819 820 H The Radon Transform where Z P.nOI x/ O.r/ı.x r nO/d 2r (H.5) D denotes the projection of the object function O.r/ onto the direction defined by the unit vector nO. The variable x appearing in this expression is now interpreted as the real variable defining the location in the nO-direction at which the rectilinear line integral through the object function O.r/ is taken. As an illustration, the vertical line integral through the object function depicted in Fig. H.1 is given by Z Z O.x;y0/dy0 D O.x0;y0/ı.x x0/dx0dy0 ` ZD D O.r/ı.x r nO/d 2r; D which is precisely the expression defined in (H.5). The two-dimensional Radon transform of a (sufficiently well behaved) function O.r/ D O.x;y/ is then defined as [2] Z RŒO .nOI x/ O.r/ı.x r nO/d 2r: (H.6) D y' x O(r) v n r x' Fig. H.1 Geometry of the vertical line integral through the object function O.r/ H The Radon Transform 821 With use of the notation (see Appendix C in Vol. 1) Z 1 ikx Fk ff.x/g D f.x/e dx; (H.7) 1 n o Z 1 1 F 1 Q Q ikx x f.k/ D f.k/e dk; (H.8) 2 1 for the Fourier transform and its inverse, respectively, where x and k are referred to as Fourier conjugate variables, then as a special case of the Fourier integral theorem F 1 F [3], x ı k D 1, that is, the subsequent application of the forward and inverse Fourier transforms (on a sufficiently well-behaved class of functions) results in the identity operator. (H.4) may then be expressed as F 1 F O.R/ D k ı k fP.nOI x/g ; (H.9) so that Fk fO.r/g D Fk fP.nOI x/g : (H.10) This important result is known as the projection slice theorem (or central slice theorem) [4]. In its N -dimensional generalization, this theorem relates the .N 1/- dimensional Fourier transforms of the projections of an object function onto hy- perplanes to the N -dimensional Fourier transform of that original function. Simply stated, this theorem states that the .N 1/-dimensional Fourier transform of a given projection of the function is equal to a “slice” through the N -dimensional Fourier transform of that function. The Fourier integral representation of the delta function given in (H.2) involves a vector k which may be expressed in polar coordinate form as being the vector with magnitude k 2 Œ0; 1/ and with direction specified by a unit vector nO which is at an angle 2 Œ0; 2 / with respect to some fixed reference direction; one then has that Z Z Z 1 2 1 d 2k D d kdk: 1 0 0 However, it is equally valid to regard the integration variable in (H.2) as being a vector with “magnitude” 2 . 1; 1/ and direction specified by the unit vector NO which is at an angle 2 Œ0; / with respect to the reference direction; in this case one has that Z Z Z 1 1 d 2k D d j jd ; 1 0 1 where the absolute value of must be employed since the differential element of area d 2k must always be nonnegative. Just as k D knO, one also has that D NO . There is then a one-to-one correspondence between the points in k-space and the points in -space. In addition, NO may be treated as a conventional unit vector with 822 H The Radon Transform its range of directions (with respect to some fixed reference direction) appropriately restricted. The identity appearing in (H.3) may then be expressed in -space as Z Z Z 1 1 1 O ı.R r/ D d d dx j jei.R N x/ı.x R NO /: 2 (H.11) 4 0 1 1 Substitution of this expression into (H.1) and using the definition of the projection of the object function onto the direction specified by the unit vector NO given in (H.5) then results in the representation of the object function as Z Z 1 n o 1 O O.R/ D d d j jei R N F P.NO I x/ : 2 k (H.12) 4 0 1 This result then yields the well-known filtered backprojection algorithm for object reconstruction [5]. It is said to be “filtered” because of the quantity j j appearing in the integrand of (H.12). The inverse of the two-dimensional Radon transformation given in (H.6) may be obtained in the following manner [2]. Since jkjDk sgn.k/, where sgn.k/ DC1 if k>0,sgn.k/ D 0 if k D 0, and sgn.k/ D 1 if k<0is the signum function , then the representation of the delta function given in (H.11) may be rewritten as Z Z Z 1 1 1 O ı.R r/ D d d dx . /ei.R N x/ı.x R NO /: 2 sgn (H.13) 4 0 1 1 Since [see (B.14) of Vol. 1] Z Z f.x/ı0.x a/dx D f 0.x/ı.x a/dx; where the prime denotes differentiation with respect to the variable x, then with f 0.x/ D sgn. /ei.R NO x/ so that f.x/ D i sgn. /ei.R NO x/, the final integral in (H.13) becomes Z O sgn. /ei.R N x/ı.x R NO /dx Z O D i sgn. /ı0.x R NO /ei.R N x/dx; and hence Z Z Z 1 1 i O ı.R r/ D d d dx . /ı0.x R NO /ei.R N x/: 2 sgn (H.14) 4 0 1 1 References 823 With the identity [2] Z Z Z f.x/ i 1 P dx D dx dksgn.k/eikxf.x/; (H.15) x 2 1 where P indicates that the Cauchy principal value of the integral is to be taken, (H.14) becomes Z Z 2 1 ı0.x R NO / ı.R r/ D P d dx : 2 (H.16) 0 1 R NO x Substitution of this expression in (H.1) then yields Z Z 1 1 P 0.NO I x/ O.r/ D P d dx ; 2 (H.17) 2 0 1 R NO x which is known as the inverse Radon transform relationship, where Z P 0.NO I x/ D O.r/ı0.x R NO /d 2r (H.18) D is the derivative of the projection with respect to the variable x. References 1. J. Radon, “Uber¨ lineare Funktionaltransformationen und Funktionalgleichungen,” Acad. Wiss. Wien, vol. 128, pp. 1083–1121, 1919. 2. H. H. Barrett, “The Radon transform and its applications,” in Progress in Optics (E. Wolf, ed.), vol. XXI, pp. 217–286, Amsterdam: North-Holland, 1984. 3. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals. London: Oxford University Press, 1939. Ch. I. 4. R. M. Mersereau and A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE, vol. 62, pp. 1319–1338, 1974. 5. A. C. Kak, “Computerized tomography with X-ray, emission, and ultrasound sources,” Proc. IEEE, vol. 67, pp. 1245–1272, 1979. Index Abel’s lament, 779, 780, 785 attenuation coefficient ˛.!/, 469 Abel, N. H., 779 attenuation coefficient ˛.Q !/Q , nonoscillatory absorption depth zd ,23 waves, 693 Airy function Ai . /, 118 American National Standards Institute (ANSI), 767 C backprojection, 86 analytic delta function ı .t/, 236 bandwidth analytic field, 232 energy, 163 definition, 232 fractional, 162 plane wave spectrum representation, 233 basal metabolic rate (BMR), 767 angular spectrum of plane waves representa- Bertilone, D. C., 14, 17 tion, 4, 32 Bessel functions J . /, 109 angular spectrum representation Born, M. and Wolf, E., 150, 180 freely-propagating wavefield, 9 branch points radiated wavefield, 32 Lorentz model anomalous dispersion, 267 double resonance, 280 ANSI Standard C95.1, 767 single resonance, 265 antenna pattern Rocard–Powles–Debye model, 294 IEEE definition, 49 Brewster’s angle, 742 anterior pre-signal velocity, 517 Brillouin precursor, 211, 384–386, 656 m algebraic decay, 596 associated Legendre polynomial P` .u/,33 asymptotic asymptotic approximation, 420, 428 approximation, 101, 784 asymptotic expansion, 418, 424, 425 expansion, 784 instantaneous oscillation frequency, 437 dominant term, 101, 784 uniform asymptotic approximation, 433, 436 power series, 785 Brillouin precursor, Debye-type dielectric sequence, 783 asymptotic approximation, 446 asymptotic expansions effective oscillation frequency, 448, 595 integration by parts, 790 temporal width, 448, 592 linear superposition of, 786 Brillouin precursor, dispersive transmission method of stationary phase, 792 line, 572 method of steepest descent, 805 Brillouin precursor, Drude model conductor, sense of Poincare,´ 626 562 asymptotic power series Brillouin precursor, gaussian, 624 derivative of, 789 Brillouin pulse, 639, 751 integration of, 788 coded sequence of, 715 product of, 787 Brillouin, L., 104 825 826 Index Cauchy-Riemann conditions, 805 MB, 292 causality, 155 SM , 292 relativistic, 391 SB, 278, 316, 339, 341 N centroid group delay Gr , 644 0, 361 centrovelocity, Poynting vector N1, 286, 361 average, 643 critical angle c , 740 instantaneous, 650 cut-off frequency !co, 309, 523 chemical reaction rate changes, 769 Clausius-Mossotti relation, 179 cold plasma transient response Dawson’s integral, 132, 137 Dvorak-Dudley representation, 766 Debye model dielectric compact temporal support, 174 asymptotic field behavior in, 385 complementary error function erfc. /, 131 effective, transmission line, 570 complementary incomplete Lipschitz-Hankel Debye, P., 805 deformable contour of integration, 104 integral J en.a; /, 765 complex analytic signal, 181 delta function pulse, 160, 454 Brillouin precursor, 439 complex envelope, 184 Sommerfeld precursor, 411 complex half-range function, 181 detectability of radar, 714 complex index of refraction, 6 detection of relocatable targets, 714 complex intrinsic impedance .!/,38 Devaney, A. J., 17, 28 complex permittivity, 5 Devaney-Wolf representation, 31 complex permittivity .!/,29 c dielectric complex phase function Debye-type, 256, 258, 261 retarded, 161 Lorentz-type, 256, 261, 262 complex phase function .!; /, 153, 251 permittivity .!/,29 complex phase function, modified, 626 response function, 28 complex wavenumber, 6 transition-type, 256, 259 conductivity dielectric permittivity electric .!/, 29, 307 mth-order moment, 191 response function, 28 fractional mixing model, 756 static , 307 0 relative, 253 configuration domain, 231 response function, 191 constitutive relations dispersion simple dispersive medium, 28 anomalous, 267 convolution normal, 267 temporal, 87 dispersion relation, 240, 253 cophasal surfaces, 733 linear, 195 Cornu spiral, 133 dispersion surface, 240 counter-stealth, 714 dispersive wave equation ! critical values nonlinear, 223 !MB, 521 distant saddle points, 262, 277, 285 !SM, 522 first approximation, 319, 354 !SB, 513 second approximation, 324, 354, 368 !SB, 341 dominant saddle point, 337 !co, 309, 523 double exponential pulse, 163, 761 critical values Drude model metal, 306 0, 255, 277, 285, 301, 304, 327, 366 Dvorak-Dudley representation, 766 1, 256, 258, 277, 285, 327, 330, 356 1, 302, 367 c , 510 effective 0 value, 597 m, 513 effective oscillation frequency s , 509 single cycle pulse, 595 0eff , 597 Einstein, A. BM , 292 special theory of relativity, 672, 682 Index 827 electric energy density, 717 rational approximations, 138 electric susceptibility e.!/, 179 sine S. /, 133 electromagnetic beam field Fresnel parameter separable, 70 complex FQ, 197 electromagnetic bullets, 81 real F , 199 electropermeabilization, 769 Fresnel reflection matrix, 736 electroporation, 769, 770 Fresnel transmission matrix, 736 energy bandwidth, 163 Fresnel-Kirchhoff diffraction integral, 18 energy transport velocity, 289, 530, 685 energy velocity nonoscillatory waves, 688 G. G. Stokes, 792 energy velocity description Gamma function .a/, 790 nonuniform model, 689 asymptotic approximation, 805 uniform model, 696, 699 gaussian beam, 75 envelope function, 157 beam waist, 76 equation of continuity, 28 divergence angle, 77 Erdelyi,´ A., 34 Rayleigh range, 77 error function erf. /, 131 spot size, 76 Euler’s constant , 777 gaussian envelope function, 177 evolved heat Q, 719 gaussian envelope pulse exciton precursor, 666 dispersion length `D , 200 Birman and Frankel, 666 group velocity, 633 experimental measurements gaussian pulse propagation Aaviksoo, Kuhl, and Ploog, 666 asymptotic description, 623 Avenel, Rouff, Varoquaux and Williams, experimental results, 636 662 group velocity approximation, 199 Choi and Osterberg,¨ 666 proper group velocity description, 636 D. D. Stancil, 661 pulse separation, 625 Falcon, Laroche, and Fauve, 662 scaling law, 626 Jeong, Dawes, and Gauthier, 669 transition to the group velocity description, Pleshko and Palocz,´ 656 630 exponential integral E1.x/, 777 geometrical optics limit, 74 Gitterman, E. and M., 728 Goos-Hanchen¨ shift, 748 Felsen, L. B., 239, 244, 679 group delay, 528 filtered backprojection algorithm, 822 group method first forerunner Havelock,T.H.,148 Brillouin’s result, 416 group velocity, 147, 528 first precursor, 384, 385 complex, 193, 195 asymptotic expansion, 395 gaussian envelope pulse, 633 Fourier conjugate variables, 821 Hamilton, Sir W. R., 147 Fourier integral nonoscillatory waves, 688 asymptotic approximation, 794 Rayleigh, Lord, 147 Fourier transform, 821 real, 199 fractional bandwidth, 162 Stokes, G. G., 147 fractional mixing model, 756 group velocity approximation Fresnel approximation, 18 Eckart, C., 149 Fresnel coefficients Lighthill, M. J., 149 p-polarization, 740 Whitham, G. B., 149 s-polarization, 739 group velocity dispersion, 529 Fresnel equations group velocity dispersion (GVD), 194, 195, generalized, 735 199 Fresnel integral group velocity method, 149 cosine C. /, 133 extended, 702 828 Index health and safety issues, 621 Laplace’s method, 800 heat density Legendre polynomials P`.u/,33 evolved, 719 Rodriques’ formula, 33 net, 723 linear dispersion approximation, 195, 529 Heaviside step function signal Lommel functions, 14 Brillouin precursor, 442 Lord Kelvin, 792 Sommerfeld precursor, 412 Lorentz model dielectric steady-state behavior, 540 asymptotic field behavior in, 383 Heaviside unit step function, 161 double-resonance, 279 Heaviside-Poynting theorem, 717 single-resonance, 264 Helmholtz equation, 4, 159 Lorentz–Lorenz formula, 179 homogeneous, isotropic, locally linear (HILL) Low Probability of Intercept (LPI), 714 temporally dispersive media constitutive relations, 1 Huygens-Fresnel principle, 18 magnetic hyperbolic tangent envelope function, 169 permeability .!/,29 hyperbolic tangent envelope spectrum, 173 response function, 28 magnetic energy density, 717 magnetostatic waves, 661 IEEE/ANSI safety standards, 642 main signal arrival, 517, 520 immature dispersion regime, 680 main signal velocity, 517, 520, 583 impulse radar, 713 masking function, 83 impulse response, 161, 454, 682 mature dispersion regime, 460, 548, 625, 679, impulse response function 680 spatial, 12 maximal distortion domain, 574 incomplete gamma function maximum permissible exposure (MPE), 767 asymptotic expansion, 791 Maxwell’s equations normalized .a;x/, 790 Bateman–Cunningham form, 80 incomplete Gamma function .a;x/, 791 frequency-domain form, 29 asymptotic expansion, 791 homogeneous, 80 incomplete Lipschitz–Hankel integral time-domain form, 28 mean angular frequency of a pulse, 224 Jen.a; /, 764 inhomogeneous wave, 733 mean angular resonance frequency !Ns , 359 instantaneous oscillation frequency middle precursor, 211, 384 Brillouin precursor, 437 uniform asymptotic approximation, 453 Sommerfeld precursor, 407 middle saddle point dominance intermediate distortion domain, 574 necessary condition for, 289 intrinsic impedance middle saddle points, 283, 286 complex .!/,43 first approximation, 361 mine detection, 756 of free space 0,80 inverse problem minimal distortion domain, 574 modern asymptotic theory, 679 material identification, 602 0 inverse source problem modified complex phase function ˚m.!; /, nonuniqueness, 85 626 normal solution, 86 molecular conformation changes, 771 time-dependent, 79 mother pulse, 753 isodiffracting wave field, 233 multipole expansion isotimic contour, 274 electromagnetic wavefield, 42 scalar wavefield, 34 multipole moments, 32, 34, 42 Kramers–Kronig relations, 253 near saddle point, 256, 277, 285, 366 Lalor, E.,´ 17 first approximation, 327 Laplace integral, 794 second approximation, 329 Index 829 near saddle points multiple resonance Lorentz model first approximation, 356, 370 dielectric, 496, 498 second approximation, 357 nonuniform asymptotic description, 469 neighborhood Rocard–Powles–Debye model dielectrics, spherical, 780 471 net heat density W3D .z/, 723 single resonance Lorentz model dielectric neutralizer function .p;q/,54 above absorption band case, 485 non-impulse radar, 714 below absorption band case, 481 nonlinear envelope equation (NEE), 228 intra-absorption band case, 486, 487 nonlinear response, 180 zero frequency case, 482 nonlinear Schrodinger¨ equation, 229 uniform asymptotic description, 476 nonoscillatory waves, 687 accuracy, 492 nonradiating sources, 84 polychromatic normal dispersion, 267 definition, 183 posterior pre-signal velocity, 517 Poynting vector, 717 Occupational Safety and Health Administra- complex, 9 tion (OSHA), 767 time-average, 10 Olver’s saddle point method, 98 Olver’s theorem, 100 pre-signal velocity Olver-type path, 103 anterior, 517, 583 optical precursor posterior, 517, 583 observation of, 664 precursor orbital angular momentun operator Ls ,39 Brillouin, 656 order O, 780 observability of orthogonality relations Aaviksoo, Lippmaa, and Kuhl, 666 spectral amplitude vectors, 39 Alfano, Birman, Ni, Alrubaiee, and Das, 667 Okawachi, Slepkov, Agha, Geraghty, paraxial approximation, 76 and Gaeta, 669 quadratic phase dispersion, 18 observation of phase delay, 526 Aaviksoo, Kuhl, and Ploog, 666 phase velocity, 148 Choi and Osterberg,¨ 666 complex, 192, 195 D. D. Stancil, 661 phase velocity vp.!/, 185, 526 Falcon, Laroche, and Fauve, 662 phase velocity approximation, 528 Jeong, Dawes, and Gauthier, 669 plane wave Pleshko and Palocz,´ 656 attenuation factor ˛.!/,6 Varoquaux, Williams and Avenel, 662 evanescent, 7, 11 Sommerfeld, 656 homogeneous, 7, 11 space–time measurement, 738 inhomogeneous, 8 precursor field propagation factor ˇ.!/,6 Brillouin, 211, 384–386 transversality relations, 5 middle, 211, 384 Poincare,´ H., 784 Sommerfeld, 210, 384, 385 polarization density, 718 polarizing angle, 742 total, 504, 506, 556 pole contribution prepulse, 514, 552 Drude model conductor, 500 projection slice theorem, 821 Heaviside spep function signal propagation factor ˇ.!/, 470 below absorption band case, 488 propagation kernel, 18 Heaviside step function signal pulse spreading above absorption band case, 490 rectangular envelope, 575 intra-absorption band case, 490 pulse synthesization, 669 830 Index quadratic dispersion approximation, 196, 199, semiconducting material 529 asymptotic field behavior in, 385 quadratic dispersion relation, 196 Sherman’s expansion, 73 Quantum Optics Workshop on Slow and Fast Sherman’s recursion formula, 57 Light, 670 Sherman, G. C., 17 quasi-static field, 687 signal arrival, 511 quasimonochromatic, 184 signal contribution, 384–386 definition, 182 signal frequency !c , 157 limit, 610, 614 signal velocity, 511 comparison of numerical and asymptotic results, 552 radiation pattern, 36, 46 comparison with energy velocity, 531 filtered, 86 main, 517, 520, 522, 583 frequency-domain, 84 measurement, 550, 554 IEEE definition, 49 prepulse, 518, 522, 583 scalar wavefield, 37 rectangular envelope pulse, 577 time-domain, 82 sound, 661 Radon transform, 83, 819, 820 signum function, 822 inverse, 823 simple polarizable dielectric, 179 raised cosine envelope signal, 617 single-sided Fourier transform, 232 Rayleigh range, 238 singular dispersion limit, 264, 651, 700 rectangle function, 163 slowly evolving wave approach, 151 reflection slowly varying envelope approximation, 150 generalized law, 735 slowly-evolving-wave approximation (SEWA), refraction 230 generalized law, 735 Snell’s law, 739 relativistic causality, 391, 682 soliton evolution reshaping delay Rr0 , 644 wave equation for, 229 residue Sommerfeld precursor, 210, 384, 385, 656 simple pole, 127 asymptotic approximation, 398, 400 resonance peak, 490, 507 asymptotic expansion, 395 Brillouin precursor, 536 instantaneous oscillation frequency, 407 Sommerfeld precursor, 534 uniform asymptotic approximation, 403, Riemann, B, 805 404 Rocard–Powles–Debye model dielectric, 293 uniform asymptotic expansion, 401 Sommerfeld precursor, Drude model conductor, 562 saddle point, 806 Sommerfeld precursor, gaussian, 624 order, 102 Sommerfeld radiation condition, 51 saddle point dynamics, 256 Sommerfeld’s Relativistic Causality Theorem, saddle point equation, 255, 316 393 saddle point method, 98 source function, 83 saddle points space–time parameter distant, 262, 277, 285, 368 retarded, 161 energy velocity equivalent !Ej , 684 space–time parameter , 154 isolated, 111 spatiotemporal Fourier-Laplace transform, 9 middle, 283, 286 special theory of relativity near, 256, 277, 285, 370 relativistic causality, 682 scalar dipole field, 91 specific absorption rate (SAR), 767 second forerunner spectral amplitude vectors Brillouin’s result, 442, 443 orthogonality relations, 39 second precursor, 384, 386 spectral distribution function, 11 asymptotic expansion, 418, 424, 425 spectrum domain, 231 self-induced transparency, 530 spherical Bessel function j`. /,33 Index 831 .C/ spherical Hankel function h` . /,34 Van Bladel envelope function, 174, 638 m spherical harmonic functions Y` . ; '/,32 vector potential Stamnes, J. J., 17 plane wave field, 154 stationary phase method m vector spherical harmonic functions Y` .˛; ˇ/, Kelvin, L., 148 39 stationary phase point, 792 velocity interior, 52 group, 147 steady-state response, 704 phase, 148 steepest descent path, 806 void detection, 756 Stirling’s formula, 805 Stokes’ phenomena, 104 Stratton, J. A., 705 strictly monochromatic field, 183 Watson’s lemma subtraction of the pole technique, 127 complex argument, 797 sum rule, 254, 262 real argument, 797 superluminal propagation wave equation Flash Gordon, 670 M. Kitano, 670 reduced, 30 superluminal pulse propagation, 151 slowly-varying envelope approximation, 186 slowly-varying-envelope (SVE), 194 thermal damage wave vector localized, 771 complex, 5, 730 total internal reflection, 740, 744 wavefield total precursor field, 504, 506 freely-propagating, 2 transient response, 704 nonoscillatory, 690 transmission line time-harmonic, 690 effective dispersion model, 570 waveform transversality relations, 731 monochromatic, 148 trapezoidal envelope, 165 nonoscillatory, 687 trapezoidal envelope function, 165 polychromatic, 148 trapezoidal envelope spectrum, 167 wavefront velocity, 406, 407 wavenumber ultrawideband complex, 5, 186, 731 definition of, 162 weak dispersion limit, 264, 319, 342, 350, 651 FCC definition, 162 absorptive equivalence relation, 655 UWB radar, 713 phasal equivalence relation, 656 undersea radar communication Weyl’s integral, 30 feasibility using the Brillouin precursor, Whittaker’s representation, 31 605 Whittaker, E. T., 23 uniform asymptotic expansion, 96 Wolf, E., 28